Geometry with an Introduction to Cosmic Topology

🧭 Overview

🧠 One-sentence thesis

A finite two-dimensional universe can exist without boundaries by having identified edges (like a torus or sphere), and the global geometry of such a universe differs from local Euclidean geometry, with topology and geometry linked by the Gauss-Bonnet equation.

📌 Key points (3–5)

• Finite without boundary: A finite universe can avoid having an edge by identifying opposite edges (like the Asteroids video game screen) or by being a closed surface like a sphere.
• Local vs global geometry: Small-scale measurements follow Euclidean rules (triangle angles sum to 180°), but large-scale geometry on curved surfaces is non-Euclidean (e.g., spherical triangles exceed 180°).
• Common confusion: The same surface can appear Euclidean locally but non-Euclidean globally—a bug on a sphere sees flat geometry nearby but curved geometry over large distances.
• Topology-geometry link: The Gauss-Bonnet equation connects a surface's shape (topology, right side) with its geometry (left side), so knowing global geometry constrains possible universe shapes.

🌐 The finite universe problem

🌐 Why boundaries are problematic

• The excerpt presents a two-dimensional universe traditionally modeled as an infinite plane (like the xy-plane).
• Cosmologists and mathematicians notice everything observable is finite, so they ask: could the universe itself be finite?
• A boundary or edge would make boundary points "physically different from the rest of space," which is unappealing.
• The central question: How can a finite universe have no boundary?

🎮 The Asteroids model—identified edges

A finite two-dimensional universe can be represented as a rectangular region with opposite edges identified.

• The excerpt uses the Asteroids video game as an analogy:
  • Moving off the top of the screen makes you reappear at the bottom.
  • Moving off the left edge makes you reappear on the right.
• This identification means the top edge is "glued" point-by-point to the bottom edge, and the left edge to the right edge.
• Result: a finite area with no boundary—you never encounter an edge, yet the total area is finite.

🍩 The torus as a physical realization

• In three dimensions, you can physically achieve the edge identification:
  1. Bend the rectangle to join top and bottom edges → produces a cylinder.
  2. Bend the cylinder to join the left and right circles → produces a torus (donut shape).
• A two-dimensional being living in the surface cannot see the 3D embedding but understands the space perfectly as "rectangle-with-edges-identified."
• The torus is one example of a finite, boundaryless two-dimensional surface.

🏐 The sphere as another example

• A sphere (like a beach ball surface) is also finite-area and has no edge.
• A bug on the sphere sees the surface as locally flat (no edges anywhere) but the total area is finite.
• Both torus and sphere solve the finite-without-boundary problem, but they have different shapes (topology).

📐 Euclidean vs non-Euclidean geometry

📐 Local Euclidean behavior

• On small scales, familiar Euclidean geometry holds:
  • Triangle angles sum to 180°.
  • The Pythagorean theorem works (builders use it to check right angles).
• The excerpt states: "small triangles have angle sum essentially equal to 180°, which is a defining feature of Euclidean geometry."
• Example: In everyday construction on a flat plane or small patch of a curved surface, Euclidean rules apply.

🌍 Global non-Euclidean behavior on a sphere

• On larger scales, geometry deviates from Euclidean rules.
• The excerpt gives a concrete example:
  • Draw a triangle on a sphere using the north pole and two points on the equator.
  • Each equator point has a 90° angle (the paths meet the equator at right angles).
  • The angle at the north pole adds additional degrees.
  • Total angle sum exceeds 180° by the amount of the north pole angle.
• This violates Euclidean geometry, so the sphere has non-Euclidean geometry on a global scale.

🔍 Don't confuse local and global

• The same surface can be:
  • Locally Euclidean: small measurements follow flat-plane rules.
  • Globally non-Euclidean: large-scale measurements reveal curvature.
• A two-dimensional being might think the universe is flat based on local experiments, but global measurements (large triangles) reveal the true geometry.

🔗 The topology-geometry relationship

🔗 Gauss-Bonnet equation

The Gauss-Bonnet equation is kA = 2πχ, where geometry is on the left side and topology is on the right side.

• The excerpt does not explain the symbols in detail but emphasizes the structure:
  • Left side (kA): represents geometry (curvature and area).
  • Right side (2πχ): represents topology (the shape's intrinsic structure, denoted χ).
• This equation links the type of geometry a surface has with its topological shape.

🧩 Why this matters for cosmology

• If a two-dimensional being can measure the global geometry of her universe (e.g., by checking large triangle angle sums), she can deduce constraints on the universe's shape.
• The excerpt states: "if a two-dimensional being can deduce what sort of global geometry holds in her world, she can greatly reduce the possible shapes for her universe."
• Different surfaces (torus, sphere, etc.) have different geometries, so geometry measurements help identify topology.

Surface	Boundary?	Global geometry	Angle sum of large triangles
Infinite plane	No (infinite)	Euclidean	Exactly 180°
Torus	No (finite)	Locally Euclidean, globally flat	180° (flat)
Sphere	No (finite)	Non-Euclidean (positively curved)	Greater than 180°

🎯 The book's goal

• The excerpt states: "a primary goal of this book is to arrive at this relationship, given by the pristine Gauss-Bonnet equation."
• Understanding how geometry and topology connect allows beings (or cosmologists) to infer the shape of their universe from geometric measurements.

🧭 Overview

🧠 One-sentence thesis

The debate over Euclid's fifth postulate led to the discovery of non-Euclidean geometries (hyperbolic and elliptic), which are now understood as special cases of Klein's transformation-based approach and may describe the global geometry of our universe.

📌 Key points (3–5)

• The parallel postulate problem: Euclid's fifth postulate was controversial because it seemed more like a theorem than a self-evident axiom, sparking centuries of attempts to prove it from the other four postulates.
• Discovery of non-Euclidean geometry: In the early 19th century, Bolyai and Lobachevsky independently showed that consistent geometries exist where the fifth postulate does not hold, proving it is not a necessary consequence of the first four.
• Three geometry types: Euclidean (exactly one parallel line; triangle angles sum to 180°), hyperbolic (at least two parallel lines; triangle angles sum to less than 180°), and elliptic (zero parallel lines; triangle angles sum to more than 180°).
• Common confusion: The Pythagorean theorem depends on the parallel postulate, so it is specific to Euclidean geometry; non-Euclidean geometries have their own variations.
• Klein's unifying approach: Instead of Euclid's additive axiom-based method, Klein's Erlangen Program defines geometry as the study of what remains unchanged under allowable transformations, making all three geometries emerge as special cases.

📜 Euclid's postulates and the controversy

📜 The five postulates

Euclid's Elements begins with five postulates:

1. One can draw a straight line from any point to any point.
2. One can produce a finite straight line continuously in a straight line (i.e., extend a line segment to any length).
3. One can describe a circle with any center and radius.
4. All right angles equal one another.
5. The parallel postulate: If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side.

🔥 Why the fifth postulate was controversial

• The first four postulates are short, simple, and intuitive.
• The fifth is long, complex, and sounds more like something to prove than to assume.
• Euclid himself may have been bothered by it—he avoided using it until the 29th proposition in his text.
• For over 2000 years, philosophers and mathematicians tried (and failed) to prove the fifth postulate from the first four.

🔄 Equivalent reformulations

Two equivalent statements emerged that are more relevant to studying geometry:

Version	Statement
5′ (Playfair's Axiom)	Given a line and a point not on the line, there is exactly one line through the point that does not intersect the given line.
5″	The sum of the angles of any triangle is 180°.

• Playfair's Axiom (named after Scottish mathematician John Playfair, 1748–1819) is the version that will be altered to produce non-Euclidean geometries.

🌍 The birth of non-Euclidean geometry

🇭🇺 Bolyai's struggle and breakthrough

• Farkas Bolyai (1775–1856) of Hungary spent much of his life trying to prove the parallel postulate from the other four and failed.
• He begged his son János (1802–1860) to abandon the problem, writing: "For God's sake, I beseech you, give it up. Fear it no less than the sensual passions because it too may take all your time and deprive you of your health, peace of mind and happiness in life."
• János continued anyway and independently discovered (along with Russian mathematician Nikolai Lobachevsky, 1792–1856) that a well-defined geometry is possible where the first four postulates hold but the fifth does not.
• This proved that the fifth postulate is not a necessary consequence of the first four.

🔍 Gauss's unpublished work

• Carl Friedrich Gauss (1777–1855) had also discovered this geometry but did not publish his work because he feared it would be too controversial for the establishment.

✅ Establishing logical consistency

• A key challenge was whether changing the parallel postulate created a system with contradictory theorems.
• In 1868, Italian mathematician Enrico Beltrami (1835–1900) showed that non-Euclidean geometry could be constructed within the Euclidean plane.
• This meant: as long as Euclidean geometry is consistent, non-Euclidean geometry is consistent as well.
• Non-Euclidean geometry was thus placed on solid logical ground.

🔺 The three geometries

🔺 Hyperbolic geometry

Postulate 5H: Given a line and a point not on the line, there are at least two lines through the point that do not intersect the given line.

• This is the geometry Bolyai and Lobachevsky discovered.
• Triangle angle sum: less than 180°.
• Example: In hyperbolic space, you can draw multiple lines through a point that never meet a given line.

🌐 Elliptic geometry

Postulate 5E: Given a line and a point not on the line, there are zero lines through the point that do not intersect the given line.

• This models geometry on the sphere.
• Triangle angle sum: greater than 180°.
• Example: On a sphere, consider a triangle formed by the north pole and two points on the equator. Each equator angle is 90°, so the total exceeds 180° by the angle at the north pole.

📐 Euclidean geometry (for comparison)

Postulate 5′ (Playfair's Axiom): Given a line and a point not on the line, there is exactly one line through the point that does not intersect the given line.

• Triangle angle sum: exactly 180°.
• This is the familiar "flat" geometry.

📊 Summary table

Geometry	Parallel lines through a point	Triangle angle sum
Euclidean	Exactly one	= 180°
Hyperbolic	At least two	< 180°
Elliptic	Zero	> 180°

🧮 The Pythagorean theorem and its dependence on the parallel postulate

🧮 Euclidean version

Theorem 1.2.1 (Pythagorean Theorem): In right-angled triangles, the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle.

• In symbols: c squared equals a squared plus b squared (where c is the hypotenuse and a, b are the legs).
• This appears as Proposition 47 at the end of Book I of Euclid's Elements.
• Key point: The Pythagorean theorem depends on the parallel postulate, so it is specific to Euclidean geometry.

🔄 Non-Euclidean variations

• Hyperbolic and elliptic geometries have their own variations of the Pythagorean theorem.
• The text mentions a unified Pythagorean theorem that covers all three geometries (presented in Chapter 7, from a recent result).
• Don't confuse: The familiar Pythagorean theorem does not hold in non-Euclidean geometries; each geometry has its own distance relationships.

🏗️ Converse for builders

• Proposition 48 of Book I gives the converse: If we measure the legs of a triangle and find that c squared equals a squared plus b squared, then the angle opposite c is right.
• This is the practical version builders use to check right angles.

🔧 Klein's Erlangen Program

🔧 Additive vs subtractive approaches

• Euclid's additive approach: Start with basic definitions and axioms, then build a sequence of results depending on previous ones.
• Klein's subtractive approach: Start with a space and a group of allowable transformations, then throw out all concepts that do not remain unchanged under these transformations.

🎯 Klein's definition of geometry

Geometry, to Klein, is the study of objects and functions that remain unchanged under allowable transformations.

• German mathematician Felix Klein (1849–1925) developed this approach, called the Erlangen Program (after the university where he worked).
• Benefit: All three geometries (Euclidean, hyperbolic, elliptic) emerge as special cases from a general space and a general set of transformations.
• Example: Einstein's theory of special relativity is derived from the notion that the laws of nature are invariant with respect to Lorentz transformations—an illustration of Klein's approach.

🌌 Applications to the universe

🌌 Local vs global geometry

• Euclidean geometry remains an excellent model for our local geometry.
• The angles of a triangle drawn on paper do add up to 180°.
• Even "galactic" triangles determined by nearby stars have angle sums indistinguishable from 180°.
• But: On a larger scale, things might be different.

🐛 The bug analogy

• A bug living in a field on Earth's surface might reasonably conclude it is living on an infinite plane.
• The bug cannot sense that its flat, visible world is just a small patch of a curved surface (Earth) in three-dimensional space.
• Likewise, our apparently Euclidean three-dimensional universe might be curving in some unseen fourth dimension, making the global geometry non-Euclidean.

🔬 Cosmological implications

• Under reasonable assumptions about space, hyperbolic, elliptic, and Euclidean geometry are the only three possibilities for the global geometry of our universe.
• Researchers analyze cosmological data to determine which geometry is ours.
• Why it matters:
  • Deducing the geometry tells us about the shape and whether the universe is finite.
  • If elliptic → must be finite in volume.
  • If Euclidean or hyperbolic → can be either finite or infinite.
  • Each geometry type corresponds to a class of possible shapes.
  • The overall geometry may be fundamentally connected to the fate of the universe.

🌟 Grand application

• The excerpt concludes: "Clearly there is no more grand application of geometry than to the fate of the universe!"

🎭 Cultural impact

📖 Dostoevsky's reference

• Non-Euclidean geometry caused a stir outside mathematics.
• Fyodor Dostoevsky included it in The Brothers Karamazov (first published 1880).
• In the novel, Ivan tells his younger brother Alyosha that if one cannot fathom non-Euclidean geometry, then one has no hope of understanding questions about God.
• This shows how the discovery was seen as philosophically significant beyond mathematics.

🧭 Overview

🧠 One-sentence thesis

Small triangles and their angle sums allow a two-dimensional observer to distinguish between different geometries on surfaces, revealing whether a surface is homogeneous and which of the three possible geometries—Euclidean, hyperbolic, or elliptic—it possesses.

📌 Key points (3–5)

• Homogeneous surfaces: a surface is homogeneous if the local geometry is the same at every point; non-homogeneous surfaces have different geometry at different locations.
• Geodesics and triangles: on a surface, "straight lines" are geodesics (shortest paths), and triangles built from geodesics reveal the local geometry through their angle sums.
• Angle sum as a diagnostic: triangles on convex regions have angle sum greater than 180°; triangles on saddle-shaped regions have angle sum less than 180°; flat regions have angle sum equal to 180°.
• Common confusion: a surface can appear non-homogeneous when drawn in the plane (e.g., with cone points) but admit a homogeneous geometry when embedded differently (e.g., on a sphere).
• Three geometries: every surface can be given one of three types of homogeneous geometry—Euclidean, hyperbolic, or elliptic.

🌍 Homogeneity and local geometry

🌍 What homogeneity means

A homogeneous surface is a space that has the same local geometry at every point.

• If you move around a homogeneous surface, the geometric properties (like triangle angle sums) remain the same everywhere.
• Example: the Euclidean plane is homogeneous—any triangle drawn anywhere on a flat page has angles summing to 180°.
• A sphere is also homogeneous: a two-dimensional bug on a sphere cannot geometrically distinguish one point from another.
• Don't confuse: homogeneity is about local geometry being the same everywhere, not about the global shape being simple.

🍩 Non-homogeneous surfaces

• A donut surface (torus in three-dimensional space) is not homogeneous.
• A two-dimensional bug on the donut can tell the difference between a convex point (outer wall) and a saddle-shaped point (inner wall) by drawing triangles and measuring angle sums.
• At a convex point, a triangle has angle sum greater than 180°.
• At a saddle-shaped point, a triangle has angle sum less than 180°.
• Example: the bug draws triangles with the same leg lengths at different locations and finds different angle sums, proving the surface is not homogeneous.

📐 Geodesics and triangles on surfaces

📐 What a geodesic is

A geodesic is the path of shortest distance between two points on a surface.

• For a two-dimensional bug living on a surface, a "straight line" from point A to point B is simply the shortest path from A to B.
• On the Euclidean plane, geodesics are ordinary Euclidean lines.
• On a sphere, geodesics follow great circles.

🌐 Great circles on the sphere

A great circle is a circle drawn on the surface of the sphere whose center (in three-dimensional space) corresponds to the center of the sphere; equivalently, a circle of maximum diameter drawn on the sphere.

• Example: the equator is a great circle; lines of longitude are great circles; most lines of latitude (except the equator) are not.
• The excerpt shows that circles a and b are great circles, but circle c is not.

🧵 Physical method for finding geodesics

• Pin a string at point A and draw the string tight on the surface to point B.
• The taut string will follow the geodesic from A to B.

🔺 Triangles on surfaces

• A triangle on a surface consists of three points and three edges connecting them.
• Each edge is a geodesic (shortest path) between its endpoints.
• Depending on the shape of the surface, geodesic triangles can have angle sum greater than, less than, or equal to 180°.

🏗️ Examples of surfaces and their geometries

🏗️ The flat torus (Example 1.3.4)

• The flat torus is the world where a ship flying off one edge reappears on the opposite edge (like a video game screen that wraps around).
• Homogeneous: at every spot, the pilot would report flat surroundings—triangle angles add to 180°.
• Locally, the geometry is Euclidean at every point.
• Globally, the world is very different from the infinite Euclidean plane: it is finite and has no boundary.
• Example: with a powerful telescope, the pilot could see the back of his own ship.

🍦 Coneland (Example 1.3.5)

• Start with a circular disk with a wedge removed (like a pizza missing a slice).
• Join the two radial edges to produce a cone.
• Let θ denote the angle subtended by the circular sector; the cone surface is denoted S(θ).
• Not homogeneous (unless θ happens to be 2π):
  • A triangle that does not contain the tip of the cone has angle sum equal to π radians (180°).
  • A triangle that does contain the tip has angle sum not equal to π radians.
• A two-dimensional bug can use this difference to conclude that S(θ) is not homogeneous.

🏔️ Saddleland (Example 1.3.7)

• Similar to Coneland, but use circle wedges with θ greater than 2π.
• Identifying the radial edges produces a saddle-shaped surface.
• To create such a wedge, tape together two wedges of equal radius.
• The surface is non-homogeneous, with different angle sums depending on whether the triangle contains the saddle point.

🔷 A non-Euclidean surface (Example 1.3.9)

• Consider a hexagon with edges identified according to labels and arrow orientation.
• When a ship flies off one edge, it reappears at the matching spot on the paired edge.
• If the hexagon is drawn in the plane:
  • The corners meet in groups of two, creating cone points.
  • Each corner angle is 120°, so two corners glued together form a 240° patch (not 360°).
  • The surface is not homogeneous: triangles containing a cone point have angle sum greater than 180°; other triangles have angle sum equal to 180°.
• Key insight: the surface does admit a homogeneous geometry if we embed it differently.
  • Stretch the hexagon onto the northern hemisphere of a sphere.
  • Each corner angle becomes 180°, and each pair of corners adds up to exactly 360°.
  • The surface becomes homogeneous with elliptic geometry (the geometry of the sphere), not Euclidean geometry.

🔍 Distinguishing geometries with circles

🔍 Circle definition on a surface

Given a point P on a surface and a real number r > 0, the circle centered at P with radius r is the set of all points r units away from P, where distance is measured along the shortest path (geodesic).

🔍 Circumference vs radius as a diagnostic

• In the Euclidean plane, the relationship between a circle's radius r and circumference C is C = 2πr, and this is true for any circle.
• In Coneland (centered at the tip of the cone):
  • The relationship is not C = 2πr.
  • The excerpt asks whether C > 2πr or C < 2πr (the answer depends on θ; if θ < 2π, then C < 2πr).
• In Saddleland (centered at the tip of the saddle):
  • The relationship is not C = 2πr.
  • The excerpt asks whether C > 2πr or C < 2πr (the answer depends on θ; if θ > 2π, then C > 2πr).
• A two-dimensional bug can use the circumference-radius relationship to screen for different geometries.

🌌 The three possible geometries

🌌 Every surface has one of three geometries

• The excerpt states: every surface can be given one of three types of homogeneous geometry: Euclidean, hyperbolic, or elliptic.
• This classification is fundamental and applies to surfaces in general.

🌌 Summary table

Geometry type	Triangle angle sum	Circumference vs radius	Example surface
Euclidean	Equal to 180°	C = 2πr	Flat plane, flat torus
Hyperbolic	Less than 180°	C > 2πr	Saddle-shaped regions
Elliptic	Greater than 180°	C < 2πr	Sphere

🌌 Motivation for studying non-Euclidean geometries

• The excerpt emphasizes that understanding these three geometries is essential for studying the geometry of surfaces and the universe.
• The classification will be revisited after developing hyperbolic and elliptic geometry in detail.
• Don't worry if it doesn't make complete sense yet—use these facts as motivation for learning about non-Euclidean geometries.

🧭 Overview

🧠 One-sentence thesis

Complex numbers can be represented geometrically as ordered pairs in the plane and manipulated through addition, multiplication, and polar form, providing a foundation for studying geometric transformations.

📌 Key points (3–5)

• Geometric definition: A complex number is an ordered pair (x, y) of real numbers, forming the complex plane C.
• Two representations: Cartesian form x + yi and polar form re^(iθ) both describe the same number but emphasize different properties.
• Key operations: Addition works componentwise (like vectors); multiplication uses i² = −1 and has a geometric interpretation via polar form.
• Common confusion: The imaginary part y in z = x + yi is a real number; "imaginary" refers to the i factor, not y itself.
• Modulus and conjugate: |z| measures distance from the origin; the conjugate z̄ flips the sign of the imaginary part and satisfies z · z̄ = |z|².

🔢 Defining the complex plane

🔢 What is a complex number

A complex number is an ordered pair (x, y) of real numbers.

• The set C = {(x, y) | x, y ∈ R} is called the complex plane.
• This is a geometric approach: each complex number corresponds to a point in the plane.
• The real numbers R are embedded in C via the identification x ↔ (x, 0).

🧩 Real and imaginary parts

For z = (x, y): x is the real part Re(z); y is the imaginary part Im(z).

• The imaginary part is itself a real number; the term "imaginary" comes from the factor i.
• Example: For z = 3 − 4i, Re(z) = 3 and Im(z) = −4 (not −4i).

🔤 The number i

• i = (0, 1) is defined by the property i² = −1.
• Any complex number (x, y) can be written as x + yi:
  • (x, y) = (x, 0) + (0, y) = x(1, 0) + y(0, 1) = x + yi.
• This expression x + yi is called the Cartesian form of the complex number.

➕ Operations in Cartesian form

➕ Addition and scalar multiplication

• Addition is componentwise:
  • (x, y) + (s, t) = (x + s, y + t).
• Scalar multiplication by a real number k:
  • k · (x, y) = (kx, ky).
• These operations match vector addition and scalar multiplication in the plane.
• Example: If z = 2 + i and w = −1 + 1.5i, then z + w = 1 + 2.5i.

✖️ Multiplication

• Complex multiplication uses i² = −1:
  • (x + yi) · (s + ti) = xs + ysi + xti + yti²
  • = (xs − yt) + (ys + xt)i.
• Example: If z = 3 − 4i and w = 2 + 7i, then
  • z · w = (3 − 4i)(2 + 7i) = 6 + 21i − 8i + 28 = 34 + 13i.
• Don't confuse: Multiplication is not componentwise; it involves cross-terms.

📏 Modulus

The modulus of z = x + yi is |z| = square root of (x² + y²).

• |z| gives the Euclidean distance from z to the origin (0, 0).
• Example: For z = 3 − 4i, |z| = square root of (9 + 16) = 5.

🔄 Conjugate

The conjugate of z = x + yi is z̄ = x − yi.

• The conjugate flips the sign of the imaginary part.
• Key property: z · z̄ = |z|² (proven in exercises).
• Example: If z = 3 − 4i, then z̄ = 3 + 4i, and z · z̄ = 34 − 13i is not correct; actually (3 − 4i)(3 + 4i) = 9 + 16 = 25 = |z|².
• Other useful properties (from exercises):
  • |w · z| = |w| · |z|
  • Conjugate of product: (wz)̄ = w̄ · z̄
  • Conjugate of sum: (z + w)̄ = z̄ + w̄
  • z + z̄ = 2 Re(z) (a real number)
  • z − z̄ = 2 Im(z) i
  • |z̄| = |z|

🌀 Polar form

🌀 From Cartesian to polar

• A point (x, y) in the plane has polar coordinates (r, θ) where:
  • x = r cos(θ)
  • y = r sin(θ)
• Substituting into x + yi:
  • x + yi = r cos(θ) + r sin(θ) i = r(cos(θ) + i sin(θ)).

🎯 Definition of e^(iθ)

For any real number θ, define e^(iθ) = cos(θ) + i sin(θ).

• This is a definition used to express complex numbers in polar form.
• Examples:
  • e^(i π/2) = cos(π/2) + i sin(π/2) = 0 + i = i
  • e^(i 0) = cos(0) + i sin(0) = 1
  • e^(i π) = cos(π) + i sin(π) = −1
• Famous equation: e^(i π) + 1 = 0.

🎯 Polar form of a complex number

If z = x + yi and (x, y) has polar form (r, θ), then z = re^(iθ) is the polar form of z.

• r = |z| is the modulus (non-negative).
• θ = arg(z) is the argument of z (the angle).
• Example: To convert z = −3 + 4i to polar form:
  • r = square root of (9 + 16) = 5
  • The angle α from the positive x-axis satisfies tan(α) = 4/3, but z is in the second quadrant, so θ = π − arctan(4/3) ≈ 2.21 radians.
  • Thus z = 5e^(2.21i).

✖️ Multiplication in polar form

Theorem: The product of two complex numbers in polar form is

• re^(iθ) · se^(iβ) = (rs) e^(i(θ + β)).

What this means:

• Multiply the moduli: r · s.
• Add the arguments: θ + β.
• This gives a geometric interpretation: multiplication scales by the modulus and rotates by the argument.

Proof sketch (from the excerpt):

• Expand using the definition e^(iθ) = cos(θ) + i sin(θ).
• Use trigonometric identities for cos(θ)cos(β) − sin(θ)sin(β) and sin(θ)cos(β) + cos(θ)sin(β).
• The result simplifies to rs(cos(θ + β) + i sin(θ + β)) = rs e^(i(θ + β)).

🧮 Practical applications

🧮 Pythagorean triples

• A Pythagorean triple (a, b, c) satisfies a² + b² = c².
• Complex numbers can generate these triples:
  • Let z = x + yi where x, y are positive integers.
  • Define a = Re(z²), b = Im(z²), c = z · z̄.
  • Then a² + b² = c² (proven in exercises).
• Example: The triple (3, 4, 5) comes from some z = x + yi; the triple (5, 12, 13) comes from another choice of z.

🎨 Visualizing complex numbers

• Complex numbers can be viewed as vectors from the origin to the point (x, y).
• Addition corresponds to vector addition (parallelogram rule).
• Subtraction z − w is the vector from w to z.
• Example: If z = 2 + i and w = −1 + 1.5i, then z + w = 1 + 2.5i is the diagonal of the parallelogram formed by z and w.

🧭 Overview

🧠 One-sentence thesis

The polar form of a complex number expresses it as r·e^(iθ), which simplifies multiplication by converting it into multiplying magnitudes and adding angles.

📌 Key points (3–5)

• What polar form is: any complex number z = x + yi can be rewritten as z = r·e^(iθ), where r is the modulus (distance from origin) and θ is the argument (angle).
• Key definition: e^(iθ) is defined as cos(θ) + i·sin(θ), linking exponential notation to trigonometry.
• Multiplication rule: multiplying two complex numbers in polar form means multiplying their moduli and adding their arguments.
• Common confusion: when r is negative, you can always convert to a non-negative r by adding π to the angle.
• Why it matters: polar form makes operations like multiplication much simpler than working with x + yi directly.

📐 From Cartesian to polar coordinates

📐 The coordinate relationships

• A point (x, y) in the plane can be represented in polar form (r, θ) using:
  • x = r·cos(θ)
  • y = r·sin(θ)
• This means any complex number x + yi can be rewritten as:
  • x + yi = r·cos(θ) + r·sin(θ)·i = r(cos(θ) + i·sin(θ))

🔢 Converting a specific example

The excerpt shows how to convert z = -3 + 4i to polar form:

• First, find the modulus: r = √(9 + 16) = 5
• Then find the angle: tan(α) = 4/3, so θ = π - tan⁻¹(4/3) ≈ 2.21 radians
• Result: -3 + 4i = 5·e^(i(π - tan⁻¹(4/3))) ≈ 5·e^(2.21i)
• Example: the angle calculation accounts for which quadrant the point lies in (here, the second quadrant).

🌀 The exponential definition

🌀 Defining e^(iθ)

For any real number θ, we define e^(iθ) = cos(θ) + i·sin(θ).

• This is a definition, not a derivation—it connects the exponential function to trigonometric functions.
• It allows us to write the polar form compactly as z = r·e^(iθ).

⭐ Famous special cases

The definition leads to several important values:

• e^(i·π/2) = cos(π/2) + i·sin(π/2) = 0 + i·1 = i
• e^(i·0) = cos(0) + i·sin(0) = 1
• e^(i·π) = cos(π) + i·sin(π) = -1

This last case produces the "all-star equation": e^(i·π) + 1 = 0, which connects e, i, π, 1, and 0.

🔧 Terminology and notation

🔧 Modulus and argument

If z = x + yi and (x, y) has polar form (r, θ), then z = r·e^(iθ) is called the polar form of z.

• Modulus: the non-negative scalar |r| (the distance from the origin)
• Argument: the angle θ, denoted arg(z)
• Don't confuse: the modulus is always non-negative, even though r can technically be negative (see below).

➕ Handling negative r

When r is negative, you can always convert to a non-negative modulus:

• If r < 0, then r·e^(iθ) = -|r|·e^(iθ) = e^(iπ)·|r|·e^(iθ) = |r|·e^(i(θ + π))
• This works because -1 = e^(iπ)
• Example: by adding π to the angle, you can always assume z = r·e^(iθ) where r ≥ 0.

✖️ Multiplication in polar form

✖️ The multiplication theorem

The product of two complex numbers in polar form is given by r·e^(iθ) · s·e^(iβ) = (rs)·e^(i(θ + β)).

How it works:

• Multiply the moduli: r·s
• Add the arguments: θ + β
• This is much simpler than multiplying (x₁ + y₁i)(x₂ + y₂i) in Cartesian form.

🔍 Proof sketch

The excerpt proves this using the definition of e^(iθ) and trigonometric identities:

1. Expand both exponentials: r(cos θ + i·sin θ)·s(cos β + i·sin β)
2. Multiply out: rs[cos θ·cos β - sin θ·sin β + (cos θ·sin β + sin θ·cos β)i]
3. Apply angle-sum identities: rs[cos(θ + β) + sin(θ + β)i]
4. Rewrite as: rs·e^(i(θ + β))

📏 Argument addition rule

From the multiplication theorem:

• arg(zw) = arg(z) + arg(w)
• Important caveat: this equation is taken modulo 2π, meaning arg(vw) = arg(v) + arg(w) + 2πk for some integer k, depending on how you choose the arguments.
• Don't confuse: angles "wrap around" every 2π, so the sum might differ by full rotations.

📊 Comparison: Cartesian vs polar

Aspect	Cartesian form (x + yi)	Polar form (r·e^(iθ))
What it shows	Horizontal and vertical components	Distance from origin and angle
Multiplication	Requires expanding (x₁ + y₁i)(x₂ + y₂i)	Simply multiply moduli and add angles
Conversion	Direct from coordinates	Requires computing r = √(x² + y²) and θ from tan(θ)
When to use	Addition/subtraction is simpler	Multiplication/division is simpler

🧭 Overview

🧠 One-sentence thesis

Division of complex numbers can be performed either by multiplying by the conjugate in Cartesian form or by dividing magnitudes and subtracting arguments in polar form, and angles between rays in the complex plane are computed using the argument of quotients.

📌 Key points (3–5)

• Division in Cartesian form: multiply numerator and denominator by the conjugate of the denominator to convert to standard form.
• Division in polar form: divide magnitudes and subtract arguments—if z = r·e^(iθ) and w = s·e^(iβ), then z/w = (r/s)·e^(i(θ−β)).
• Angle measurement convention: counterclockwise angles are positive, clockwise angles are negative, and angles are defined only up to multiples of 2π.
• Common confusion: the angle from L₁ to L₂ is the negative of the angle from L₂ to L₁, i.e., ∠(L₁, L₂) = −∠(L₂, L₁).
• Computing angles between rays: use the argument of the quotient of the difference vectors.

➗ Division of Complex Numbers

➗ Definition and basic concept

Division of the complex number z by w ≠ 0, denoted z/w, is the complex number u that satisfies the equation z = w · u.

• Division is not guessing; there are systematic methods in both Cartesian and polar forms.
• Example: 1/i = −i because 1 = i · (−i).

🔢 Division in Cartesian form

Method: Multiply both numerator and denominator by the conjugate of the denominator.

• The conjugate of a + bi is a − bi.
• This technique eliminates the imaginary part from the denominator.

Example: To compute (2 + i)/(3 + 2i):

• Multiply top and bottom by (3 − 2i):
  • Numerator: (2 + i)(3 − 2i) = (6 + 2) + (−4 + 3)i = 8 − i
  • Denominator: (3 + 2i)(3 − 2i) = 9 + 4 = 13
• Result: (8 − i)/13 = 8/13 − (1/13)i

📐 Division in polar form

Method: Divide magnitudes and subtract arguments.

• If z = r·e^(iθ) and w = s·e^(iβ) with w ≠ 0, then:
  • 1/w = (1/s)·e^(−iβ)
  • z/w = (r/s)·e^(i(θ−β))
• The argument of the quotient: arg(z/w) = arg(z) − arg(w) (modulo 2π).

Example: (1 + i)/(−3 + 3i):

• Convert to polar: 1 + i = √2·e^(iπ/4) and −3 + 3i = √18·e^(i3π/4)
• Divide: (√2/√18)·e^(i(π/4 − 3π/4)) = (1/3)·e^(−iπ/2) = −(1/3)i

Don't confuse: Polar division subtracts arguments; polar multiplication adds them.

📐 Angle Measurement Between Rays

📐 Angle convention and notation

∠(L₁, L₂) denotes the angle between rays L₁ and L₂, measured from L₁ to L₂.

Sign convention:

• Counterclockwise rotation: positive angle
• Clockwise rotation: negative angle
• Angles are well-defined only up to multiples of 2π

Key property: ∠(L₁, L₂) = −∠(L₂, L₁)

🧮 Computing angles between rays

Formula: If z₀ is the common initial point, z₁ is any point on L₁, and z₂ is any point on L₂, then:

• ∠(L₁, L₂) = arg((z₂ − z₀)/(z₁ − z₀))
• Equivalently: arg(z₂ − z₀) − arg(z₁ − z₀)

Example: Rays L₁ and L₂ emanate from 2 + 2i. L₁ proceeds along y = x and L₂ along y = 3 − x/2.

• Choose z₁ = 3 + 3i (on L₁) and z₂ = 4 + i (on L₂).
• Compute: ∠(L₁, L₂) = arg(2 − i) − arg(1 + i)
  • = −arctan(1/2) − π/4
  • ≈ −71.6°
• Interpretation: the angle from L₁ to L₂ is 71.6° in the clockwise direction.

🔺 Angle determined by three points

Notation: ∠uvw denotes the angle θ from ray vu to ray vw (v is the vertex).

Formula: ∠uvw = arg((w − v)/(u − v))

Special case: If u = 1 (on the positive real axis), v = 0 (the origin), and z is any point in ℂ, then ∠uvz = arg(z).

🔍 Key Properties and Relationships

🔍 Magnitude and conjugate properties

From the exercises and context:

Property	Statement
Magnitude of quotient	|z/w| = |z|/|w|
Conjugate of quotient	(z/w)̄ = z̄/w̄
Polar form of conjugate	If z = r·e^(iθ), then z̄ = r·e^(−iθ)

🔄 Converting negative radius in polar form

Issue: When representing z in polar form as z = r·e^(iθ), we may assume r is non-negative.

Solution: If r < 0, then:

• r·e^(iθ) = −|r|·e^(iθ)
• = e^(iπ) · |r|·e^(iθ) (since −1 = e^(iπ))
• = |r|·e^(i(θ + π))

Conclusion: By adding π to the angle if necessary, we can always assume z = r·e^(iθ) where r ≥ 0.

🧭 Overview

🧠 One-sentence thesis

Complex expressions provide compact representations of geometric objects like lines and circles in the complex plane, and solving complex equations (especially quadratics) always yields solutions in ℂ, unlike the real case where solutions may not exist.

📌 Key points (3–5)

• Lines in ℂ: Any line ax + by + d = 0 can be written as αz + ᾱz̄ + d = 0 (where α is complex, d is real), or equivalently as |z − γ| = |z − β| (points equidistant from γ and β).
• Circles in ℂ: A circle centered at z₀ with radius r is described by |z − z₀| = r.
• Quadratic equations always have solutions: Unlike real quadratics, complex quadratics αz² + βz + γ = 0 always have one or two solutions using the quadratic formula with complex square roots.
• Common confusion: Finding √(a + bi) requires either solving a system (equate real and imaginary parts after squaring) or using polar form with half-angle formulas—don't assume real square-root rules apply directly.
• Inequalities describe regions: Expressions like |z| < r or Im(z) < Re(z) describe geometric regions (interiors, half-planes, rays) in the complex plane.

📐 Lines in the complex plane

📐 Standard line equation in complex form

Line equation: ax + by + d = 0 can be represented as αz + ᾱz̄ + d = 0, where α = ½(a − bi) is a complex constant and d is a real number.

Why this works:

• For any complex β = s + ti, the expression βz + β̄z̄ equals 2sx − 2ty (a linear combination of x and y).
• Setting β = ½(a − bi) recovers the standard form ax + by + d = 0.
• Conversely, any equation αz + ᾱz̄ + d = 0 (α complex, d real) determines a line.

Example: The line y = mx + b (with m ≠ 0) can be written as (m + i)z + (m − i)z̄ + 2b = 0.

📏 Lines as equidistant loci

Theorem: Any line in ℂ can be expressed as |z − γ| = |z − β| for suitable points γ and β, and the set of all points equidistant from two distinct points forms a line.

How the proof works:

• Start with |z − γ|² = |z − β|² (equidistant condition).
• Expand using (z − γ)(z̄ − γ̄) = (z − β)(z̄ − β̄).
• Simplify to (β − γ)z + (β̄ − γ̄)z̄ + (|γ|² − |β|²) = 0, which has the form αz + ᾱz̄ + d = 0.
• Conversely, given a line, choose γ and β so the line is the perpendicular bisector of segment γβ.

Geometric interpretation: A line is the perpendicular bisector of any segment whose endpoints lie symmetrically on either side.

🔢 Quadratic equations in ℂ

🔢 Square roots of complex numbers

Square root: If z₀ = r₀e^(iθ₀) in polar form (r₀ ≥ 0), then √z₀ = ±√r₀ e^(iθ₀/2).

Key insight:

• Solving z² = z₀ means finding z = re^(iθ) such that r²e^(i2θ) = r₀e^(iθ₀).
• This requires r² = r₀ and 2θ = θ₀ (modulo 2π).
• As long as r₀ > 0, there are two solutions (the ± sign).

Example: To find √i, note i = 1·e^(iπ/2), so √i = ±e^(iπ/4) = ±(√2/2 + (√2/2)i).

🧮 General quadratic formula

Complex quadratic: αz² + βz + γ = 0 (α, β, γ complex) has solutions z = (−β ± √(β² − 4αγ)) / (2α).

Important difference from real case:

• In ℝ, a quadratic may have no real solutions (e.g., z² + 2z + 4 = 0).
• In ℂ, every quadratic has one or two solutions because every complex number has a square root.

Example: z² + 2z + 4 = 0 gives z = (−2 ± √(−12)) / 2 = −1 ± √3 i (using √(−1) = i).

🛠️ Two methods to compute √(a + bi)

🛠️ Method 1: Cartesian (system of equations)

Set x + yi = √(a + bi) and square both sides:

• (x + yi)² = a + bi
• x² − y² + 2xyi = a + bi

This gives three equations:

1. x² − y² = a
2. 2xy = b
3. x² + y² = |a + bi| (from modulus)

Solve for x and y.

Example: For √(8 + 6i):

• x² − y² = 8, 2xy = 6, x² + y² = 10
• Adding (1) and (3): x² = 9 → x = ±3
• Substituting: y = 1 or y = −1
• Result: √(8 + 6i) = ±(3 + i)

🛠️ Method 2: Polar (half-angle formula)

If a + bi = re^(iθ), then √(a + bi) = ±√r e^(iθ/2).

To find θ/2, use:

• tan(θ/2) = sin(θ) / (1 + cos(θ))
• Determine sin(θ) and cos(θ) from the right triangle with hypotenuse r.

Example: For √(8 + 6i) = √(10e^(iθ)):

• sin(θ) = 3/5, cos(θ) = 4/5
• tan(θ/2) = (3/5) / (1 + 4/5) = 1/3
• So √(8 + 6i) = k(3 + i) for some scalar k
• Since |√(8 + 6i)| = √10 and |k(3 + i)| = √10, we have k = ±1
• Result: √(8 + 6i) = ±(3 + i)

Don't confuse: Both methods must agree; choose based on convenience (Cartesian for simple arithmetic, polar for angles).

🧩 Worked example: solving z² − (3 + 3i)z = 2 − 3i

Rewrite as z² − (3 + 3i)z − (2 − 3i) = 0.

Apply quadratic formula with α = 1, β = −(3 + 3i), γ = −(2 − 3i):

• z = (3 + 3i ± √((3 + 3i)² + 4(2 − 3i))) / 2
• z = (3 + 3i ± √(8 + 6i)) / 2

Using √(8 + 6i) = ±(3 + i):

• z = (3 + 3i ± (3 + i)) / 2
• Solutions: z = 3 + 2i or z = i

⭕ Circles in the complex plane

⭕ Circle equation

Circle: |z − z₀| = r describes the circle centered at z₀ with radius r > 0.

Why this works:

• Let z = x + yi and z₀ = h + ki.
• |z − z₀| = |(x − h) + (y − k)i| = √((x − h)² + (y − k)²)
• So |z − z₀| = r is equivalent to (x − h)² + (y − k)² = r².

Example: |z − 3 − 2i| = 3 describes all points 3 units away from (3, 2), i.e., a circle of radius 3 centered at (3, 2).

🗺️ Regions defined by complex inequalities

🗺️ Interpreting inequalities and equations

Expression	Geometric region	Explanation
|1/z| > 2	Interior of circle |z| < 1/2	Taking reciprocals reverses inequality
Im(z) < Re(z)	Half-plane below line y = x	Setting z = x + yi gives y < x
Im(z) = |z|	Ray on positive imaginary axis	y = √(x² + y²) → x = 0 and y ≥ 0
|z + i| = 3	Circle centered at −i, radius 3	Standard circle form
|z + i| = |z − i|	Real axis (perpendicular bisector)	Points equidistant from i and −i

Example: Im(z) < Re(z) means y < x (all points below the line y = x).

Example: Im(z) = |z| means y = √(x² + y²). Squaring: y² = x² + y² → x² = 0 → x = 0. Then y = |y|, so y ≥ 0. This is the positive imaginary axis (a ray).

Don't confuse: Equations describe curves (lines, circles, rays); inequalities describe regions (interiors, half-planes).

📋 Summary: Lines and circles in ℂ

📋 Quick reference

Lines:

• Standard form: ax + by + d = 0
• Complex form: αz + ᾱz̄ + d = 0 (α = ½(a − bi), d real)
• Equidistant form: |z − γ| = |z − β| (perpendicular bisector of segment γβ)

Circles:

• Standard form: (x − h)² + (y − k)² = r²
• Complex form: |z − z₀| = r (center z₀, radius r > 0)

Key takeaway: Complex notation unifies and simplifies geometric descriptions—lines and circles have compact, elegant forms in ℂ.

🧭 Overview

🧠 One-sentence thesis

General linear transformations of the complex plane—including translations, rotations, dilations, and reflections—are one-to-one and onto functions that preserve geometric structure such as lines, circles, and angles.

📌 Key points (3–5)

• What a transformation is: a function from a set to itself that is both one-to-one (no two inputs map to the same output) and onto (every element is the image of some input).
• General linear transformations T(z) = az + b (where a ≠ 0) encompass translations, rotations, and dilations as special cases, and they preserve lines, circles, and angles.
• Euclidean isometries preserve distances between all pairs of points; translations, rotations, and reflections are isometries, while dilations (unless k = 1) are not.
• Common confusion: one-to-one vs onto—a function can be one without the other; a transformation must be both, which guarantees an inverse exists.
• Reflections as building blocks: any translation is a composition of two reflections across parallel lines, and any rotation is a composition of two reflections across intersecting lines; any Euclidean isometry is at most three reflections.

🔑 Core definitions

🔑 One-to-one and onto

One-to-one (1-1): A function f : A → B is one-to-one if whenever a₁ ≠ a₂ in A, then f(a₁) ≠ f(a₂) in B.

Onto: A function f : A → B is onto if for any b in B there exists an element a in A such that f(a) = b.

• One-to-one means distinct inputs always produce distinct outputs (no collisions).
• Onto means every possible output is actually reached by some input (full coverage).
• Example: In the schematic, f is onto but not one-to-one (two inputs map to the same output); g is one-to-one but not onto (some outputs are never reached).

🔑 Transformation

Transformation: A transformation on a set A is a function T : A → A that is both one-to-one and onto.

• Because it is both one-to-one and onto, a transformation has an inverse T⁻¹ characterized by T⁻¹(T(a)) = a and T(T⁻¹(a)) = a for all a in A.
• The inverse "undoes" T: T⁻¹(w) = z if and only if T(z) = w.
• Don't confuse: A function can be one-to-one or onto without being a transformation; only when it is both does it have an inverse on the same set.

🧩 General linear transformations

🧩 Definition and structure

General linear transformation: T(z) = az + b, where a, b are complex constants and a ≠ 0.

• This is the most general form that includes translations, rotations, and dilations as special cases.
• The condition a ≠ 0 is essential; if a = 0, the map collapses all points to b and is not one-to-one.

🧩 Proof that T(z) = az + b is a transformation

Onto: Given any w in ℂ, solve w = az + b for z to get z = (w − b)/a (valid since a ≠ 0). Then T((w − b)/a) = a · [(w − b)/a] + b = w.

One-to-one: Prove the contrapositive: if T(z₁) = T(z₂), then az₁ + b = az₂ + b, so az₁ = az₂, hence z₁ = z₂ (dividing by a ≠ 0).

• Example: This proof technique—proving the contrapositive "if Q then P" instead of "if not P then not Q"—is logically equivalent and often simpler.

🔄 Special cases of general linear transformations

🔄 Translation by b

Formula: T_b(z) = z + b

• Every point in the plane is moved by the vector b (same direction, same distance).
• Visualization: The origin moves to b; any two distinct points v and w move to T_b(v) and T_b(w), which remain distinct (one-to-one) and every point is the image of some pre-image (onto).
• Onto proof: Given w, let z = w − b; then T_b(z) = (w − b) + b = w.
• One-to-one proof (contrapositive): If T_b(z₁) = T_b(z₂), then z₁ + b = z₂ + b, so z₁ = z₂.
• Fixed points: If b ≠ 0, translation has no fixed points (no point stays in place).

🔄 Rotation by θ about the origin

Formula: R_θ(z) = e^(iθ) z

• Multiplying by e^(iθ) rotates the complex number z by angle θ counterclockwise (if θ > 0) or clockwise (if θ < 0).
• If z = r e^(iβ), then R_θ(z) = e^(iθ) · r e^(iβ) = r e^(i(θ + β)).
• Fixed points: Only the origin is fixed (R_θ(0) = 0).

🔄 Rotation by θ about any point z₀

Formula: R(z) = e^(iθ)(z − z₀) + z₀

• Three-step journey: translate so z₀ goes to the origin (T_{−z₀}), rotate by θ about the origin (R_θ), translate back (T_{z₀}).
• Composition: R = T_{z₀} ∘ R_θ ∘ T_{−z₀}.
• The composition of transformations is itself a transformation (Theorem 3.1.5).

🔄 Dilation by factor k > 0

Formula: T(z) = kz

• Stretches (if k > 1) or shrinks (if 0 < k < 1) points along rays from the origin.
• If z = x + yi, then T(z) = kx + kyi, which lies on the same line through the origin as z.
• Fixed points: Only the origin is fixed.

🪞 Reflections

🪞 Reflection across a line L

Formula: r_L(z) = e^(iθ) z̄ + b, where θ is a real angle and b is a complex constant.

• Points on L are sent to themselves; if z is not on L, it is sent to z* such that L is the perpendicular bisector of segment zz*.
• Special case: Reflection about the real axis is simply r_L(z) = z̄ (conjugation).
• General case: For any other line, rotate/translate the line to the real axis, take the conjugate, then reverse the rotation/translation.
• Example: Reflection about y = x + 5 is r_L(z) = iz − 5 + 5i (derived by translating by −5i, rotating by −π/4, conjugating, rotating by π/4, translating by 5i).

🪞 Reflections as building blocks

Theorem 3.1.17:

• A translation is the composition of reflections about two parallel lines.
• A rotation about z₀ is the composition of reflections about two lines intersecting at z₀.

Transformation	Composition of reflections
Translation by b	r₂ ∘ r₁, where L₁ and L₂ are parallel, perpendicular to b, separated by distance ‖b‖/2
Rotation by θ about z₀	r₂ ∘ r₁, where L₁ and L₂ intersect at z₀ at angle θ/2

• Don't confuse: The angle between the two lines is θ/2, not θ; the composition doubles the angle.
• Any Euclidean isometry can be expressed as the composition of at most three reflections (Theorem 3.1.20).

🎯 Geometric preservation properties

🎯 Lines and circles preserved

Theorem 3.1.9: General linear transformations map lines to lines and circles to circles.

• Proof sketch for lines: A line L is described by αz + ᾱz̄ + d = 0. Under T(z) = az + b, substitute z = (w − b)/a into the line equation; the result is another line equation in w.
• Reflections also send lines to lines and circles to circles (Theorem 3.1.19).

🎯 Angle preservation

Angle between curves: The angle ∠(r₁, r₂) measured from curve r₁ to r₂ at their intersection point is the angle between their tangent lines at that point.

Preserves angles: A transformation T preserves angles at z₀ if ∠(r₁, r₂) = ∠(T(r₁), T(r₂)) for all smooth curves r₁, r₂ intersecting at z₀.

Preserves angle magnitudes: |∠(r₁, r₂)| = |∠(T(r₁), T(r₂))| (the size but not necessarily the sign/orientation).

Theorem 3.1.12: General linear transformations preserve angles.

• Proof idea: It suffices to check lines. If L₁ and L₂ intersect at z₀, then ∠(L₁, L₂) = arg((z₂ − z₀)/(z₁ − z₀)). Under T(z) = az + b, the angle becomes arg((az₂ + b − az₀ − b)/(az₁ + b − az₀ − b)) = arg((z₂ − z₀)/(z₁ − z₀)), which is the same.
• Reflections preserve angle magnitudes (Theorem 3.1.19).

🎯 Distance preservation (Euclidean isometry)

Euclidean isometry: A transformation T of ℂ such that |T(z) − T(w)| = |z − w| for any points z, w in ℂ.

• Isometries preserve the Euclidean distance between every pair of points.
• Which general linear transformations are isometries? T(z) = az + b is an isometry if and only if |a| = 1.
  • Proof: |T(z) − T(w)| = |az + b − (aw + b)| = |a(z − w)| = |a| · |z − w|. This equals |z − w| exactly when |a| = 1.
• Translations (a = 1) and rotations (|e^(iθ)| = 1) are isometries; dilations with k ≠ 1 are not.
• Reflections are also Euclidean isometries (Theorem 3.1.19).

🔧 Practical tools

🔧 Image of a set

Image of D under T: T(D) = {a ∈ A | a = T(x) for some x ∈ D}.

• The image T(D) is the collection of all outputs T(x) as x ranges over D.
• Example: The disk D = {z | |z − 2i| ≤ 1} under T(z) = 2z + (4 − i) maps to the disk T(D) centered at 4 + 3i with radius 2 (the center 2i maps to 2·2i + 4 − i = 4 + 3i; the radius scales by |a| = 2).

🔧 Fixed points

Fixed point: An element a in A such that T(a) = a.

• To find fixed points of T(z) = az + b, solve z = az + b for z, which gives z = b/(1 − a) (provided a ≠ 1).
• Example: T(z) = 2z + (4 − i) has fixed point z = (4 − i)/(1 − 2) = −4 + i.
• Translations (b ≠ 0) have no fixed points; rotations and dilations about the origin have one fixed point (the origin); general linear transformations (a ≠ 1) have one fixed point.

🔧 Composition of transformations

Theorem 3.1.5: If T and S are transformations of A, then S ∘ T is also a transformation of A.

• Onto proof: Given c in A, since S is onto there exists b such that S(b) = c; since T is onto there exists a such that T(a) = b. Then S ∘ T(a) = S(b) = c.
• One-to-one proof (contrapositive): If S(T(a₁)) = S(T(a₂)), then T(a₁) = T(a₂) (since S is 1-1), hence a₁ = a₂ (since T is 1-1).
• This theorem justifies that compositions like rotation about any point (T_{z₀} ∘ R_θ ∘ T_{−z₀}) are themselves transformations.

🌀 Introduction to inversion

🌀 Inversion in a circle

Inversion in circle C (centered at z₀ with radius r): sends a point z ≠ z₀ to the unique point z* on the ray from z₀ through z such that |z − z₀| · |z* − z₀| = r².

• z* is called the symmetric point to z with respect to C.
• Notation: i_C(z) = z*.
• Inversion is a transformation on the set ℂ − {z₀} (all complex numbers except the center).
• The excerpt notes that inversion can be extended to include z₀ (discussed in the next section) and that i_C fixes all points on the circle C (since if |z − z₀| = r, then |z* − z₀| = r²/r = r, so z* = z).
• Don't confuse: Inversion is not a general linear transformation; it is a new type of transformation that reflects across a circle rather than a line.

🧭 Overview

🧠 One-sentence thesis

Inversion in a circle is a transformation that reflects points across the circle by mapping each point to a symmetric point on the same ray from the center, preserving angle magnitudes and mapping clines (circles or lines) to clines, which makes it fundamental for visualizing non-Euclidean geometry.

📌 Key points (3–5)

• What inversion does: reflects a point z across a circle C to a symmetric point z* on the same ray from the center, satisfying |z − z₀| · |z* − z₀| = r².
• Clines to clines: inversion maps circles and lines (collectively called "clines") to clines; a cline through the center becomes a line, otherwise it becomes a circle.
• Angle preservation: inversion preserves the magnitude of angles between curves (though it may reverse their sign).
• Symmetry preservation: if two points are symmetric with respect to a cline, their images under inversion remain symmetric with respect to the image cline.
• Common confusion: lines and circles behave differently under inversion depending on whether they pass through the center of the circle of inversion—lines through the center invert to themselves (if orthogonal), while other clines swap between line and circle forms.

🎯 The inversion transformation

🎯 Definition and construction

Inversion in circle C: Given a circle C with center z₀ and radius r, inversion sends a point z ≠ z₀ to the unique point z* on the ray from z₀ through z such that |z − z₀| · |z* − z₀| = r².

• The point z* is called the symmetric point to z with respect to C.
• Notation: i_C(z) = z*.
• The transformation is defined on all complex numbers except the center z₀ (extended in Section 3.3 to include z₀ and ∞).

🔄 Basic behavior

• Points on the circle: all points on C are fixed (mapped to themselves).
• Inside ↔ outside swap: points inside the circle map to points outside, and vice versa.
• Distance from center: the closer z gets to the center z₀, the farther i_C(z) moves from the circle.

📐 Formula for unit circle

For the unit circle S¹ (center 0, radius 1), the symmetric point equation simplifies:

• |z| · |z*| = 1.
• Since z* lies on the same ray through the origin, z* = kz for some positive real k.
• Solving gives k = 1/|z|², so z* = z / |z|².
• Because |z|² = z · z̄, inversion in the unit circle is i*_S¹(z) = 1/z̄.

📐 Formula for arbitrary circle

For a circle C centered at z₀ with radius r:

i*_C(z) = r² / (z̄ − z̄₀) + z₀

(Derived by composing inversion in the unit circle with translations and dilations.)

🔗 Clines: unifying lines and circles

🔗 What is a cline?

Cline: a Euclidean circle or line.

• The word "cline" (pronounced 'Klein') treats lines and circles as different manifestations of the same class of objects.
• This unified view is essential because inversion can map lines to circles and vice versa.

🧮 Algebraic equation

Any cline can be described by:

czz̄ + αz + ᾱz̄ + d = 0

where z = x + yi, α is a complex constant, and c, d are real numbers.

Condition	Type	Details
c = 0	Line	Equation reduces to a linear form
c ≠ 0 and |α|² > cd	Circle	Center at (−Re(α)/c, Im(α)/c), radius √(|α|² − cd)/c²

🏗️ Constructing a cline through three points

Theorem 3.2.4: There exists a unique cline through any three distinct points.

• If the three points are collinear, the unique cline is that line.
• Otherwise, construct perpendicular bisectors of two segments; their intersection is the center of the unique circle through the three points.

🔄 How inversion transforms clines

🔄 Clines map to clines

Theorem 3.2.6: Inversion in a circle maps clines to clines.

• If the cline passes through the center of the circle of inversion → its image is a line.
• If the cline does not pass through the center → its image is a circle.

Why this matters: Lines and circles are interchangeable under inversion, which is crucial for non-Euclidean geometry.

Proof sketch (for inversion in unit circle):

• Start with cline equation czz̄ + αz + ᾱz̄ + d = 0.
• Substitute w = 1/z̄ (the image point).
• Multiply the original equation by 1/(z · z̄) to get c + α·w + ᾱ·w̄ + dww̄ = 0.
• This is again a cline equation (with c and d swapped roles).
• If d = 0 (original cline through origin) → image is a line; if d ≠ 0 → image is a circle.

📏 Example: inverting figures

The excerpt shows inversions of a circle, the letter 'M', and a grid:

• A circle not through the center inverts to another circle.
• Line segments in 'M' (not through the center) invert to arcs of circles.
• This visual behavior confirms the theorem.

⊥ Orthogonality and self-inversion

⊥ Orthogonal clines

Two clines are orthogonal if they intersect at right angles.

• A line is orthogonal to a circle ↔ the line passes through the center of the circle.

🔁 Clines orthogonal to the circle of inversion

Corollary 3.2.9: Inversion in C takes clines orthogonal to C to themselves.

Theorem 3.2.8: A cline through z (where z ≠ z₀ and z not on C) is orthogonal to C ↔ it passes through z*, the symmetric point to z.

Proof idea:

• For a line through z: it passes through z* ↔ it passes through the center z₀ ↔ it is orthogonal to C.
• For a circle D through z: use the power of a point (Lemma 3.2.7). The circle D is orthogonal to C ↔ r² = s² − k² (where s is distance from z₀ to center of D, k is radius of D). This condition is equivalent to z* lying on D.

🔋 Power of a point (Lemma 3.2.7)

If C has radius r centered at o, and p is outside C, and a line through p intersects C at m and n, then:

|p − m| · |p − n| = s² − r²

where s = |p − o|.

• The quantity s² − r² is the power of the point p with respect to C.
• Proof uses Pythagorean theorem on perpendicular bisector of segment mn.

📐 Angle and symmetry preservation

📐 Angle magnitude preservation

Theorem 3.2.10: Inversion in a cline preserves angle magnitudes.

• The angle between two curves at their intersection point has the same magnitude before and after inversion.
• The sign of the angle may reverse (clockwise ↔ counterclockwise).

Proof sketch:

• The angle between two curves r₁ and r₂ at z equals the angle between their tangent lines L₁ and L₂.
• Construct two circles C₁ and C₂ tangent to L₁ and L₂, both orthogonal to the circle of inversion C.
• These circles pass through both z and z*.
• The angle at z* (after inversion) equals the angle at z in magnitude.
• For angles on the circle C, use a concentric circle and compose with a dilation.

Don't confuse: angle magnitude is preserved, but orientation (sign) may flip.

🔗 Symmetry point preservation

Theorem 3.2.12: Inversion preserves symmetry points.

If p and q are symmetric with respect to cline D, then i_C(p) and i_C(q) are symmetric with respect to i_C(D).

Proof idea:

• Construct two clines E and F through p and q, both orthogonal to D.
• Inversion maps E and F to clines orthogonal to i_C(D).
• The images i_C(p) and i_C(q) lie on both image clines, so they are symmetric with respect to i_C(D).

🎯 Applications

🎯 Apollonian circles (Theorem 3.2.14)

Given two distinct points p, q and a positive real k > 0, the set D of all points z satisfying:

|z − p| = k|z − q|

is a cline.

• If k = 1, D is the perpendicular bisector (a line).
• If k ≠ 1, D is a circle.

Proof sketch:

• Invert about a circle C centered at p with radius 1.
• Show that triangles Δpz*q* and Δpqz are similar.
• Derive that |z* − q*| is constant, so i_C(D) is a circle.
• Since inversion preserves clines and p is not on i_C(D), D itself is a circle.

Circles of Apollonius: the family of clines obtained by varying k; points p and q are symmetric with respect to each cline in the family.

🔗 Symmetric points for two non-intersecting clines (Theorem 3.2.16)

If two clines do not intersect and at least one is a circle, there exist two points p and q symmetric with respect to both clines.

Proof sketch:

• Case 1 (line L and circle C): construct perpendicular from center of C to L; build auxiliary circles to find two points on the perpendicular that are symmetric to both.
• Case 2 (two circles C₁ and C₂): invert one circle to a line, apply Case 1, then invert back (using symmetry preservation).
• Exception: concentric circles require extending inversion to include the center (done in Section 3.3).

🛠️ Construction techniques

🛠️ Constructing the symmetric point (inside the circle)

For z inside circle C centered at z₀:

1. Draw ray from z₀ through z.
2. Construct perpendicular to this ray at z; let t be an intersection with C.
3. Construct radius z₀t.
4. Construct perpendicular to this radius at t; z* is where this perpendicular meets the ray.

🛠️ Constructing the symmetric point (outside the circle)

For z outside circle C centered at z₀:

1. Construct circle with diameter z₀z; let t be an intersection with C.
2. Construct perpendicular to z₀z through t; z* is where this perpendicular meets segment z₀z.

🌐 Extension to the extended plane (preview)

🌐 The point at infinity

• Points close to z₀ invert to points far from z₀ (magnitudes → ∞).
• Define a new point ∞ (the point at infinity).
• The extended plane C⁺ = C ∪ {∞}.

🌐 Extended inversion

Inversion in circle C centered at z₀ with radius r extends to C⁺:

• i_C(z₀) = ∞
• i_C(∞) = z₀
• i_C(z) = r² / (z̄ − z̄₀) + z₀ for z ≠ z₀, ∞

Convention: ∞ is on every line in the extended plane; reflection across any line fixes ∞.

Why this matters: treating z₀ and ∞ as symmetric points completes the inversion transformation and is essential for non-Euclidean geometry (developed in later sections).

🧭 Overview

🧠 One-sentence thesis

The extended plane C⁺ adjoins a single point at infinity to the complex plane, enabling inversion and other transformations to be defined everywhere and revealing that lines and circles form a unified family of curves (clines) that all pass through ∞.

📌 Key points (3–5)

• Why add ∞: inversion maps points near the center z₀ to points far away, and vice versa; adding ∞ makes inversion a complete transformation with no gaps.
• Where ∞ lives: ∞ is approached along any line in any direction; by convention, every line in C⁺ passes through ∞.
• How transformations behave at ∞: general linear transformations (az + b) fix ∞, but more general transformations (az + b)/(cz + d) can send ∞ to a finite point and vice versa.
• Common confusion: lines vs circles—in C⁺, lines are just circles that pass through ∞; both are "clines" and behave uniformly under inversion.
• Stereographic projection: the extended plane can be identified with a sphere, giving a geometric picture of ∞ as the north pole and preserving angles everywhere, including at ∞.

🔄 Inversion and the point at infinity

🔄 Why inversion needs ∞

• Inversion in a circle C centered at z₀ with radius r maps points by the formula r²/(z − z₀) + z₀.
• Points close to z₀ are mapped far away: as a sequence approaches z₀, its image magnitudes go to ∞.
• Conversely, points with large magnitudes are mapped close to z₀: sequences marching off to ∞ invert to sequences approaching z₀.
• Without ∞, inversion is undefined at z₀ and has no target for sequences going to ∞.

➕ Defining the extended plane

Extended plane C⁺: the complex plane C with one additional point ∞ adjoined.

• Inversion can now be extended to all of C⁺:
  • If z ≠ z₀ and z ≠ ∞, use the usual formula r²/(z − z₀) + z₀.
  • If z = z₀, then iC(z₀) = ∞.
  • If z = ∞, then iC(∞) = z₀.
• Symmetric points: z₀ and ∞ are defined to be symmetric with respect to the circle of inversion C.

🌐 Where is ∞?

• You approach ∞ by proceeding in either direction along any line in the complex plane.
• More generally, if {zₙ} is a sequence with |zₙ| → ∞ as n → ∞, then lim zₙ = ∞.
• Convention: ∞ is on every line in the extended plane.
• Reflection across any line fixes ∞.

🔧 Transformations on C⁺

🔧 General linear transformations fix ∞

Theorem 3.3.1: Any general linear transformation T(z) = az + b (with a ≠ 0) extended to C⁺ fixes ∞.

• Why: as |zₙ| → ∞, we have |azₙ + b| → ∞ as well (by limit methods from calculus).
• Therefore T(∞) = ∞.

📊 Fixed-point counts in C⁺

Transformation	Fixed points in C⁺
Translation Tᵦ(z) = z + b	One point: ∞
Rotation Rθ(z) = e^(iθ)z	Two points: 0 and ∞
Dilation T(z) = kz	Two points: 0 and ∞
Reflection rₗ(z) across line L	All points on L (including ∞)

🔀 Transformations that move ∞

• More general transformations of the form T(z) = (az + b)/(cz + d) can send ∞ to a finite point and send a finite point to ∞.
• Finding where T sends ∞: take a sequence marching off to ∞ (e.g., 1, 2, 3, ...) and compute the limit of T(zₙ).
• Finding which point maps to ∞: find the input that makes the denominator zero.

Example: For T(z) = (i + 1)/(z + 2i):

• T(−2i) = ∞ (denominator is zero).
• T(∞) = 0 (the sequence T(1), T(2), T(3), ... has constant numerator and unbounded denominator, so the quotient → 0).

Example: For T(z) = (iz + 3i + 1)/(2iz + 1):

• T(i/2) = ∞.
• T(∞) = 1/2.

🔗 Clines and key results extended to C⁺

🔗 Clines through three points

• Key result: There exists a unique cline through any three distinct points in C⁺.
• Special case: If one of the three points is ∞, the unique cline is the line through the other two points.
• Why this matters: lines are just circles that pass through ∞; in C⁺, lines and circles are unified as "clines."

🔗 Properties preserved under inversion

The following properties from the previous section now hold for all points in C⁺:

• Theorem 3.2.8 applies to all points z not on C, including z = z₀ or z = ∞.
• Angle preservation: Inversion about a cline preserves angle magnitudes at all points in C⁺.
• Symmetry preservation: Inversion preserves symmetry points for all points in C⁺ (even if p or q is ∞).
• Symmetry for non-intersecting clines: Theorem 3.2.16 now holds for all clines that do not intersect, including concentric circles. For concentric circles, the points symmetric to both are ∞ and the common center.

🌍 Stereographic projection

🌍 The unit 2-sphere

Unit 2-sphere S²: the set of all points in 3-space that are one unit from the origin: S² = {(a, b, c) ∈ ℝ³ | a² + b² + c² = 1}.

🌍 Definition of stereographic projection

• Let N = (0, 0, 1) be the north pole on the sphere.
• For any point P ≠ N on the sphere, φ(P) is the point on the ray NP that lives in the xy-plane.
• Algebraic formula: For P = (a, b, c) with c ≠ 1, the line through N and P is r(t) = ⟨0, 0, 1⟩ + t⟨a, b, c − 1⟩.
• This line intersects the xy-plane when z = 0, which occurs at t = 1/(1 − c).
• Therefore φ((a, b, c)) = a/(1 − c) + (b/(1 − c))i.
• Where does φ send N? To ∞. A sequence of points on S² approaching N has image points in C with magnitudes approaching ∞.

🌍 Why stereographic projection matters

Theorem 3.3.5: Stereographic projection preserves angles. If two curves on the sphere intersect at angle θ, their image curves in C⁺ also intersect at angle θ.

• This theorem guides how we define angles at ∞.

📐 Angles at ∞

📐 Defining angles at ∞

• If two curves in C⁺ intersect at ∞, the angle at which they intersect is defined to equal the angle at which their pre-image curves under stereographic projection intersect on the sphere.
• Parallel lines: two parallel lines intersect at ∞ at angle 0.
• Lines intersecting at two points: if two lines intersect at a finite point p and also at ∞, the angle at ∞ equals the negative of the angle at p.

📐 Inversion preserves angles everywhere

• As a consequence of stereographic projection and angle preservation, inversion about a circle preserves angle magnitudes at all points in C⁺, including at ∞.

🧭 Overview

🧠 One-sentence thesis

Möbius transformations are the unique class of functions that map the extended complex plane onto itself by composing an even number of inversions, and they preserve angles, map clines to clines, and are completely determined by where they send any three distinct points.

📌 Key points (3–5)

• Definition and condition: A Möbius transformation has the form T(z) = (az + b)/(cz + d) and is a valid transformation of the extended plane if and only if the determinant ad − bc is not zero.
• Built from inversions: Every Möbius transformation is the composition of an even number of inversions (reflections or circle inversions), which explains why they preserve angles and map clines to clines.
• Fundamental uniqueness: There is exactly one Möbius transformation that maps any three distinct points to any other three distinct points (Fundamental Theorem).
• Common confusion: The determinant condition ad − bc ≠ 0 is necessary and sufficient; if the determinant is zero, the function is not one-to-one or onto, so it fails to be a transformation.
• Cross ratio tool: The cross ratio (z, w; u, v) provides an algebraic way to construct the unique Möbius transformation sending w → 1, u → 0, and v → ∞, and it remains invariant under any Möbius transformation.

🔍 Definition and determinant condition

🔍 What is a Möbius transformation

A function T(z) = (az + b)/(cz + d) where a, b, c, d are complex constants is called a Möbius transformation if ad − bc ≠ 0. These are also called fractional linear transformations.

• The complex number ad − bc is called the determinant of T, denoted Det(T).
• The determinant condition is both necessary and sufficient for T to be a transformation (one-to-one and onto) of the extended complex plane C⁺.

✅ Why the determinant matters (Theorem 3.4.1)

Forward direction (ad − bc ≠ 0 ⇒ T is a transformation):

• To show one-to-one: if T(z₁) = T(z₂), cross-multiply and simplify to get (ad − bc)z₁ = (ad − bc)z₂; since ad − bc ≠ 0, divide it out to conclude z₁ = z₂.
• To show onto: given any w in C⁺, solve (az + b)/(cz + d) = w for z. The determinant condition ensures a and c are not both zero, so the equation can be solved: z = (−dw + b)/(cw − a).

Reverse direction (ad − bc = 0 ⇒ T is not a transformation):

• Case 1: If ad = 0, then bc = 0 as well, so at least two of {a, b, c, d} are zero. For example, if a = c = 0, then T(z) = b/d is constant, hence neither one-to-one nor onto.
• Case 2: If ad ≠ 0, then all four constants are non-zero and a/c = b/d. Then T(0) = b/d and T(∞) = a/c are equal, so T is not one-to-one.

Don't confuse: The determinant being zero does not mean the formula is undefined; it means the function fails to be a valid transformation (it collapses points or misses targets).

🔄 Inverse and composition (Theorems 3.4.2 and 3.4.3)

• The inverse of T(z) = (az + b)/(cz + d) is T⁻¹(z) = (−dz + b)/(cz − a), which is itself a Möbius transformation with the same determinant.
• The composition of two Möbius transformations is again a Möbius transformation.

🔗 Connection to inversions

🔗 Möbius transformations as even compositions (Theorem 3.4.4)

A transformation of C⁺ is a Möbius transformation if and only if it is the composition of an even number of inversions.

Why this works:

• Any general linear transformation T(z) = az + b is already the composition of an even number of inversions: it is a dilation (two inversions about concentric circles) plus a rotation (two reflections) plus a translation (two reflections).
• For the general case T(z) = (az + b)/(cz + d) with c ≠ 0, rewrite by long division:
  • T(z) = (a/c) + [(bc − ad)/c]/(cz + d).
  • This is the composition T₃ ∘ T₂ ∘ T₁, where T₁(z) = cz + d (linear), T₂(z) = 1/z (inversion in the unit circle followed by reflection), and T₃(z) = [(bc − ad)/c]z + (a/c) (linear).
  • Each of T₁, T₂, T₃ is an even number of inversions, so T is as well.

Converse: If T is the composition of two inversions, it is a Möbius transformation (checked case-by-case for two circle inversions, one circle inversion plus one line reflection, and two line reflections). Any even number of inversions is then half as many Möbius transformations, hence a Möbius transformation by closure under composition.

🎨 Inherited properties (Theorem 3.4.5)

Because Möbius transformations are built from inversions:

• They map clines to clines (since inversions do).
• They preserve angles (since an even number of inversions preserves angle orientation and magnitude).

🎯 Fixed points and uniqueness

🎯 How many fixed points (Theorem 3.4.6)

Any Möbius transformation T: C⁺ → C⁺ fixes 1, 2, or all points of C⁺.

How to find fixed points:

• Solve (az + b)/(cz + d) = z, which gives the quadratic cz² + (d − a)z − b = 0.
• If c ≠ 0, the quadratic has 1 or 2 solutions.
• If c = 0 and a ≠ d, then T(z) = (az + b)/d fixes ∞ and one finite point z = b/(d − a), so 2 fixed points.
• If c = 0 and a = d, then the equation reduces to 0 = −b, so b = 0 and T(z) = z (the identity), which fixes every point.

Example: A non-identity Möbius transformation cannot fix three distinct points (this fact is used in the uniqueness proof below).

🏆 Fundamental Theorem (Theorem 3.4.8)

There is a unique Möbius transformation taking any three distinct points of C⁺ to any three distinct points of C⁺.

Construction (existence):

1. Build a Möbius transformation T that sends z₁ → 1, z₂ → 0, z₃ → ∞ (assuming z₃ ≠ ∞):
   • Invert about a circle centered at z₃ to send z₃ → ∞.
   • Translate to send the image of z₂ to 0.
   • Rotate and dilate about the origin to send the image of z₁ to 1.
   • Reflect across the real axis (to make the total number of inversions even).
2. Similarly, build S that sends w₁ → 1, w₂ → 0, w₃ → ∞.
3. The composition S⁻¹ ∘ T sends z₁ → w₁, z₂ → w₂, z₃ → w₃.

Uniqueness:

• Suppose U and V both send zᵢ → wᵢ for i = 1, 2, 3.
• Then V⁻¹ ∘ U fixes z₁, z₂, z₃ (three points).
• By Theorem 3.4.6, the only Möbius transformation fixing more than two points is the identity.
• So V⁻¹ ∘ U = identity, hence U = V.

Don't confuse: This theorem does not say "any function sending three points to three points is Möbius"; it says "there is exactly one Möbius transformation that does so."

🧮 Cross ratio

🧮 Definition and use (Definition 3.4.10)

The cross ratio of four complex numbers z, w, u, v (where w, u, v are distinct) is: (z, w; u, v) = [(z − u)/(z − v)] · [(w − v)/(w − u)].

• When z is a variable and w, u, v are distinct constants, T(z) = (z, w; u, v) is the unique Möbius transformation sending w → 1, u → 0, v → ∞.
• If one of the points is ∞, the formula reduces by canceling terms with ∞. For example, if u = ∞, then T(z) = (z − v)/(w − v).

🔧 Building a transformation (Example 3.4.11)

Problem: Find the Möbius transformation sending 1 → 3, i → 0, 2 → −1.

Method:

1. Write T(z) = (z, 1; i, 2) (sends 1 → 1, i → 0, 2 → ∞).
2. Write S(w) = (w, 3; 0, −1) (sends 3 → 1, 0 → 0, −1 → ∞).
3. The desired map is S⁻¹ ∘ T, found by setting the cross ratios equal:
   • (z, 1; i, 2) = (w, 3; 0, −1).
   • Solve for w in terms of z.
4. Result: V(z) = (−3z + 3i)/[(7 − 4i)z + (−8 + 5i)].
5. Check: V(1) = 3, V(i) = 0, V(2) = −1. ✓

🔒 Invariance (Theorem 3.4.12)

If z₀, z₁, z₂, z₃ are four distinct points and T is any Möbius transformation, then: (z₀, z₁; z₂, z₃) = (T(z₀), T(z₁); T(z₂), T(z₃)).

Why: Define S(z) = (z, z₁; z₂, z₃). Then S ∘ T⁻¹ sends T(z₁) → 1, T(z₂) → 0, T(z₃) → ∞, so S ∘ T⁻¹(z) = (z, T(z₁); T(z₂), T(z₃)). Plug in z = T(z₀) on both sides to get the result.

📐 Application: four points on a cline (Example 3.4.13)

• Four distinct points z, w, u, v lie on the same cline if and only if (z, w; u, v) is a real number.
• Example: The points 1, i, −1, −i lie on the circle |z| = 1. Check: (1, i; −1, −i) = 2 (real). ✓

🔄 Symmetry and mapping clines

🔄 Preservation of symmetry (Corollary 3.4.14)

• If z and z* are symmetric with respect to cline C, and T is any Möbius transformation, then T(z) and T(z*) are symmetric with respect to T(C).
• This follows from the fact that inversions preserve symmetry and Möbius transformations are compositions of inversions.

🗺️ Mapping one cline to another (Theorem 3.4.15)

Given any two clines C₁ and C₂, there exists a Möbius transformation T that maps C₁ onto C₂.

Construction:

1. Pick a point p₁ on C₁ and two points q₁, q₁* symmetric with respect to C₁.
2. Pick a point p₂ on C₂ and two points q₂, q₂* symmetric with respect to C₂.
3. Build the Möbius transformation sending p₁ → p₂, q₁ → q₂, q₁* → q₂*.
4. By the symmetry preservation property, this T maps C₁ to C₂.

Example scenario: To map the circle |z| = 4 to the line 3x + y = 4, choose three points on the circle, three points on the line, and apply the Fundamental Theorem.

🧭 Overview

🧠 One-sentence thesis

Visualizing Möbius transformations through their fixed points and associated families of clines reveals their geometric behavior as combinations of dilations, rotations, and translations along specific coordinate systems.

📌 Key points (3–5)

• Type I and Type II clines: Given two points p and q, Type I clines pass through both points; Type II clines are those for which p and q are symmetric (circles of Apollonius).
• Normal forms simplify understanding: Instead of the standard a, b, c, d coefficients, normal forms express transformations in terms of fixed points, dilation factors, and rotation factors, making geometric behavior transparent.
• Three classes by fixed-point behavior: Elliptic (rotation only, no dilation), hyperbolic (dilation only, no rotation), and parabolic (one fixed point, pushing along mutually tangent clines).
• Common confusion—loxodromic vs. special cases: A general transformation with two fixed points is loxodromic (both dilation and rotation); elliptic and hyperbolic are special cases when one component vanishes.
• Pole and inverse pole relationship: For transformations fixing two finite points p and q, the pole (sent to infinity) and inverse pole (image of infinity) satisfy p + q = z∞ + w∞.

🎯 Type I and Type II clines

🔵 Type I clines

Type I cline of p and q: a cline that goes through both p and q.

• These are curves connecting the two points.
• Example: For points 0 and ∞, Type I clines are precisely lines through the origin.
• They form one "axis" of a coordinate system for the plane.

🔴 Type II clines

Type II cline of p and q: a cline with respect to which p and q are symmetric.

• Also called circles of Apollonius.
• Example: For points 0 and ∞, Type II clines are circles centered at the origin (inversion in a circle takes the center to ∞).
• They form the other "axis" of the coordinate system.

⊥ Orthogonality and preservation

• Any Type II cline intersects any Type I cline at right angles.
• Möbius transformations preserve both types: if C is a Type I cline of p and q, then T(C) is a Type I cline of T(p) and T(q); similarly for Type II clines.
• Together, Type I and Type II clines create a general coordinate system for the plane—every point z lies at the intersection of exactly one Type I and one Type II cline, meeting at right angles.

🔧 Normal form for two fixed points

📐 Deriving the normal form

When a Möbius transformation T fixes two distinct finite points p and q (neither is ∞), the transformation can be rewritten in normal form:

Normal form (two fixed points): T(z) − p / T(z) − q = r e^(iθ) · (z − p) / (z − q)

• Here r is a dilation factor and θ is a rotation factor.
• This form is far more illuminating than the standard (az + b)/(cz + d) because each constant has clear geometric meaning.

🧩 The three-leg journey interpretation

The normal form arises from the composition T = S⁻¹ ∘ U ∘ S, where:

• S(z) = (z − p)/(z − q) maps p to 0 and q to ∞.
• U(z) = r e^(iθ) z is a simple dilation-rotation fixing 0 and ∞.
• S⁻¹ maps back.

How a point z moves under T:

1. S sends z (on the intersection of a Type I and Type II cline of p and q) to S(z) (on a line through 0 and a circle centered at 0).
2. U dilates S(z) along the line by factor r, then rotates around a circle by angle θ.
3. S⁻¹ sends the result back to the intersection of Type I and Type II clines of p and q.

Shortcut insight: T pushes points along Type I clines of p and q (by dilation r) and along Type II clines of p and q (by rotation θ).

🔍 Special case: fixing 0 and ∞

If T fixes 0 and ∞, then T(z) = r e^(iθ) z.

• Dilation by r pushes points along lines through the origin (Type I clines of 0 and ∞).
• Rotation by θ pushes points along circles centered at the origin (Type II clines of 0 and ∞).
• If r > 1, points move toward ∞; if 0 < r < 1, toward 0; if r = 1, no dilation occurs.

🌀 Three classes of transformations

🌀 Elliptic transformations

Elliptic Möbius transformation: a transformation where |r e^(iθ)| = 1, so there is no dilation.

• Points rotate around Type II clines of p and q.
• No net movement toward or away from the fixed points.
• Example: Points swirl around circles of Apollonius.

➡️ Hyperbolic transformations

Hyperbolic Möbius transformation: a transformation where θ = 0, so there is no rotation.

• Points move along Type I clines of p and q.
• All points either move away from p and toward q, or vice versa, depending on whether r > 1 or r < 1.
• The pole z∞, inverse pole w∞, and the two fixed points p and q all lie on the same Euclidean line.

🌪️ Loxodromic transformations

• A general transformation with both dilation (r ≠ 1) and rotation (θ ≠ 0).
• Combines the behaviors of elliptic and hyperbolic maps.
• Don't confuse: Loxodromic is the general case; elliptic and hyperbolic are special cases when one component vanishes.

🎯 Pole and inverse pole

📍 Definitions

Pole (z∞): the finite point sent to ∞ by T, i.e., T(z∞) = ∞.

Inverse pole (w∞): the image of ∞ under T, i.e., T(∞) = w∞.

• These exist when T fixes two finite points p and q and is not the identity.

🔗 Key relationship (Lemma 3.5.7)

For a Möbius transformation fixing distinct finite points p and q, with pole z∞ and inverse pole w∞:

p + q = z∞ + w∞

• This relationship is derived by plugging z = z∞ and z = ∞ into the normal form and solving.
• It provides a simple algebraic constraint linking the four points.

📝 Explicit formula (Theorem 3.5.8)

If T fixes p and q, sends z∞ to ∞, and sends ∞ to w∞, then:

T(z) = w∞ z − pq / z − z∞

• This formula is derived from the normal form using the relationship p + q = z∞ + w∞.
• Example: Fix −1 and 1, send i to ∞. By the lemma, ∞ maps to −i, so T(z) = (−iz + 1)/(z − i).

🪂 Parabolic transformations (one fixed point)

🎯 When there is only one fixed point

If T fixes only one point p ≠ ∞ (and no other), the normal form is:

Normal form (one fixed point): 1/(T(z) − p) = 1/(z − p) + d

• This is derived by conjugating T with S(z) = 1/(z − p), which maps p to ∞.
• The conjugated map U fixes only ∞, so U(z) = z + d (a translation).

🌊 Geometric behavior

Parabolic Möbius transformation: a transformation fixing just one point, pushing points along clines that are mutually tangent at that point.

• The map U(z) = z + d pushes points along parallel lines in the direction of d; these lines all meet at ∞ and are mutually tangent there.
• S⁻¹ takes this system to clines mutually tangent at p.
• The single line in this system passes through p and T(∞).
• Example: T(z) = (7z − 12)/(3z − 5) has fixed point z = 2 (solving (z − 2)² = 0). Using T(0) = 2.4, we find d = 3, so the normal form is 1/(T(z) − 2) = 1/(z − 2) + 3.

🔍 Special case: fixing only ∞

If T fixes only ∞, then T(z) = z + d for some complex constant d.

• This is a pure translation.
• Translations are the only Möbius transformations that fix ∞ and no other point.

📊 Summary table of transformation types

Type	Fixed points	Normal form	Geometric behavior
Elliptic	Two distinct (p, q)	|r e^(iθ)| = 1	Rotation around Type II clines; no dilation
Hyperbolic	Two distinct (p, q)	θ = 0	Dilation along Type I clines; points move toward one fixed point, away from the other
Loxodromic	Two distinct (p, q)	Both r ≠ 1 and θ ≠ 0	Combination of dilation and rotation
Parabolic	One (p)	1/(T(z)−p) = 1/(z−p) + d	Points move along clines mutually tangent at p
Translation	One (∞ only)	T(z) = z + d	Points move along parallel lines

🔬 Worked examples

🧪 Example: Hyperbolic transformation

Consider T(z) = (6 + 3i)z + (2 − 3i) / (z + 3).

Fixed points: Solve T(z) = z:

• (6 + 3i)z + (2 − 3i) = z² + 3z
• z² − (3 + 3i)z − (2 − 3i) = 0
• Solutions: z = i and z = 3 + 2i

Normal form: T(z) − i / T(z) − (3 + 2i) = λ · (z − i) / (z − (3 + 2i))

Finding λ: Use T(−3) = ∞:

• 1 = λ · (−3 − i) / (−3 − (3 + 2i))
• λ = 2

Classification: Since λ = 2 is real and positive (no rotation, only dilation), T is hyperbolic.

Behavior: Points move along Type I clines (lines through i and 3 + 2i) away from i and toward 3 + 2i.

Pole and inverse pole: z∞ = −3, w∞ = 6 + 3i. Note that i + (3 + 2i) = (3 + 3i) = −3 + (6 + 3i), confirming the lemma.

🧪 Example: Loxodromic transformation

Fix i and 0, send 1 to 2.

Normal form: T(z) − i / T(z) = λ · (z − i) / z

Finding λ: Use T(1) = 2:

• (2 − i)/2 = λ(1 − i)
• λ = 3/4 + i/4

In polar form: r = √10/4, θ = arctan(1/3)

Classification: Since both r ≠ 1 and θ ≠ 0, T is loxodromic.

Behavior: Points move along Type I clines of i and 0 by scale factor r and along Type II clines by angle θ.

🧭 Overview

🧠 One-sentence thesis

A geometry is defined by a space paired with a group of transformations, and figures are congruent if one can be mapped to the other by an allowable transformation in that group.

📌 Key points (3–5)

• What makes a group of transformations: a collection must contain the identity, be closed under composition, and contain inverses for every transformation.
• How geometry is defined: a geometry is a pair (S, G) where S is a set and G is a group of transformations on S; congruence means one figure can be transformed into another using a transformation from G.
• Invariant sets and functions: a set of figures is invariant if transformations in G always map figures in the set back into the set; a function is invariant if it gives the same value before and after any transformation in G.
• Common confusion: not every invariant set is minimally invariant—a minimally invariant set has no proper subset that is also invariant, which happens exactly when all figures in the set are congruent to each other.
• Why it matters: Klein's Erlangen Program approach means geometry studies what remains unchanged under allowable transformations, unifying Euclidean, hyperbolic, and elliptic geometries as special cases.

🔧 What is a group of transformations

🔧 The three required properties

Group of transformations: A collection G of transformations of a set A that satisfies three properties: (1) contains the identity transformation, (2) closed under composition, and (3) contains the inverse of every transformation in G.

• Identity: G must include the transformation T(a) = a for all a in A.
• Closure: if T and S are in G, then the composition T ∘ S must also be in G.
• Inverses: if T is in G, then T⁻¹ must also be in G.

The excerpt notes that associativity of composition is automatic for transformations, so it does not need checking.

🔄 Example: translations of the plane

The collection T of all translations T_b(z) = z + b (for all b in C) forms a group:

Property	Why T satisfies it
Identity	T₀(z) = z + 0 = z is in T
Closure	T_b ∘ T_c(z) = z + (b + c) = T_(b+c)(z), which is in T
Inverses	T_b has inverse T₋_b since T_b ∘ T₋_b(z) = z and T₋_b ∘ T_b(z) = z

Example: translating by b and then by c is the same as translating by b + c; the inverse of "translate by b" is "translate back by −b."

🔄 Example: finite group of rotations

The set H = {R₀, R_(π/2), R_π, R_(3π/2)} of four rotations about the origin forms a group:

• Identity: R₀(z) = z.
• Closure: composing two rotations gives another rotation in H (e.g., R_(3π/2) ∘ R_π = R_(π/2)).
• Inverses: each rotation has an inverse in H (e.g., R⁻¹_(π/2) = R_(3π/2)).

🪞 Example: two-element reflection group

The set G = {identity, r} where r(z) = z̄ (reflection across the real axis) is a group because r ∘ r equals the identity.

🌐 What is a geometry

🌐 The pair (S, G)

Geometry: A pair (S, G) where S is a set and G is a group of transformations on S.

• Point: an element of S.
• Figure: any subset A of S.
• Congruent figures: figures A and B are congruent (written A ≅ B) if there exists a transformation T in G such that T(A) = B.

The excerpt allows treating individual points as figures (writing a ≅ b instead of {a} ≅ {b}).

🔍 Congruence depends on the group

Different groups lead to different notions of congruence:

• In the four-rotation geometry (C, H): the circle |z − i| = 0.5 is congruent to three other circles (its images under the four rotations), but not to the circle |z| = 0.5.
• In the reflection geometry (C, {1, r}): the point 3 + i is congruent to 3 − i but not to −3 + i.
• In translational geometry (C, T): only figures that differ by a translation are congruent; you cannot rotate or dilate.

Don't confuse: congruence is not absolute—it depends entirely on which transformations are allowed in G.

📐 Example: translational geometry

In the geometry (C, T) where T is the group of all translations:

• Allowable moves: only translations (shifting position).
• Not allowed: rotations, dilations, reflections.
• Among the figures in Figure 4.1.7, only H and L are congruent because they have the same shape and size and differ only by a translation.

🔒 Invariant sets and functions

🔒 What makes a set invariant

Invariant set: A collection D of figures in a geometry (S, G) such that for any figure A in D and any transformation T in G, the image T(A) is also in D.

• The set "stays closed" under all transformations in G.
• Example: in translational geometry, the set of all lines is invariant because translating any line gives another line.
• However, two figures in an invariant set need not be congruent (e.g., lines with different slopes are both in the set of all lines but are not congruent).

📏 What makes a function invariant

Invariant function: A function f defined on an invariant set D such that f(B) = f(T(B)) for any figure B in D and any transformation T in G.

• The function's value does not change when you apply a transformation.
• Example: in translational geometry, the slope function on lines is invariant because translating a line does not change its slope.

🎯 Minimally invariant sets

Minimally invariant set: An invariant set D such that no proper subset of D is also an invariant set.

• Key theorem: An invariant set D is minimally invariant if and only if any two figures in D are congruent to each other.
• This means a minimally invariant set contains exactly those figures that are "the same" under the allowed transformations.

Example: in translational geometry, the set of all lines with slope 8 is minimally invariant (any two such lines are congruent by translation), but the set of all lines is not minimally invariant (it contains proper invariant subsets, like all lines with a fixed slope).

🔨 How to construct a minimally invariant set

The proof of Theorem 4.1.10 gives a recipe:

• Start with any figure A in the geometry (S, G).
• Form the set A = {T(A) | T ∈ G} (all images of A under transformations in G).
• This set A is minimally invariant.

Example: if A is a single line in translational geometry, then A is the set of all lines parallel to A (all translations of A).

🏛️ Euclidean geometry

🏛️ The group E

Euclidean geometry: The geometry (C, E) where E consists of all transformations of the form T(z) = e^(iθ) z + b, where θ is a real number and b is in C.

• These are exactly the general linear transformations T(z) = az + b where |a| = 1.
• The group E includes rotations and translations but not dilations (because |a| = 1 means no scaling).
• The excerpt notes that checking E is a group is left as an exercise.

📏 Euclidean distance as an invariant

Euclidean distance: The distance between two points z₁ and z₂ is defined as |z₁ − z₂|.

The excerpt begins to show that Euclidean distance is an invariant function in Euclidean geometry (the argument is cut off, but the claim is that for any T in E, the distance between any two points equals the distance between their images).

🧭 Overview

🧠 One-sentence thesis

Möbius geometry preserves angles and clines (circles and lines) but not Euclidean distance, making it the foundation for hyperbolic and elliptic geometries where Euclid's first four postulates hold but the fifth fails.

📌 Key points (3–5)

• What Möbius geometry is: the geometry (C⁺, M), where M is the group of all Möbius transformations on the extended complex plane.
• What is preserved: angles, clines (circles and lines), cross ratio, and symmetry points—but not Euclidean distance.
• Key invariant set: all clines form a minimally invariant set, meaning any two clines are congruent in Möbius geometry.
• Common confusion: Euclidean distance is not preserved—the transformation T(z) = 1/z changes distances even though it preserves angles.
• Why it matters: Möbius geometry provides the framework for hyperbolic and elliptic geometries, where angles remain comparable (Euclid's 4th postulate) but parallel behavior differs (5th postulate fails).

🔧 Definition and group structure

🔧 What Möbius geometry is

Möbius Geometry is the geometry (C⁺, M), where M denotes the group of all Möbius transformations.

• C⁺ is the extended complex plane (the complex plane plus the point at infinity).
• M is the collection of all Möbius transformations, which have the form T(z) = (az + b)/(cz + d).
• The excerpt notes that M is indeed a group because:
  • The inverse of a Möbius transformation is a Möbius transformation.
  • The composition of two Möbius transformations is a Möbius transformation.
  • The identity map T(z) = z is a Möbius transformation (with a = d = 1 and b = c = 0).

🧩 Key properties of Möbius transformations

The excerpt recasts results from Chapter 3 as geometric facts:

• Uniqueness: Any transformation in M is uniquely determined by the image of three points.
• Fixed points: If T in M is not the identity map, then T fixes exactly 1 or 2 points.
• Symmetry: Möbius transformations preserve symmetry points.

🔄 What Möbius geometry preserves

✅ Invariants: what stays the same

Invariant	Meaning
Clines	All circles and lines (clines) are congruent; the set of all clines is minimally invariant
Cross ratio	The cross ratio of four points is preserved under any Möbius transformation
Angle measure	Angles between curves are preserved
Symmetry points	Points symmetric with respect to a cline remain symmetric after transformation

✅ Clines as a minimally invariant set

• Any two clines are congruent in Möbius geometry (Theorem 3.4.15 from the text).
• The set of all clines is minimally invariant (Theorems 3.4.5 and 3.4.15).
  • This means: if you take any cline and apply any Möbius transformation, you get another cline; and no proper subset of all clines has this property.
• Example: A circle and a line are congruent in Möbius geometry—there exists a Möbius transformation mapping one to the other.

❌ What is not preserved: Euclidean distance

Euclidean distance is not an invariant function of Möbius geometry.

• The excerpt provides a concrete counterexample:
  • Let p = 2 and q = 3 (two points on the real axis).
  • Euclidean distance: d(p, q) = |p − q| = 1.
  • Apply T(z) = 1/z: T(p) = 1/2 and T(q) = 1/3.
  • New distance: d(T(p), T(q)) = |1/2 − 1/3| = 1/6 ≠ 1.
• Don't confuse: Even though distance changes, angles are still preserved—this is crucial for the geometric interpretation.

🌐 Why Möbius geometry matters

🌐 Connection to non-Euclidean geometries

The excerpt emphasizes the historical motivation:

• Humanity sought a geometry where Euclid's first 4 postulates hold but the 5th postulate (parallel postulate) fails.
• Euclid's 4th postulate states that all right angles equal one another.
  • This requires that angles can be compared by transforming one onto another without changing the angle.
  • The excerpt states: "Transformations do not change angles."
• Möbius geometry preserves angles, so it provides the foundation for hyperbolic and elliptic geometries, which are "subgeometries" of Möbius geometry.

🌐 Angles vs. distance

• Why angle preservation is "a good thing": If Ralph holds a right angle in one location and Randy holds one elsewhere, a transformation should map one onto the other and show they are equal.
• Why distance non-preservation is acceptable: The excerpt notes that "our old-fashioned notion of distance goes out the window," but this is not a problem for the geometric goals—angles are what matter for the postulates.

📐 Practical implications

📐 Working with Möbius geometry

• To show two figures are congruent: find a Möbius transformation that maps one to the other.
• To find a minimally invariant set: start with any figure A and form the set {T(A) | T ∈ M} (all images of A under all Möbius transformations).
• To check if a property is invariant: verify that it holds for any figure B and its image T(B) under any Möbius transformation T.

📐 Example: clines

• Example: The set of all clines is minimally invariant.
  • If you take any cline (say, a circle centered at the origin) and apply all possible Möbius transformations, you generate all clines.
  • No smaller set of clines has this property—you cannot exclude any cline and still have an invariant set.

🧭 Overview

🧠 One-sentence thesis

The Poincaré disk model defines hyperbolic geometry as the unit disk D with transformations H that preserve it, and every such transformation can be built from two reflections about clines orthogonal to the boundary circle.

📌 Key points (3–5)

• The model's definition: hyperbolic geometry is the pair (D, H), where D is the open unit disk and H consists of all Möbius transformations that send D to itself.
• Reflections generate all transformations: every transformation in H is the composition of two hyperbolic reflections (inversions in clines orthogonal to the circle at infinity).
• Three types of transformations: depending on whether the two clines of inversion intersect zero, one, or two times, transformations are classified as translations, parallel displacements, or rotations.
• Common confusion: the "circle at infinity" S¹∞ (the unit circle) bounds D but is not part of the hyperbolic plane itself; transformations in H send both D and S¹∞ to themselves.
• Why it matters: this model provides a concrete way to study hyperbolic geometry using Möbius transformations, which preserve angles and other key properties.

🌐 The hyperbolic plane and its boundary

🌐 What D and H are

The Poincaré disk model for hyperbolic geometry is the pair (D, H) where D consists of all points z in C such that |z| < 1, and H consists of all Möbius transformations T for which T(D) = D.

• D is called the hyperbolic plane: the open unit disk (all complex numbers z with absolute value less than 1).
• H is called the transformation group in hyperbolic geometry: all Möbius transformations that map D to itself.
• The excerpt notes that H does form a group of transformations (worked out in exercises).

⭕ The circle at infinity

The unit circle is called the circle at infinity, denoted by S¹∞.

• S¹∞ is the boundary of D (the set of points z with |z| = 1).
• It is not included in the hyperbolic plane D but bounds it.
• Any Möbius transformation that sends D to itself also sends S¹∞ to itself.
• Don't confuse: S¹∞ is used extensively in constructions but is not part of the geometry's "space."

🪞 Hyperbolic reflections

🪞 What a hyperbolic reflection is

An inversion in a cline C that is orthogonal to S¹∞ is called a reflection of the hyperbolic plane, or a hyperbolic reflection.

• Consider a cline C that is orthogonal (meets at right angles) to the circle at infinity S¹∞.
• Inverting about C sends S¹∞ to itself and also sends the interior D to itself.
• Example: a Euclidean line through the origin meets the unit circle at right angles, so inversion about such a line is a hyperbolic reflection.

🔧 Reflections generate all transformations in H

Theorem 5.1.3: Any Möbius transformation in H is the composition of two reflections of the hyperbolic plane.

• The proof constructs two hyperbolic reflections that together produce any given transformation T in H.
• Case 1: If T fixes the origin (z₀ = 0), then T is a rotation about the origin, which is the composition of two reflections about Euclidean lines through the origin.
• Case 2: If z₀ ≠ 0, the proof constructs a circle C orthogonal to S¹∞ that inverts z₀ to 0, then a second reflection (about a line through the origin) to complete the transformation.
• Lemma 5.1.5: Given any point z₀ in D and any point z₁ on S¹∞, there exists a transformation in H that sends z₀ to 0 and z₁ to 1.

🔀 Three types of transformations

🔀 Classification by intersection of clines

The excerpt categorizes transformations in H by whether the two clines of inversion intersect zero, one, or two times.

Intersection count	Type of transformation	Fixed points	Clines of motion
Two (inside D)	Rotation of the hyperbolic plane	Point p in D (and p* outside)	Type II clines of p and p*
One (on S¹∞)	Parallel displacement	Point p on S¹∞	Horocycles (circles tangent to S¹∞ at p)
Zero	Translation of the hyperbolic plane	Two points p and q on S¹∞	Type I clines of p and q

🔄 Rotation about a point

• If the two clines of inversion L₁ and L₂ intersect inside D at point p, they also intersect outside D at the symmetric point p*.
• The resulting transformation fixes p (and p*) and rotates points in D around p along type II clines of p and p*.
• Example: Figure 5.1.7(a) shows an 'M' figure rotated about point p; successive applications T(M) and T²(M) move the figure along circular arcs centered at p.

↔️ Parallel displacement

• If the clines of inversion intersect exactly once, it must be at a point p on the unit circle S¹∞.
• The resulting map moves points along horocycles: circles in D that are tangent to the unit circle at p.
• Don't confuse: horocycles are not the same as type I or type II clines; they are a special class of circles tangent to S¹∞.

➡️ Translation

• If the clines of inversion do not intersect, there are two points p and q symmetric with respect to both clines.
• These two fixed points must live on the unit circle S¹∞.
• The transformation pushes points along type I clines of p and q.
• Example: Figure 5.1.7(b) shows an 'M' figure translated; the figure moves along arcs that connect the two fixed points p and q on S¹∞.

🛠️ Constructing transformations

🛠️ Building a circle orthogonal to S¹∞

The proof of Theorem 5.1.3 includes a construction for finding a circle C orthogonal to S¹∞ that inverts a given point z₀ to the origin:

1. Draw circle C₁ with diameter from 0 to z*₀ (the point symmetric to z₀ with respect to S¹∞).
2. Let p be a point of intersection of C₁ and the unit circle S¹∞.
3. Construct circle C through p centered at z*₀.
4. Since angle ∠0pz*₀ is a right angle, S¹∞ is orthogonal to C.
5. Inversion about C sends z₀ to 0 and z*₀ to ∞.

🎯 Completing the transformation

• After the first inversion sends z₀ to 0, a second reflection is needed to send some point z₁ on S¹∞ to 1.
• The first inversion sends z₁ to some point z'₁ on S¹∞.
• Reflect across the line through the origin that bisects the angle ∠10z'₁; this sends z'₁ to 1 and leaves 0 and ∞ fixed.
• Composing these two inversions yields the desired transformation T.
• Since a Möbius transformation is uniquely determined by the image of three points, this construction produces exactly T.

🧭 Overview

🧠 One-sentence thesis

Hyperbolic geometry is built from hyperbolic lines (clines orthogonal to the boundary circle), and its parallel postulate differs fundamentally from Euclidean geometry: through a point not on a given line, there exist two distinct parallel lines, and parallelism is not transitive.

📌 Key points (3–5)

• Hyperbolic lines: portions of clines inside the disk D that meet the boundary circle S¹∞ at right angles; they are the analogs of Euclidean lines and are generated by hyperbolic reflections.
• Parallel lines redefined: two hyperbolic lines are parallel if they share exactly one ideal point (a point on the boundary circle).
• Non-Euclidean parallel postulate: given a point and a line not through it, there exist two distinct hyperbolic lines through the point parallel to the given line.
• Common confusion: parallelism is not transitive in hyperbolic geometry—if L is parallel to M and M is parallel to N, L need not be parallel to N (unlike in Euclidean geometry).
• Congruence: any two hyperbolic lines are congruent under transformations in H, and hyperbolic circles are type II clines centered at a point.

📏 Hyperbolic lines and their properties

📏 What is a hyperbolic line?

Hyperbolic line in (D, H): the portion of a cline inside D that is orthogonal to the circle at infinity S¹∞.

• The boundary circle S¹∞ is called the "circle at infinity."
• Points on S¹∞ are called ideal points.
• Hyperbolic lines are the fundamental "straight" objects in hyperbolic geometry, just as Euclidean lines are in Euclidean geometry.
• Why orthogonality matters: transformations in H are generated by hyperbolic reflections (inversions about clines orthogonal to S¹∞), so these clines play the role of "lines."

🔍 Which hyperbolic lines are Euclidean lines?

• A Euclidean line intersects a circle at right angles if and only if it passes through the center of the circle.
• Since S¹∞ is centered at the origin, the only hyperbolic lines that are also Euclidean lines are those passing through the origin.
• Symmetric-point argument: any Euclidean line passes through ∞; to be orthogonal to S¹∞, it must also pass through the point symmetric to ∞ with respect to the unit circle, which is 0.

✅ Uniqueness and existence

Theorem: There exists a unique hyperbolic line through any two distinct points in the hyperbolic plane.

• Construction: given points p and q in D, construct p* (the point symmetric to p with respect to S¹∞). There is exactly one cline through p, q, and p*, and this cline is orthogonal to S¹∞.
• This hyperbolic line is unique because there is only one cline through three given points.

🔄 Congruence of hyperbolic lines

Theorem: Any two hyperbolic lines are congruent in hyperbolic geometry.

• Proof idea: any hyperbolic line L can be mapped to the hyperbolic line on the real axis by a transformation in H.
  • Pick a point p on L and one of its ideal points v.
  • By an earlier lemma, there exists a transformation T in H that sends p to 0, v to 1, and p* to ∞.
  • Then T(L) is the portion of the real axis inside D.
• Since any hyperbolic line is congruent to the real-axis line, and H is a group, any two hyperbolic lines are congruent to each other.

🔀 Parallel lines in hyperbolic geometry

🔀 Definition of parallel

Two hyperbolic lines are parallel if they share exactly one ideal point.

• This is very different from the Euclidean definition (lines that never meet).
• In hyperbolic geometry, parallel lines "meet" at the boundary, at an ideal point.

🚫 The hyperbolic parallel postulate

Theorem: Given a point z₀ and a hyperbolic line L not through z₀, there exist two distinct hyperbolic lines through z₀ that are parallel to L.

• Proof sketch (for z₀ at the origin):
  • L has two ideal points, call them u and v.
  • Since L does not pass through the origin, the Euclidean segment uv is not a diameter of the unit circle.
  • Construct one Euclidean line through 0 and u, and another through 0 and v.
  • Each of these lines is a hyperbolic line through 0, and each shares exactly one ideal point with L, so each is parallel to L.
• The result holds for general z₀ (left as an exercise).

⚠️ Parallelism is not transitive

• In Euclidean geometry: if L is parallel to M and M is parallel to N, then L is parallel to N.
• In hyperbolic geometry: this is not the case.
• Example: the two hyperbolic lines through 0 that are parallel to L (sharing ideal points u and v, respectively) are not parallel to each other—they intersect at 0.
• Don't confuse: "parallel" in hyperbolic geometry means "sharing one ideal point," not "never intersecting inside D."

🔺 Hyperbolic triangles

🔺 What is a hyperbolic triangle?

• Three points in D that are not all on a single hyperbolic line determine a hyperbolic triangle.
• The sides of the triangle are portions of hyperbolic lines (arcs of clines orthogonal to S¹∞).

🔺 Are all hyperbolic triangles congruent?

• No.
• Transformations in H are Möbius transformations, which preserve angles.
• Triangles with different angles cannot be congruent.
• Example: a hyperbolic triangle Δpqr has sides that are arcs; if two triangles have different angle measures, no transformation in H can map one to the other.

⭕ Hyperbolic circles

⭕ Definition

Hyperbolic circle centered at p: a Euclidean circle C inside D that is a type II circle of p and p* (the point symmetric to p with respect to the unit circle).

• Hyperbolic circles are Euclidean circles, but their "center" in the hyperbolic sense (p) is not the Euclidean center.
• Why type II clines?: under hyperbolic rotation about p, points move along type II clines of p and p*. If distance is invariant under transformations in H, all points on a given type II cline must be the same hyperbolic distance from p.

⭕ Existence

Theorem: Given any points p and q in D, there exists a hyperbolic circle centered at p through q.

• Construction: construct p*, then find the type II cline of p and p* that passes through q.
• This type II cline lies entirely within D because S¹∞ is also a type II cline of p and p*, and distinct type II clines cannot intersect.

⭕ Tangency and perpendicularity

• A hyperbolic line L can be tangent to a hyperbolic circle C at a point p.
• In that case, L is perpendicular to the hyperbolic segment from the center to p (left as an exercise).

🛠️ Construction techniques

🛠️ Constructing a hyperbolic line through two points

1. Given point p in D, construct p* (symmetric to p with respect to the unit circle).
2. Given a second point q in D, construct the cline through p, q, and p*. Call it C.
3. C intersects the unit circle at right angles (because it passes through p and p*, which are symmetric with respect to S¹∞).
4. The portion of C inside D is the unique hyperbolic line through p and q.
5. Mark the two ideal points where C meets S¹∞.

🛠️ Constructing a hyperbolic circle

Given points p and q in D (with q not on the line through p and p*):

1. Find the center o of the Euclidean circle through p, p*, and q.
2. Construct the segment oq.
3. Construct the perpendicular to oq at q; it intersects the Euclidean line through p and p* at a point o′.
4. Construct the Euclidean circle centered at o′ through q.
5. This circle is the hyperbolic circle through q centered at p.

🛠️ Perpendicular from a point to a line

• Given a point and a hyperbolic line not passing through it, there exists a hyperbolic line through the point that is perpendicular to the given line.
• This perpendicular is unique (left as an exercise).

🔑 Key distinctions and confusions

Concept	Euclidean geometry	Hyperbolic geometry
Lines	Euclidean lines	Clines orthogonal to S¹∞
Parallel lines	Never intersect	Share exactly one ideal point
Number of parallels through a point	Exactly one	Exactly two
Transitivity of parallelism	Yes (if L ‖ M and M ‖ N, then L ‖ N)	No
Circles	Euclidean circles	Euclidean circles (type II clines)
Congruence of lines	All lines congruent	All hyperbolic lines congruent

Common confusion: In hyperbolic geometry, "parallel" does not mean "never intersecting inside D." It means "sharing one ideal point on the boundary." Two hyperbolic lines can intersect inside D and still not be parallel (they share zero ideal points), or they can be parallel (share one ideal point), or they can be "ultraparallel" (share zero ideal points and do not intersect inside D).

🧭 Overview

🧠 One-sentence thesis

Hyperbolic distance is derived by requiring that transformations preserve distance, that small distances match Euclidean proportions, and that the shortest path between two points lies on the hyperbolic line connecting them, leading to a logarithmic distance formula and an arc-length differential that reflects the geometry's non-Euclidean nature.

📌 Key points (3–5)

• Core derivation strategy: the distance function is built by listing five required features (positivity, shortest-path on hyperbolic lines, additivity, invariance under transformations in H, and local proportionality to Euclidean distance), then using calculus and cross ratios to derive the formula.
• Distance from the origin: hyperbolic distance from 0 to z is ln((1 + |z|) / (1 − |z|)), which approaches infinity as |z| approaches 1 (the boundary circle).
• General distance formula: distance between p and q can be computed via the cross ratio of p, q, and the two ideal points of the hyperbolic line through them, or by transforming one point to the origin and applying the simpler formula.
• Arc-length differential: the hyperbolic length of a smooth curve r(t) is the integral of (2 / (1 − |r(t)|²)) |r′(t)| dt, which is invariant under transformations in H.
• Common confusion: hyperbolic circles and perpendicular bisectors behave differently from Euclidean ones—three noncollinear points need not determine a unique hyperbolic circle, and the boundary of the disk is infinitely far away in hyperbolic distance despite being finite in Euclidean distance.

📐 Deriving the distance function

📋 Five required features

The excerpt lists five properties that the hyperbolic distance function d_H(p, q) ought to satisfy:

1. Positivity: distance between two distinct points should be positive.
2. Shortest path on hyperbolic lines: the shortest path between two points should lie on the hyperbolic line connecting them.
3. Additivity on lines: if p, q, r are three points on a hyperbolic line with q between the other two, then d_H(p, q) + d_H(q, r) = d_H(p, r).
4. Invariance under transformations: distance should be preserved by transformations in H (the group of hyperbolic isometries). In other words, d_H(p, q) = d_H(T(p), T(q)) for any transformation T in H.
5. Local proportionality: in the limit for small distances, hyperbolic distance should be proportional to Euclidean distance.

Why the last feature matters: "One theme of this text is that locally, on small scales, non-Euclidean geometry behaves much like Euclidean geometry."

• Small hyperbolic segments approximate Euclidean segments to first order.
• Small hyperbolic triangles look like Euclidean triangles.
• Hyperbolic angles correspond to Euclidean angles.

🔍 Distance from the origin

The derivation starts by finding the distance from the origin to a point z in D (the unit disk).

Step-by-step approach:

• Rotate z to the positive real axis so z becomes x = |z|.
• Find a hyperbolic line L that reflects x to the origin (constructed in Theorem 5.1.3).
• Consider a nearby point x + h on the positive real axis; under the reflection it maps to w = −h / (1 − x² − hx).
• By invariance, d_H(x, x + h) = d_H(0, w).
• By additivity, d_H(0, x) + d_H(x, x + h) = d_H(0, x + h).
• Define d(x) = d_H(0, x); then d(x + h) − d(x) = d(w).
• Divide by h and take the limit as h → 0 to get d′(x) = lim (d(w) / h).

Key limit: as h → 0, w → 0, so by the local proportionality assumption, lim (d(w) / |w|) = k for some constant k. The excerpt sets k = 2 "as this makes length and area formulas look very nice later on."

Differential equation:

• d′(x) = lim [2 · (h / (1 − x² − hx)) / h] = 2 / (1 − x²).
• Integrate using partial fractions: ∫ (2 / (1 − x²)) dx = ∫ (1/(1 − x) + 1/(1 + x)) dx = −ln(1 − x) + ln(1 + x).
• Result: d(x) = ln((1 + x) / (1 − x)).

Hyperbolic distance from 0 to z: The hyperbolic distance from 0 to a point z in D is d_H(0, z) = ln((1 + |z|) / (1 − |z|)).

Important consequence: as |z| → 1 (approaching the boundary circle at infinity), d_H(0, z) → ∞. This satisfies Euclid's postulate that "one can produce a hyperbolic segment to any finite length."

🔗 Connection to cross ratios

The expression (1 + x) / (1 − x) corresponds to the cross ratio of the points 0, x, 1, and −1 (the ideal points of the hyperbolic line through 0 and x on the real axis).

• (0, x; 1, −1) = [(0 − 1)/(0 + 1)] · [(x + 1)/(x − 1)] = (1 + x) / (1 − x).
• Thus d_H(0, x) = ln((0, x; 1, −1)).

🧮 General distance formula

🌐 Using cross ratios

To find the distance between arbitrary points p and q in D:

• There exists a transformation T in H that takes p to the origin and q to some point x on the positive real axis.
• By invariance, d_H(p, q) = d_H(T(p), T(q)) = d_H(0, x).
• By invariance of cross ratio, d_H(0, x) = ln((0, x; 1, −1)) = ln((p, q; u, v)), where u and v are the ideal points of the hyperbolic line through p and q.
• Important: u is the ideal point you head toward going from p to q, and v is the ideal point you head toward going from q to p.

Working formula for d_H(p, q): One may compute the hyperbolic distance between p and q by first finding the ideal points u and v of the hyperbolic line through p and q and then using the formula d_H(p, q) = ln((p, q; u, v)).

🛠️ Practical computation

In practice, finding ideal points can be difficult. A simpler approach:

• Use a transformation T(z) = (z − p) / (1 − p̄z) that sends p to 0.
• T sends q to T(q) = (q − p) / (1 − p̄q).
• By invariance, d_H(p, q) = d_H(0, T(q)) = ln((1 + |T(q)|) / (1 − |T(q)|)).

Theorem 5.3.3: Given two points p and q in D, the hyperbolic distance between them is

d_H(p, q) = ln([ |1 − p̄q| + |q − p| ] / [ |1 − p̄q| − |q − p| ]).

Example 5.3.4:

• For p = (1/2)i, q = (1/2) + (1/2)i, the distance d_H(p, q) ≈ 1.49 units.
• For z = 0.95 e^(i5π/6), w = −0.95, the distance d_H(z, w) ≈ 4.64 units.

📏 Arc-length and curve measurement

🧵 Smooth curves

Definition 5.3.5: A smooth curve is a differentiable map from an interval of real numbers to the plane r: [a, b] → C such that r′(t) exists for all t and never equals 0.

• Write r(t) = x(t) + iy(t), so r′(t) = x′(t) + iy′(t).

📐 Euclidean arc-length differential (review)

In Euclidean geometry:

• Approximate the length of a small segment by Δs = |r(t + Δt) − r(t)|.
• As Δt → 0, obtain ds = |r′(t)| dt = √((dx/dt)² + (dy/dt)²) dt.
• Example: the circumference of a circle with radius a is ∫₀^(2π) a dt = 2πa.

🌀 Hyperbolic arc-length differential

For a smooth curve r(t) in D:

• Approximate the hyperbolic length of a tiny portion from r(t) to r(t + Δt) by d_H(r(t), r(t + Δt)).
• Transform r(t) to 0 via T(z) = (z − r(t)) / (1 − r(t)z̄).
• For small Δt, d_H(r(t), r(t + Δt)) ≈ 2 · |T(r(t + Δt))|.
• Simplify: 2 · |(r(t + Δt) − r(t)) / (1 − r(t)r̄(t + Δt))| = 2 · (|(r(t + Δt) − r(t)) / Δt|) / |1 − r(t)r̄(t + Δt)| · |Δt|.
• As Δt → 0, the numerator → |r′(t)| and the denominator → 1 − |r(t)|².

Definition 5.3.6: If r: [a, b] → D is a smooth curve in the hyperbolic plane, define the length of r, denoted L(r), to be L(r) = ∫ₐᵇ (2 / (1 − |r(t)|²)) |r′(t)| dt.

Theorem 5.3.7: Arc-length is an invariant of hyperbolic geometry. If r is a smooth curve in D and T is any transformation in H, then L(r) = L(T(r)).

• Hyperbolic reflections also preserve arc-length (Exercise 5.3.6).
• All hyperbolic reflections and transformations in H are hyperbolic isometries: they preserve hyperbolic distance.

🔵 Hyperbolic circles

Corollary 5.3.8: All points on a hyperbolic circle centered at p are equidistant from p.

Proof sketch: If u and v are on the same hyperbolic circle centered at p, there exists a hyperbolic rotation fixing p that maps u to v. By invariance, d_H(p, u) = d_H(p, v).

🛤️ Geodesics and the triangle inequality

🎯 Hyperbolic lines are shortest paths

Theorem 5.3.9: Hyperbolic lines are geodesics; that is, the shortest path between two points in (D, H) is along the hyperbolic segment between them.

Proof sketch:

• First show that the geodesic from 0 to a point c on the positive real axis is the real axis itself.
• Consider an arbitrary smooth curve r(t) = x(t) + iy(t) from 0 to c with x(t) nondecreasing.
• Compute L(r) = ∫ₐᵇ (2 / (1 − [x(t)² + y(t)²])) √(x′(t)² + y′(t)²) dt.
• The hyperbolic line segment from 0 to c is r₀(t) = x(t) + 0i, with length L(r₀) = ∫ₐᵇ (2 / (1 − x(t)²)) √(x′(t)²) dt.
• Compare integrands directly: L(r) ≥ L(r₀).
• Since transformations in H preserve arc-length and hyperbolic lines, the shortest path between any two points is along the hyperbolic line through them.

📊 The distance function is a metric

Corollary 5.3.10: The hyperbolic distance function is a metric on the hyperbolic plane. For any points p, q, u in D:

Property	Statement	Why it holds
Positivity	d_H(p, q) ≥ 0, and d_H(p, q) = 0 iff p = q	The quotient in the ln is always ≥ 1; equals 1 iff p = q
Symmetry	d_H(p, q) = d_H(q, p)	The formula is symmetric in p and q
Triangle inequality	d_H(p, q) + d_H(q, u) ≥ d_H(p, u)	Hyperbolic lines are geodesics

🔬 Examples and special cases

📍 Comparing two paths

Example 5.3.11: Two paths from p = 0.5i to q = 0.5 + 0.5i:

1. Hyperbolic segment (solid): d_H(p, q) ≈ 1.49 units (computed via Theorem 5.3.3).
2. Euclidean segment (dashed): parameterized by r(t) = t + (1/2)i for 0 ≤ t ≤ 1/2.
   • r′(t) = 1.
   • L(r) = ∫₀^(1/2) (2 / (1 − (t² + 1/4))) dt ≈ 1.52 units.

Conclusion: The hyperbolic segment is shorter, as expected.

⊥ Perpendicular bisectors

Example 5.3.12: For any two points p and q in D:

• Construct the hyperbolic circle centered at p through q and the hyperbolic circle centered at q through p.
• The hyperbolic line L through the two intersection points is the perpendicular bisector of segment pq.
• Hyperbolic reflection about L swaps p and q.
• Since reflections preserve distance, each point z on L satisfies d_H(z, p) = d_H(z, q).

Important difference from Euclidean geometry: In hyperbolic geometry, three noncollinear points need not determine a unique hyperbolic circle. The perpendicular bisectors of the three sides of a triangle may not intersect at a single point (Figure 5.3.13 shows three perpendicular bisectors that do not meet).

🌊 Perpendicular distance to a cline

Exercise 5.3.3 concept: Suppose L is a hyperbolic line and C is any cline through the ideal points of L. For any point z on L, its perpendicular distance to C is the length of the hyperbolic segment from z to C that meets C at right angles. The perpendicular distance from C to L is the same at every point of L (by invariance of distance under hyperbolic transformations).

🔑 Key exercises and further results

🧪 Selected exercises

1. Exercise 5.3.1: Prove that under the inversion described, w = −h / (1 − x² − hx).
2. Exercise 5.3.2: Determine a point in D whose hyperbolic distance from the origin is 2,003,007.4 units. (Hint: solve ln((1 + |z|) / (1 − |z|)) = 2,003,007.4 for |z|; note that |z| will be very close to 1.)
3. Exercise 5.3.4: Determine d_H(p, q) for p = 0.5, q = 0.25 + 0.5i using Theorem 5.3.3.
4. Exercise 5.3.5: Prove Theorem 5.3.7 (arc-length invariance under transformations in H).
5. Exercise 5.3.6: Prove that hyperbolic reflections preserve arc-length by first showing it for reflection about the real axis, then using Theorem 5.3.7 for general hyperbolic lines.
6. Exercise 5.3.7: Prove that the set C of all points z in D such that d_H(z, z₀) = r (for fixed z₀ in D and r > 0) is a Euclidean circle.

🚫 Don't confuse

• Euclidean vs hyperbolic distance: the same two points have different distances depending on the geometry; hyperbolic distance grows without bound as you approach the boundary circle, even though the Euclidean distance to the boundary is finite.
• Hyperbolic circles vs Euclidean circles: a hyperbolic circle (constant hyperbolic distance from a center) is a Euclidean circle, but its Euclidean center is not the hyperbolic center.
• Three-point circles: in Euclidean geometry, three noncollinear points always determine a unique circle; in hyperbolic geometry, they may not (the perpendicular bisectors may not intersect).

🧭 Overview

🧠 One-sentence thesis

In hyperbolic geometry, area is an invariant computed via a special differential formula, and remarkably, the area of any triangle equals π minus the sum of its interior angles—meaning no triangle can have area as large as π, even though side lengths can be arbitrarily large.

📌 Key points (3–5)

• Area formula: Area in the hyperbolic plane (D, H) is given by a double integral involving the factor 4r / (1 − r²)² in polar coordinates, and area is preserved under hyperbolic transformations.
• Triangle area formula: A hyperbolic triangle with interior angles α, β, γ has area A = π − (α + β + γ), which is always less than π.
• Ideal triangles: An ideal triangle (all three vertices at ideal points) has area exactly π; a 2/3-ideal triangle (two vertices at ideal points) with interior angle α has area π − α.
• Common confusion: Unlike Euclidean geometry, larger triangles in hyperbolic geometry do not have larger areas—area is bounded by π, while side lengths can grow without bound.
• Hyperbolic trigonometry: Hyperbolic triangles obey laws of cosines and sines that relate angles and side lengths using hyperbolic sine and cosine functions (sinh and cosh).

📐 Area in hyperbolic geometry

📐 The area differential and invariance

Area of a region R in (D, H): Given by the double integral over R of 4r / (1 − r²)² dr dθ in polar coordinates.

• This formula is difficult to evaluate directly except in simple cases.
• Why it matters: Area is an invariant of hyperbolic geometry—a region's area does not change as it moves around the hyperbolic plane under transformations in H.
• Example: A circle centered at the origin with hyperbolic radius a has area 4π sinh²(a/2), where sinh(x) = (eˣ − e⁻ˣ)/2 is the hyperbolic sine function.

📐 Hyperbolic sine and cosine

Hyperbolic sine: sinh(x) = (eˣ − e⁻ˣ)/2
Hyperbolic cosine: cosh(x) = (eˣ + e⁻ˣ)/2

• These functions appear naturally in hyperbolic area and distance formulas.
• They satisfy cosh²(x) − sinh²(x) = 1, analogous to the Pythagorean identity for circular trig functions.
• Example: The area of a hyperbolic circle with radius r is 4π sinh²(r/2).

🔺 Triangle area formulas

🔺 Ideal triangles

Ideal triangle: A triangle whose three vertices are all ideal points (points on the circle at infinity).

• Key fact: All ideal triangles are congruent—they form a minimally invariant set in (D, H).
• Area: Every ideal triangle has area exactly π.
• Proof sketch: An ideal triangle can be partitioned into two 2/3-ideal triangles by drawing a perpendicular from one ideal point; each 2/3-ideal triangle has interior angle π/2, so total area is π/2 + π/2 = π.

🔺 2/3-ideal triangles

2/3-ideal triangle: A triangle with two vertices at ideal points and one vertex in the hyperbolic plane.

• Area formula: A 2/3-ideal triangle with interior angle α has area π − α.
• The proof uses the upper half-plane model (mentioned but not detailed in this excerpt).
• Example: If the interior angle is π/2, the area is π − π/2 = π/2.

🔺 General hyperbolic triangles

Theorem: A hyperbolic triangle with interior angles α, β, γ has area A = π − (α + β + γ).

• How to see it: Extend the sides of the triangle to ideal points to form an ideal triangle (area π). The ideal triangle contains the original triangle plus three 2/3-ideal triangles. Each 2/3-ideal triangle has area π minus its interior angle. Subtracting these regions and solving gives A = π − (α + β + γ).
• Remarkable consequence: The angle sum α + β + γ is always less than π, and the area is always less than π.
• Don't confuse: In Euclidean geometry, angle sum is always π and area can be arbitrarily large; in hyperbolic geometry, angle sum is always less than π and area is bounded by π, even though side lengths can be arbitrarily large.

📏 Hyperbolic trigonometry

📏 First hyperbolic law of cosines

For a hyperbolic triangle with angles α, β, γ and opposite side lengths a, b, c:

cosh(c) = cosh(a) cosh(b) − sinh(a) sinh(b) cos(γ)

• This relates the hyperbolic lengths of the sides to one angle.
• Special case (right triangle): If γ = π/2, then cos(γ) = 0, so cosh(c) = cosh(a) cosh(b). This is the hyperbolic hypotenuse theorem.

📏 Second hyperbolic law of cosines

cosh(c) = [cos(α) cos(β) + cos(γ)] / [sin(α) sin(β)]

• This relates the hyperbolic length of one side to all three angles.
• Application (angle of parallelism): In a 1/3-ideal triangle (one vertex at an ideal point), if γ = 0 and β = π/2, then cosh(c) = 1/sin(α), where c is the distance from a point to a perpendicular line and α is the angle of parallelism.

📏 Hyperbolic law of sines

sinh(a) / sin(α) = sinh(b) / sin(β) = sinh(c) / sin(γ)

• Analogous to the Euclidean law of sines, but using hyperbolic sine for the side lengths.
• The proofs of these laws are left as exercises in the excerpt.

🏗️ Special figures in hyperbolic geometry

🏗️ No hyperbolic squares

• Key fact: Four-sided figures with four right angles and four hyperbolic line segments as sides do not exist in hyperbolic geometry.
• Why: If such a figure existed, its angle sum would be 2π. Dividing it along a diagonal into two triangles would give a total angle sum of 2π, meaning at least one triangle has angle sum ≥ π, which is impossible.
• Don't confuse: A two-dimensional explorer can make a journey that would trace a square in Euclidean geometry (travel distance a, turn right 90°, repeat four times), but in hyperbolic geometry this journey does not return to the starting point.

🏗️ Blocks (pseudo-rectangles)

• A block is a four-sided figure with four right angles and opposite sides of equal length, but not all sides are hyperbolic line segments.
• Construction: Use two perpendicular hyperbolic lines and two cline arcs through ideal points.
• Example: Through points a and −a on the real axis, construct perpendicular hyperbolic lines; then construct cline arcs through points on these lines and the ideal points 1 and −1.

🏗️ Right-angled hexagons

Theorem: For any triple (a, b, c) of positive real numbers, there exists a right-angled hexagon in (D, H) with alternate side lengths a, b, c. All such hexagons are congruent.

• Construction: Start with a vertex at the origin, place the next vertex at distance a on the positive real axis, then construct perpendiculars and use hyperbolic circles to place vertices at the required distances.
• Why it matters: Unlike squares, right-angled regular polygons with more than four sides do exist in hyperbolic geometry.

🏗️ Inscribed circle in an ideal triangle

• Example: If you inscribe a circle in any ideal triangle, the points of tangency form an equilateral triangle with side lengths 2 ln(φ), where φ = (1 + √5)/2 is the golden ratio.
• This is computed by choosing a convenient ideal triangle (with ideal points −1, 1, i) and finding the inscribed circle by hyperbolic reflection.

🚀 Exploring the hyperbolic plane

🚀 What a pilot would observe

• Homogeneity: Any two points in the hyperbolic plane are congruent, so a pilot could not distinguish between points geometrically.
• Geodesics: Light travels along hyperbolic lines (the shortest paths), so the pilot's line of sight follows these curves.
• Infinite and boundaryless: The pilot would view the hyperbolic plane as infinite with no boundary, and could in theory make an orbit of arbitrary radius around any point.

🚀 Testing for hyperbolic geometry

• Triangle angle sum: Triangles in hyperbolic geometry have angle sums less than 180°, but this is only noticeable for large triangles.
• Example from the excerpt: One triangle has angle sum about 130°, another about 22°.
• Challenge: Whether an explorer could map out a large enough triangle depends on how much ground can be covered relative to the size of the universe.

🧭 Overview

🧠 One-sentence thesis

The upper half-plane model provides an alternative representation of hyperbolic geometry that simplifies certain computations, such as calculating triangle areas, by mapping the Poincaré disk to the complex upper half-plane via a specific Möbius transformation.

📌 Key points (3–5)

• What the model is: the space of all complex numbers with positive imaginary part, with transformations that preserve this space.
• How to transfer between models: a Möbius transformation V maps the Poincaré disk to the upper half-plane, preserving clines and angles.
• Key structural change: ideal points move from the unit circle to the real axis; hyperbolic lines become clines perpendicular to the real axis.
• Common confusion: distance and length formulas look different in the upper half-plane model, but they preserve the same hyperbolic distances because the transformation preserves cross ratios.
• Why it matters: the upper half-plane model makes area calculations easier, leading to the simple formula that a 2/3-ideal triangle with angle α has area π − α.

🔄 The two models and the transfer map

🔄 Definition of the upper half-plane model

Upper half-plane model of hyperbolic geometry: the space U consists of all complex numbers z such that Im(z) > 0, with transformation group U consisting of all Möbius transformations that send U to itself.

• The space U is called the upper half-plane of the complex numbers.
• This is a different representation of the same hyperbolic geometry as the Poincaré disk model (D, H).

🔀 The transformation V

The Möbius transformation that maps the disk to the upper half-plane is built from two inversions:

1. Invert about the circle C centered at i passing through −1 and 1.
2. Reflect about the real axis.

The resulting transformation is:

• From disk to upper half-plane: w = V(z) = (−iz + 1) / (z − i)
• From upper half-plane to disk: z = V⁻¹(w) = (iw + 1) / (w + i)

Convention for this section: z denotes a point in the disk D; w denotes a point in the upper half-plane U.

🎯 What the transformation does

Since V is a Möbius transformation, it preserves:

• Clines (circles and lines)
• Angles

Key structural changes:

• Ideal points on the unit circle S¹∞ move to the real axis.
• Hyperbolic lines in the disk become clines that intersect the real axis at right angles.
• The point ∞ in the upper half-plane corresponds to −i in the disk.

📏 Distance in the upper half-plane model

📏 Definition of hyperbolic distance

Hyperbolic distance in the upper half-plane: dᵤ(w₁, w₂) is defined as the hyperbolic distance between the pre-images of w₁ and w₂ in the disk model.

If w₁ and w₂ have pre-images z₁ and z₂ in the disk, then:

• dᵤ(w₁, w₂) = dₕ(z₁, z₂) = ln((z₁, z₂; u, v))
• where u and v are the ideal points of the hyperbolic line through z₁ and z₂.

Since cross ratios are preserved under Möbius transformations:

• dᵤ(w₁, w₂) = ln((w₁, w₂; p, q))
• where p, q are the ideal points of the hyperbolic line in the upper half-plane through w₁ and w₂.

🧮 Example: distance between ri and si

For r > s > 0, the distance between ri and si on the positive imaginary axis:

• The hyperbolic line is the positive imaginary axis with ideal points 0 and ∞.
• dᵤ(ri, si) = ln((ri, si; 0, ∞)) = ln(r/s).

🗺️ Example: distance between any two points

To find the distance between arbitrary points w₁ and w₂ in U:

1. Map them back to the disk: z₁ = V⁻¹(w₁), z₂ = V⁻¹(w₂).
2. Apply a transformation S(z) = e^(iθ) (z − z₁)/(1 − z̄₁z) that sends z₁ to 0 and z₂ to the positive imaginary axis at ki, where k = |S(z₂)|.
3. Apply V to get V ∘ S ∘ V⁻¹, which sends w₁ to i and w₂ to ((1+k)/(1−k))i.
4. The distance is: dᵤ(w₁, w₂) = ln(1 + k) − ln(1 − k).

📐 Arc-length and area in the upper half-plane

📐 Arc-length differential

Starting from the disk model's arc-length differential ds = 2|dz|/(1 − |z|²), and using z = V⁻¹(w) = (iw + 1)/(w + i), the derivation yields:

Arc-length differential in the upper half-plane: ds = |dw| / Im(w)

Length of a smooth curve r(t) for a ≤ t ≤ b in the upper half-plane model: L(r) = ∫ᵃᵇ |r'(t)| / Im(r(t)) dt

🧮 Example: length of a horizontal curve

For the horizontal curve r(t) = t + ki where a ≤ t ≤ b:

• r'(t) = 1 and Im(r(t)) = k
• L(r) = ∫ᵃᵇ (1/k) dt = (b − a)/k

📦 Area differential

From the arc-length differential comes the area differential:

Area of a region R in the upper half-plane model (in Cartesian coordinates): A(R) = ∫∫ᴿ (1/y²) dx dy

🔺 Triangle area formula

🔺 Area of a 2/3-ideal triangle

A 2/3-ideal triangle has two ideal points and one interior vertex.

Example calculation: Consider the triangle with ideal points 1, ∞ and interior vertex w on the unit circle.

• If w = e^(i(π−α)) where 0 < α < π, the interior angle at w is α.
• The area integral becomes: A = ∫₁^(cos(π−α)) ∫_(√(1−x²))^∞ (1/y²) dy dx
• Simplifying: A = ∫₁^(cos(π−α)) 1/√(1−x²) dx
• Using the trig substitution cos(θ) = x: A = ∫₀^(π−α) dθ = π − α

🎯 General theorem

Theorem 5.5.10: The area of a 2/3-ideal triangle having interior angle α is equal to π − α.

Why this works generally:

• Any 2/3-ideal triangle is congruent to one of the form 1 w ∞ where w is on the upper half of the unit circle.
• Since transformations preserve angles and area, the formula applies to all 2/3-ideal triangles.

Don't confuse: This is a specific result for 2/3-ideal triangles (two ideal points, one interior vertex), not for all hyperbolic triangles. The area depends only on the interior angle, not on side lengths.

🧭 Overview

🧠 One-sentence thesis

Stereographic projection maps diametrically opposed points on the sphere to pairs of points in the extended plane that satisfy the equation z · w = −1, defining the concept of antipodal points in the extended plane.

📌 Key points (3–5)

• Diametrically opposed points on the sphere: two distinct points on the same line through the sphere's center, also called antipodal points.
• How stereographic projection transfers antipodal structure: the map φ takes diametrically opposed points (a, b, c) and (−a, −b, −c) on S² to related points in the extended plane C⁺.
• Antipodal points in the extended plane: two points z and w in C⁺ are antipodal if z · w = −1; by convention, 0 and ∞ are also antipodal.
• Common confusion: antipodal points in C⁺ are not arbitrary pairs; they must satisfy the specific equation z · w = −1, and each point has a unique antipodal partner.
• Formula for the antipodal partner: for any z in C⁺, its unique antipodal point z_a is −1/z (when z ≠ 0, ∞), ∞ (when z = 0), or 0 (when z = ∞).

🌐 Geometric setup on the sphere

🌐 The unit 2-sphere S²

The unit 2-sphere S² consists of all points (a, b, c) in R³ for which a² + b² + c² = 1.

• This is the standard sphere of radius 1 centered at the origin in three-dimensional space.
• Points on S² are triples of real numbers satisfying the sphere equation.

🔄 Stereographic projection φ

The stereographic projection map φ : S² → C⁺ is defined by:

• φ(a, b, c) = a/(1 − c) + b/(1 − c) i if c ≠ 1;
• φ(a, b, c) = ∞ if c = 1.

• This map transfers information about the sphere onto the extended plane C⁺ (the complex plane plus a point at infinity).
• The north pole (c = 1) maps to ∞; all other points map to finite complex numbers.
• Why it matters: stereographic projection is the bridge between spherical geometry and the extended plane, preserving certain geometric properties.

🔗 Diametrically opposed points on S²

Two distinct points on a sphere are called diametrically opposed points if they are on the same line through the center of the sphere. Diametrically opposed points on the sphere are also called antipodal points.

• If P = (a, b, c) is on S², then the point diametrically opposed to it is −P = (−a, −b, −c).
• These are "opposite" points on the sphere, like the north and south poles.
• Example: (1, 0, 0) and (−1, 0, 0) are diametrically opposed.

🎯 Antipodal points in the extended plane

🎯 Definition in C⁺

Two points z and w in C⁺ are called antipodal points if they satisfy the equation z · w = −1. Furthermore, we set 0 and ∞ to be antipodal points in C⁺.

• This is the transferred notion of antipodal structure from the sphere to the extended plane.
• The equation z · w = −1 is the key algebraic condition.
• By convention, the pair (0, ∞) is defined to be antipodal, even though the equation z · w = −1 does not apply directly to infinity.
• Terminology: if z and w are antipodal points, we say w is antipodal to z, and vice versa.

🧮 Formula for the unique antipodal partner

Each z in C⁺ has a unique antipodal point z_a, given by:

Case	Antipodal partner z_a
z ≠ 0, ∞	−1/z
z = 0	∞
z = ∞	0

• How to compute: for a nonzero finite z, take the negative reciprocal: z_a = −1/z.
• Why this works: if z · w = −1, then w = −1/z.
• Example: if z = 2, then z_a = −1/2, and indeed 2 · (−1/2) = −1.
• Example: if z = i, then z_a = −1/i = i (since −1/i = −i/i² = −i/(−1) = i). Wait, let's recalculate: −1/i = −1/i · (−i)/(−i) = i/1 = i. Actually, i · i = −1, so z_a = i is correct.

🔍 Geometric interpretation

• The excerpt notes that −1/z = (−1/|z|²) · z̄ (where z̄ is the complex conjugate of z).
• This means z_a is a scaled version of z̄: it is the conjugate of z, scaled by −1/|z|².
• Implication: z and z_a lie on the same line through the origin in the complex plane (when viewed as R²), but on opposite sides and at reciprocal distances (adjusted by the negative sign).
• Don't confuse: z_a is not simply −z; it is −1/z, which involves both inversion and negation.

🔗 Connection to stereographic projection

🔗 How φ maps antipodal pairs

• The excerpt states: "φ maps diametrically opposed points of the sphere to points in the extended plane that satisfy a particular equation."
• That equation is z · w = −1.
• Why this matters: the algebraic condition z · w = −1 in C⁺ captures the geometric relationship of being diametrically opposed on S².
• Example: if P = (a, b, c) on S² maps to z = φ(P), then −P = (−a, −b, −c) maps to w = φ(−P), and z · w = −1.

🧩 Uniqueness and symmetry

• Each point z in C⁺ has exactly one antipodal partner z_a.
• The relationship is symmetric: if w is antipodal to z, then z is antipodal to w.
• This mirrors the symmetry of diametrically opposed points on the sphere.

🧭 Overview

🧠 One-sentence thesis

Elliptic geometry is the second type of non-Euclidean geometry that might describe the geometry of the universe, developed using the sphere as a guide and relying on the concept of antipodal points transferred to the extended plane via stereographic projection.

📌 Key points (3–5)

• What elliptic geometry is: the second type of non-Euclidean geometry (alongside hyperbolic geometry) that could describe the universe's geometry.
• The sphere as guide: two-dimensional elliptic geometry is developed by studying the sphere and transferring information to the extended plane.
• Antipodal points: diametrically opposed points on the sphere map to points in the extended plane that satisfy a specific equation (z · w = −1).
• Common confusion: antipodal points on the sphere (geometric opposition through the center) versus antipodal points in the extended plane (algebraic relation z · w = −1); stereographic projection connects these two notions.
• Chapter structure: begins with stereographic projection review, develops elliptic geometry in Sections 6.2 and 6.3, then reflects on established geometry before moving to surfaces.

🌐 The sphere and stereographic projection

🌐 The unit 2-sphere

The unit 2-sphere S² consists of all points (a, b, c) in R³ for which a² + b² + c² = 1.

• This is the standard sphere of radius 1 centered at the origin in three-dimensional space.
• The sphere serves as the guide for developing two-dimensional elliptic geometry.

🗺️ Stereographic projection

The stereographic projection map φ : S² → C⁺ is defined by:

• φ(a, b, c) = a/(1−c) + b/(1−c) i if c ≠ 1
• φ(a, b, c) = ∞ if c = 1

• This map transfers information about the sphere onto the extended plane C⁺ (the complex plane plus the point at infinity).
• The north pole (where c = 1) maps to infinity.
• All other points on the sphere map to finite complex numbers.
• The chapter begins with a review of this projection because it is the key tool for translating spherical geometry into planar terms.

🔄 Antipodal points on the sphere

🔄 Diametrically opposed points

Two distinct points on a sphere are called diametrically opposed points if they are on the same line through the center of the sphere. Diametrically opposed points on the sphere are also called antipodal points.

• If P = (a, b, c) is on S², then the point diametrically opposed to it is −P = (−a, −b, −c).
• These are the "opposite ends" of a diameter through the sphere's center.
• Example: the north and south poles are antipodal; any point and its reflection through the origin are antipodal.

🧮 How stereographic projection maps antipodal points

• The excerpt states that φ maps diametrically opposed points of the sphere to points in the extended plane that satisfy a particular equation.
• This observation motivates the algebraic definition of antipodal points in the extended plane.

🧮 Antipodal points in the extended plane

🧮 Definition in C⁺

Two points z and w in C⁺ are called antipodal points if they satisfy the equation z · w = −1. Furthermore, we set 0 and ∞ to be antipodal points in C⁺. If z and w are antipodal points, we say w is antipodal to z, and vice versa.

• This is an algebraic characterization: two complex numbers are antipodal if their product is −1.
• The special case: 0 and ∞ are defined to be antipodal by convention (since the product is not well-defined in the usual sense).
• This definition captures the image under stereographic projection of diametrically opposed points on the sphere.

🔢 Finding the unique antipodal point

Each z in C⁺ has a unique antipodal point, denoted z_a, given by:

Case	Formula
z ≠ 0, ∞	z_a = −1/z
z = 0	z_a = ∞
z = ∞	z_a = 0

• For finite nonzero z, the antipodal point is −1/z.
• The excerpt notes that −1/z = (−1/|z|²) · z̄ (where z̄ is the complex conjugate), so z_a is a scaled version of z̄.
• This means z and z_a lie on the same line through the origin in the complex plane, but on opposite sides and at reciprocal distances (scaled by −1/|z|²).

🔍 Don't confuse: sphere vs plane

• On the sphere: antipodal points are geometric—diametrically opposed through the center.
• In the extended plane: antipodal points are algebraic—related by z · w = −1.
• Stereographic projection φ connects these two notions: if P and −P are antipodal on S², then φ(P) and φ(−P) are antipodal in C⁺.

📖 Chapter roadmap

📖 Structure and focus

• Section 6.1: Review of stereographic projection and antipodal points (the current section).
• Sections 6.2 and 6.3: Development of elliptic geometry itself.
• Section 6.4: Reflection on what has been established in geometry before moving to geometry on surfaces in Chapter 7.
• The chapter focuses on two-dimensional elliptic geometry, using the sphere as the guiding model.

📖 Context: non-Euclidean geometries

• Elliptic geometry is the second type of non-Euclidean geometry (the first being hyperbolic geometry, covered in Chapter 5).
• Both are candidates for describing the geometry of the universe, contrasting with Euclidean (flat) geometry.
• The sphere's intrinsic curvature provides the geometric intuition for elliptic geometry, just as the hyperbolic plane's negative curvature guided hyperbolic geometry.

🧭 Overview

🧠 One-sentence thesis

Elliptic geometry has its own formulas for measuring length, area, and distance that differ from Euclidean geometry by a sign and lead to triangles whose angles always sum to more than 180°.

📌 Key points (3–5)

• Arc-length and area formulas: elliptic geometry uses specific integral formulas that differ from hyperbolic geometry by a single sign.
• Distance between points: the elliptic distance takes the minimum of two possible segment lengths along the unique elliptic line connecting them, with an upper bound of π/2.
• Elliptic circles: defined as all points at a fixed geodesic path length from a center; they may consist of portions of two distinct clines in the disk model.
• Triangle area formula: the area of an elliptic triangle equals (α + β + γ) − π, meaning angles always sum to more than 180°.
• Common confusion: unlike Euclidean or hyperbolic geometry, there is no upper bound on journey length along an elliptic line, but the distance between any two points cannot exceed π/2.

📏 Length and area definitions

📏 Arc-length formula

Length of a smooth curve r : [a, b] → P²: L(r) = integral from a to b of (2 times the absolute value of r′(t)) divided by (1 + the square of the absolute value of r(t)), with respect to t.

• This formula applies to smooth curves in the projective plane P² with transformation group S.
• The key difference from hyperbolic geometry is a single sign change in the formula.
• This sign difference is consistent with the algebraic descriptions of transformations in the respective geometries.

Invariance: Arc-length is an invariant of elliptic geometry (Theorem 6.3.2), meaning transformations in S preserve the length of curves.

📐 Area formula

Area of a figure R in polar coordinates: A(R) = double integral over R of (4r) divided by ((1 + r²)²), with respect to r and θ.

• Area is also an invariant of elliptic geometry, following from the invariance of arc-length.
• The entire projective plane P² has finite area equal to 2π (unlike hyperbolic space, which has infinite area).

🎯 Distance between points

🎯 Two segments, one distance

The elliptic distance between points p and q is defined as the minimum of two possible segment lengths:

• There is exactly one elliptic line through p and q.
• A traveler at p can reach q by going in either of two directions along this line.
• The distance d_S(p, q) is the length of the shorter of these two segments.

Example: Imagine a bug at point p wanting to walk to point q along an elliptic line—it can proceed in either direction, and the distance is whichever route is shorter.

📊 Distance formula

Elliptic distance: d_S(p, q) = min{2 arctan(|q − p| / |1 + p̄q|), 2 arctan(|1 + p̄q| / |q − p|)}

Special case: For the origin and a point x on the positive real axis (0 < x ≤ 1):

• d_S(0, x) = 2 arctan(x)
• This is derived by parameterizing the segment r(t) = t for 0 ≤ t ≤ x and computing the arc-length integral.

Upper bound: The distance between any two points in (P², S) cannot exceed π/2.

Don't confuse: Although distance is bounded by π/2, there is no upper bound on how long a journey along an elliptic line can be—a traveler can do "laps" for any r > 0.

🔄 Verification with the sphere model

• d_S(0, 1) = 2 arctan(1) = π/2
• Via stereographic projection, the elliptic segment from 0 to 1 corresponds to one-quarter of a great circle on the unit sphere.
• A great circle on the unit sphere has circumference 2π, so one-quarter has length π/2, confirming the formula.

⭕ Elliptic circles

⭕ Definition and construction

Elliptic circle centered at z₀ with radius r: the set of all points z in P² such that there exists a geodesic path of length r from z₀ to z.

• Geodesic paths follow elliptic lines.
• Each transformation T in S fixes two antipodal points and pushes points along type II clines of those points.
• Since transformations preserve distance, these type II clines determine elliptic circles—all points on them are equidistant from the center.

🎨 Visual representation in the disk model

• Elliptic circles centered at the origin correspond to Euclidean circles centered at the origin.
• A Euclidean circle centered at the origin with Euclidean radius a (0 < a < 1) corresponds to an elliptic circle with elliptic radius 2 arctan(a).
• For a general center p and point q, the elliptic circle may consist of portions of two distinct clines that live in the closed unit disk.

Example: If a type II cline through p, p^a, and q extends outside the unit disk, the elliptic circle is represented by the portions of two type II clines (through q and q^a) that lie inside the disk.

📏 Circumference formula

Theorem 6.3.7: An elliptic circle with elliptic radius r < π/2 has circumference C = 2π sin(r).

• For small r, this approaches the Euclidean formula 2πr (the limit as r → 0⁺ of (2π sin(r))/(2πr) = 1).
• Circles with radius r ≥ π/2 have unusual shapes—they may not look like circles at all!

Special cases:

• When r = π/2, the circle becomes an elliptic line.
• When r = π, the circle is a single point.

🔺 Triangle area and lunes

🌙 Lunes

Lune: a region in P² bounded between two elliptic lines.

• Two elliptic lines trap a region because antipodal points on the unit circle are identified.
• A traveler in the shaded region can visit all shaded points without crossing the boundary walls.

Lemma 6.3.9: The area of a lune with angle α is 2α.

Proof sketch:

• Move the vertex of the lune to the origin with one leg on the real axis.
• Half the lunar region is described in polar coordinates by 0 ≤ r ≤ 1 and 0 ≤ θ ≤ α.
• Compute the area integral: A = 2 times the integral from 0 to α of the integral from 0 to 1 of (4r)/((1 + r²)²) dr dθ.
• Using substitution u = 1 + r², this simplifies to 2α.

🔺 Triangle area formula

Theorem 6.3.12: In elliptic geometry (P², S), the area of a triangle with angles α, β, γ is A = (α + β + γ) − π.

Derivation:

• Each corner of the triangle determines a lune.
• The three lunes cover the entire projective plane, with the triangle covered three times.
• Sum of three lune areas = area of entire projective plane + 2 × area of triangle.
• 2π + 2·A(Δpqr) = 2α + 2β + 2γ
• Solving: A(Δpqr) = (α + β + γ) − π

Key consequence: The angles of any triangle in elliptic geometry sum to more than 180° (more than π radians).

Example: If a tax collector observes a triangle with angles 92°, 62°, and 27°, convert to radians and compute A = (α + β + γ) − π to find the area.

📐 Trigonometry in elliptic geometry

📐 Law of cosines

Theorem 6.3.14a (Elliptic law of cosines): For an elliptic triangle with side lengths a, b, c and angles α, β, γ:

cos(c) = cos(a) cos(b) + sin(a) sin(b) cos(γ)

Proof strategy:

• Position the triangle conveniently: one corner at the origin, one on the positive real axis at r, one at q = ke^(iγ).
• Express a = d_S(0, r) = 2 arctan(r), so cos(a) = (1 − r²)/(1 + r²) and sin(a) = 2r/(1 + r²).
• Similarly for b: cos(b) = (1 − k²)/(1 + k²) and sin(b) = 2k/(1 + k²).
• For c, use the distance formula and expand |1 + rke^(iγ)|² and |ke^(iγ) − r|².
• Use e^(iγ) + e^(−iγ) = 2 cos(γ) to simplify.
• The formula holds for any triangle because transformations preserve angles and distances.

📐 Law of sines

Theorem 6.3.14b (Elliptic law of sines):

sin(a)/sin(α) = sin(b)/sin(β) = sin(c)/sin(γ)

Proof strategy:

• Construct the circle containing side c, which goes through r, q, and their antipodal points.
• Let p be the center of the circle and R its Euclidean radius.
• Use midpoints m_r and m_q of segments connecting points to their antipodals.
• Form right triangles Δpm_r r and Δpm_q q.
• From these triangles: sin(β) = (1 + r²)/(2rR) and sin(α) = (1 + k²)/(2kR).
• Compare ratios to show sin(a)/sin(α) = sin(b)/sin(β).
• Transform the triangle to show sin(c)/sin(γ) matches the common ratio.

📐 Right triangle theorem

Corollary 6.3.16 (Elliptic hypotenuse theorem): In a right triangle with side lengths a, b and hypotenuse c:

cos(c) = cos(a) cos(b)

• This follows from the law of cosines when γ = π/2 (so cos(γ) = 0).

🌐 Connection to spherical geometry

🌐 Distance on the sphere

For spherical geometry (C⁺, S), the distance formula simplifies:

d_S(p, q) = 2 arctan(|q − p| / |1 + p̄q|)

• This formula applies to all positive real numbers x in C⁺ (not just those in the unit disk).
• The distance d_S(p, q) corresponds to the actual distance on the unit 2-sphere between φ⁻¹(p) and φ⁻¹(q), where φ is stereographic projection.

Verification: For x > 0, the distance on the sphere between φ⁻¹(0) and φ⁻¹(x) equals arccos((1 − x²)/(1 + x²)), which can be shown to equal 2 arctan(x).

🌐 Small-radius approximations

For small elliptic radius r:

• Circumference: lim (r → 0⁺) of (2π sin(r))/(2πr) = 1, so the Euclidean formula 2πr is a good approximation.
• Area: lim (r → 0⁺) of (4π sin²(r/2))/(πr²) = 1, so the Euclidean formula πr² is a good approximation.

This shows that locally, elliptic geometry resembles Euclidean geometry, just as hyperbolic geometry does.

🧭 Overview

🧠 One-sentence thesis

Elliptic geometry satisfies Euclid's first four postulates but fails the fifth postulate in a different way than hyperbolic geometry does, because any two elliptic lines always intersect.

📌 Key points (3–5)

• What this section does: shows that elliptic geometry (P₂, S) satisfies Euclid's postulates 1–4 but not the fifth.
• How the fifth postulate fails: in elliptic geometry, given a line and a point not on it, there is no line through the point that avoids the given line (all elliptic lines intersect).
• Common confusion: elliptic vs hyperbolic failure of the fifth postulate—hyperbolic geometry has infinitely many parallel lines through a point; elliptic geometry has zero parallel lines.
• Why postulates 2 and 3 still hold in finite space: elliptic space has no boundary, so lines can be extended indefinitely and circles of any radius can be drawn, even though the total area is finite.

📐 Euclid's five postulates

📐 The postulates restated

The excerpt lists Euclid's five postulates:

1. One can draw a straight line from any point to any point.
2. One can produce a finite straight line continuously in a straight line.
3. One can describe a circle with any center and radius.
4. All right angles equal one another.
5. Given a line and a point not on the line, there is exactly one line through the point that does not intersect the given line.

• The fifth postulate is also known as the parallel postulate.
• The excerpt emphasizes that elliptic geometry satisfies 1–4 but fails 5.

✅ Postulates 1–4: satisfied in elliptic geometry

✅ Postulate 1: drawing a line between two points

• What the excerpt says: the first postulate is satisfied in (P₂, S) by Theorem 6.2.11.
• This means any two points in elliptic space can be connected by a straight line.

✅ Postulate 2: extending a line indefinitely

• Why it holds even in finite space: elliptic space has no boundary.
• You can walk along an elliptic straight line for as long as you want; you will eventually return to your starting point and continue making laps.
• The key is that "indefinitely" does not require infinite total space—it requires no edge or stopping point.

✅ Postulate 3: circles of any radius

• Why it holds: in the previous section, circles were defined about any point in P₂.
• Since you can walk an arbitrarily long distance from any point, you can describe a circle of any radius.
• Again, the finite total area does not prevent arbitrarily large radii (the circle will wrap around the space).

✅ Postulate 4: all right angles equal

• Why it holds: Möbius transformations preserve angles, and the maps in S are special Möbius transformations.
• Therefore, right angles remain congruent under the transformations that define elliptic geometry.

❌ Postulate 5: the parallel postulate fails

❌ Why the fifth postulate fails in elliptic geometry

In elliptic geometry, any two elliptic lines intersect (Theorem 6.2.13).

• What this means: given a line and a point not on the line, there is not a single line through the point that does not intersect the given line.
• In fact, there are zero such lines—every line through the point will intersect the given line.

🔄 Comparing elliptic and hyperbolic failures

Geometry	Does the fifth postulate hold?	How it fails
Euclidean	Yes	Exactly one parallel line through the point
Hyperbolic	No	Infinitely many lines through the point that do not intersect the given line
Elliptic	No	Zero lines through the point that do not intersect the given line (all lines intersect)

• Don't confuse: both hyperbolic and elliptic geometries violate the parallel postulate, but in opposite directions.
• Hyperbolic: too many parallels.
• Elliptic: no parallels at all.

🎯 The elliptic version of the fifth postulate

• The excerpt notes that "the elliptic version of the fifth postulate differs from the hyperbolic version."
• Elliptic geometry replaces the parallel postulate with the statement that any two lines intersect.
• This is the defining feature that distinguishes elliptic geometry from both Euclidean and hyperbolic geometries.

🧭 Overview

🧠 One-sentence thesis

Curvature is the intrinsic geometric property of a space that dictates how circles and triangles behave differently from Euclidean predictions, and it can be measured by examining how the ratio of a circle's circumference to its radius deviates as the radius changes.

📌 Key points (3–5)

• What curvature measures: how drastically a surface bends away from its tangent plane at a point; it is an intrinsic property that doesn't change if the surface is bent without stretching.
• Three fundamental types: positive curvature (surface on one side of tangent plane, like a sphere), negative curvature (saddle-shaped, tangent plane cuts through), and zero curvature (surface has a line agreeing with tangent plane, like a cylinder).
• How to detect curvature: a two-dimensional bug can measure it by comparing actual circle circumferences to the Euclidean prediction c = 2πr; deviations reveal the curvature.
• Common confusion: all three geometries (hyperbolic, Euclidean, elliptic) look nearly identical on small scales—small circles and triangles behave almost Euclidean—but the second derivative of the circumference ratio distinguishes them.
• Mathematical definition: curvature k equals negative three times the limit of the second derivative of the ratio c/(2πr) as radius r approaches zero.

📐 Understanding curvature for curves and surfaces

📏 Curvature of a curve

The curvature of a curve at a point is a measure of how drastically the curve bends away from its tangent line.

• The radius of curvature at a point corresponds to the radius of the circle that best approximates the curve at that point.
• The relationship: curvature k = 1/r, where r is the radius of the approximating circle.
• This concept is often studied in multivariable calculus courses.

🌐 Curvature of a surface

The curvature of a surface at a particular point is a measure of how drastically the surface bends away from its tangent plane at the point.

Three fundamental types:

Type	Description	Visual characteristic	Example
Positive	Surface lives entirely on one side of tangent plane (near the point)	Cup-shaped	Sphere
Negative	Tangent plane cuts through the surface	Saddle-shaped	Hyperbolic plane
Zero	Surface has a line along which it agrees with tangent plane	Flat or rolled	Cylinder, Euclidean plane

Don't confuse: A cylinder has zero curvature even though it is "curved" in space—curvature is intrinsic, not about how the surface sits in three-dimensional space.

🔍 Gauss's Theorem Egregium and intrinsic curvature

🎯 Intrinsic property

• Gauss called his theorem about curvature the Theorem Egregium.
• Key insight: curvature is an intrinsic property of the surface.
• The curvature doesn't change if the surface is bent without stretching.
• A two-dimensional inhabitant living in the space can determine the curvature by taking measurements—they don't need to "see" the surface from outside.

🐛 What a bug would observe

• A two-dimensional bug living in hyperbolic, projective, or Euclidean plane would notice that small circles have circumference related to radius by the Euclidean formula c ≈ 2πr.
• In Euclidean geometry, this formula applies to all circles.
• In non-Euclidean cases, the bug would notice significant differences for large circles:
  • Positive curvature (cup-shaped point): large circles have circumference less than 2πr predicted by Euclidean geometry.
  • Negative curvature (saddle-shaped point): large circles have circumference greater than 2πr.

Example: A large chunk of orange peel fractures if pressed flat onto a table because the orange peel has positive curvature, and its actual circumference is less than the Euclidean prediction would allow when flattened.

📊 Measuring curvature through circle circumferences

📈 Circumference formulas in different geometries

The actual circumference c for circles of radius r differs across geometries:

Geometry	Circumference formula	Curvature
Hyperbolic plane	c = 2π sinh(r)	k = -1
Euclidean plane	c = 2πr	k = 0
Elliptic plane	c = 2π sin(r)	k = 1

📉 The ratio c/(2πr)

• In all three cases, the ratio c/(2πr) approaches 1 as r shrinks to 0.
• The first derivative of the ratio also approaches 0 as r → 0⁺ in all three cases.
• The second derivative of the ratio distinguishes these geometries.

Don't confuse: The first derivative being zero means all geometries look flat at small scales; only the second derivative reveals the intrinsic curvature.

🧮 Formal definition of curvature

Definition 7.1.4: Suppose a circle of radius r about a point p is drawn in a space, and its circumference is c. The curvature of the space at p is k = -3 times the limit as r approaches 0⁺ of the second derivative with respect to r of [c/(2πr)].

In words:

• Take the ratio of actual circumference to Euclidean prediction: c/(2πr)
• Compute the second derivative of this ratio with respect to r
• Take the limit as r → 0⁺
• Multiply by -3

This working definition captures how the space deviates from Euclidean geometry at infinitesimal scales.

🌍 Examples: Computing curvature

🔵 Curvature of a sphere (Example 7.1.5)

Consider a sphere with radius s:

• A circle centered at the north pole with surface radius r has Euclidean radius x = s sin(r/s).
• The circumference is c = 2πs sin(r/s).
• Using the power series expansion: sin(r/s) = (r/s) - (r³)/(6s³) + (r⁵)/(120s⁵) - ...
• After substitution: c/(2πr) = 1 - r²/(6s²) + r⁴/(120s⁴) - ...
• The second derivative: -1/(3s²) + (12r²)/(120s⁴) - ...
• As r → 0⁺, all terms with r vanish, leaving: k = 1/s²

Key insight: Because the sphere is homogeneous, curvature is the same at all points. A sphere of radius s has constant curvature k = 1/s².

🌀 Curvature of the hyperbolic plane (Example 7.1.6)

• Hyperbolic geometry is homogeneous, so curvature is the same at all points.
• The circumference of a circle is c = 2π sinh(r).
• Using the power series: sinh(r) = r + r³/3! + r⁵/5! + ...
• After substitution: c/(2πr) = 1 + r²/3! + r⁴/5! + ...
• The second derivative: 1/3 + (12r²)/5! + ...
• As r → 0⁺, all terms with r vanish, leaving: k = -1

The hyperbolic plane in (D, H) has constant curvature k = -1.

⚪ Curvature of the Euclidean plane (Exercise 2)

• In Euclidean geometry, c = 2πr exactly for all circles.
• The ratio c/(2πr) = 1 for all r (constant).
• The second derivative of a constant is 0.
• Therefore: k = 0

🟣 Curvature of the projective plane (Exercise 1)

• In elliptic geometry, c = 2π sin(r).
• Following similar steps as the sphere example with s = 1 yields: k = 1

🔄 Elliptic geometry with varying curvature

🎚️ Scaling the sphere

• One may model elliptic geometry on spheres of varying radii.
• A change in radius causes a change in curvature and in the relationship between triangle area and angle sum.
• For any real number k > 0, construct a sphere centered at the origin with radius 1/√k.
• According to Example 7.1.5, this sphere has constant curvature k.

🗺️ The model (P²ₖ, Sₖ)

• Space P²ₖ: the closed disk in the complex plane of radius 1/√k, with antipodal points of the boundary identified.
• Antipodal points with respect to S²ₖ satisfy: z_a = -1/(kz).
• Transformations Sₖ: Möbius transformations that preserve antipodal points.
• A transformation T is in Sₖ if and only if: when z_a = -1/(kz), then T(z_a) = -1/(kT(z)).

Elliptic geometry with curvature k: the geometry (P²ₖ, Sₖ) with k > 0.

Note: (P²₁, S₁) is precisely the elliptic geometry studied in Chapter 6 (the unit sphere case).

🔀 Transformations and lines

• Transformations in Sₖ correspond to rotations of the sphere S²ₖ.
• They have the form: T(z) = e^(iθ) (z - z₀)/(1 + kz₀z).
• Lines in elliptic geometry with curvature k are clines with the property: if they go through z, they also go through z_a = -1/(kz).

Example: A bug convinced she lives in elliptic geometry who finds a triangle with three right angles and area 3π (instead of the expected π/2 for the unit sphere) might conclude she lives on a larger sphere—one with different curvature k.

🧭 Overview

🧠 One-sentence thesis

Elliptic geometry can be modeled on spheres of any positive curvature k by scaling the radius to 1/√k, which changes both the curvature and the relationship between triangle area and angle sum.

📌 Key points (3–5)

• Constructing elliptic geometry with curvature k > 0: use a sphere of radius 1/√k centered at the origin, then apply stereographic projection to the extended plane.
• Antipodal points: two points z and z_a in the extended plane are antipodal with respect to the sphere S²_k when z_a = −1/(kz).
• Lines in this geometry: clines that pass through both z and its antipodal point z_a; these correspond to great circles on the sphere.
• Triangle area formula: for a triangle with angles α, β, γ, the area is (1/k)(α + β + γ − π), showing that angle sum exceeds π and the excess depends on curvature.
• Common confusion: the space P²_k is a scaled version of the projective plane from Chapter 6; when k = 1, it reduces to the previously studied geometry.

🌐 Building the model

🌐 The sphere and stereographic projection

• Start with a sphere S²_k centered at the origin in three-dimensional space with radius 1/√k.
• Define stereographic projection φ_k: S²_k → C⁺ by the formula:
  • φ_k(a, b, c) = a/(1 − c√k) + b/(1 − c√k) i when c ≠ 1/√k
  • φ_k(a, b, c) = ∞ when c = 1/√k
• This maps the sphere down to the extended complex plane.

🔗 Antipodal points

Two points z and z_a in C⁺ are antipodal with respect to S²_k when they satisfy z_a = −1/(kz).

• Diametrically opposed points on the sphere S²_k map via φ_k to antipodal points in the plane.
• Example: if z is a point in the plane, its antipodal partner z_a is determined by the equation z_a = −1/(kz).
• Don't confuse: antipodal points are not just "opposite" in a Euclidean sense; they are related by this specific algebraic relationship involving k.

📐 The space and transformations

• The space P²_k is the closed disk in C of radius 1/√k, with antipodal points on the boundary identified.
• This is a scaled version of the projective plane from Chapter 6.
• The group of transformations S_k consists of Möbius transformations that preserve antipodal points: T ∈ S_k if and only if whenever z_a = −1/(kz), then T(z_a) = −1/(kT(z)).
• Transformations in S_k correspond to rotations of the sphere S²_k and have the form T(z) = e^(iθ)(z − z₀)/(1 + kz₀z).

📏 Lines and measurement

📏 Lines in elliptic geometry with curvature k

Lines in elliptic geometry with curvature k are clines with the property that if they pass through z, they also pass through z_a = −1/(kz).

• These lines correspond precisely to great circles on the sphere S²_k.
• Example: a line through the origin must also pass through ∞ (its antipodal point), so it is a Euclidean line in the plane.

📏 Arc-length formula

• The arc-length of a smooth curve r in P²_k is given by the integral:
  • L(r) = integral from a to b of (2|r'(t)|)/(1 + k|r(t)|²) dt
• Arc-length is an invariant under transformations in S_k.
• The shortest path between two points is along the elliptic line through them.
• The greatest possible distance between two points in (P²_k, S_k) is π/(2√k).

📏 Radius of curvature

• The quantity s = 1/√k is called the radius of curvature for the geometry.
• It is the radius of the disk on which the geometry is modeled.
• Example: a larger s (smaller k) means a "flatter" geometry; as k approaches 0, the geometry approaches Euclidean.

📐 Area formulas

📐 Area of a region

• The area of a region R given in polar form is computed by:
  • A(R) = double integral over R of (4r)/((1 + kr²)²) dr dθ
• Useful antiderivative from the proof: integral of (4r)/((1 + kr²)²) dr = −2/(k(1 + kr²)) + C

📐 Area of a lune

A lune is a 2-gon whose sides are elliptic lines in (P²_k, S_k).

• A lune with interior angle α has area 2α/k.
• The proof uses the area integral with the lune positioned so one side is the real axis.
• Special case: a lune with angle π covers the entire disk of radius 1/√k, so the total area of P²_k is 2π/k.
• This matches half the surface area of a sphere of radius 1/√k.

📐 Area of a triangle

• For a triangle with angles α, β, and γ, the area is:
  • A = (1/k)(α + β + γ − π)
• The proof uses the area of three lunes and the total area of P²_k, as in the case k = 1.
• The quantity (α + β + γ − π) is the angle excess of the triangle.
• Example: on Earth's surface (approximately spherical with radius 6375 km), k ≈ 1/6375² km⁻², so a triangle with angles α, β, γ has area A = 6375²(α + β + γ − π) km².

🌍 Real-world application

🌍 Triangles on Earth

• Earth's surface is approximately spherical with radius about 6375 km.
• The geometry on Earth can be modeled by (P²_k, S_k) where k = 1/6375² km⁻².
• To find the area of a triangle formed by three cities (e.g., Paris, New York, Rio):
  1. Measure the angles α, β, γ at each vertex using a globe, protractor, and string.
  2. The string follows a geodesic (shortest path) when pulled taut.
  3. Compute A = (1/k)(α + β + γ − π) = 6375²(α + β + γ − π) km².

🔄 Connection to other geometries

🔄 Relationship to Chapter 6

• When k = 1, the geometry (P²₁, S₁) is precisely the elliptic geometry studied in Chapter 6.
• The general case (P²_k, S_k) is a scaled version of that geometry.
• All formulas generalize from the k = 1 case by replacing 1 with k and adjusting accordingly.

🔄 Limiting behavior as k → 0⁺

• As k approaches 0 from above, elliptic geometry approaches Euclidean geometry.
• The elliptic distance d_k(0, x) from 0 to x (where 0 < x ≤ s) is d_k(0, x) = 2s arctan(x/s); as k → 0⁺, this approaches 2x (twice Euclidean distance).
• The circumference of a circle with elliptic radius r is C = 2πs sin(r/s); as k → 0⁺, this approaches 2πr (Euclidean circumference).
• The area of a circle with elliptic radius r is A = 4πs² sin²(r/(2s)); as k → 0⁺, this approaches πr² (Euclidean area).
• Don't confuse: the factor of 2 in the distance formula is an artifact of the model; the key is that the formulas approach Euclidean forms.

🧭 Overview

🧠 One-sentence thesis

Hyperbolic geometry with curvature k < 0 can be scaled to any negative curvature value, producing a family of geometries that share identical formulas with elliptic geometry but exhibit opposite geometric behavior (e.g., triangle angle sums less than π, infinite area, and circumferences greater than 2πr).

📌 Key points (3–5)

• Scaling the model: For each negative k, the space D_k is an open disk of radius 1/√|k|, and the geometry (D_k, H_k) generalizes the standard Poincaré disk model.
• Identical formulas to elliptic case: Arc-length, area, and transformation formulas for hyperbolic geometry with curvature k match those for elliptic geometry with curvature k, despite opposite geometric properties.
• Triangle area formula: In hyperbolic geometry with curvature k, the area of a triangle with angles α, β, γ is A = (1/k)(α + β + γ - π), which is negative since angle sums are less than π and k is negative.
• Common confusion—sign of k vs. type of geometry: The sign of k determines the type of geometry (negative → hyperbolic, positive → elliptic, zero → Euclidean), while the magnitude of |k| determines the radius of curvature; don't confuse the curvature constant with the geometric behavior.
• Observable consequence—angle of parallelism: In hyperbolic geometry, the angle of parallelism θ (the angle at which a distant line appears to approach an ideal point) depends on distance d to the line via Lobatchevsky's formula: tan(θ/2) = exp(-√|k| d), a relationship with no Euclidean analogue.

🌐 Constructing the scaled hyperbolic model

🌐 The space D_k and circle at infinity

For each negative number k < 0, the space D_k is the open disk of radius 1/√|k| centered at the origin in the complex plane C.

• D_k consists of all complex numbers z such that |z| < 1/√|k|.
• The circle at infinity is the boundary circle |z| = 1/√|k|.
• The quantity s = 1/√|k| is called the radius of curvature; it is the radius of the disk on which the geometry is modeled.
• Example: If k = -1/100, then s = 10, and the disk has Euclidean radius 10.

🔄 The transformation group H_k

The group H_k consists of all Möbius transformations that send D_k to itself.

• Transformations in H_k have the form T(z) = (e^(iθ) (z - z_0)) / (1 + k z_0 z̄), where θ is any real number and z_0 is a point in D_k.
• This is exactly the same formula as for the elliptic group S_k, despite the different sign of k.
• The defining property: if z and z* are symmetric with respect to the circle at infinity, then T(z) and T(z*) are also symmetric with respect to the circle at infinity.
• The symmetric point to z with respect to the circle at infinity is z* = -1/(kz).

📏 Straight lines in (D_k, H_k)

• Straight lines are the clines (circles or lines) in the extended complex plane that are orthogonal to the circle at infinity.
• Equivalently, a straight line is a cline with the property that if it goes through z, it also goes through the symmetric point -1/(kz).
• This characterization mirrors the elliptic case but applies to the hyperbolic disk.

📐 Measurement formulas

📐 Arc-length formula

The arc-length of a smooth curve r in D_k is L(r) = ∫[a to b] (2|r'(t)|) / (1 + k|r(t)|²) dt.

• This formula is identical to the elliptic arc-length formula.
• The factor 2 in the numerator and the denominator (1 + k|r(t)|²) appear in both hyperbolic and elliptic cases.
• Don't confuse: the formula looks the same, but k is negative in the hyperbolic case and positive in the elliptic case, leading to different geometric behavior.

📐 Area formula

The area of a region R given in polar coordinates is A(R) = ∫∫_R (4r) / (1 + kr²)² dr dθ.

• Again, this formula is identical to the elliptic area formula.
• The excerpt notes that this formula is "a bear to use" in practice.
• A useful antiderivative: ∫ (4r) / (1 + kr²)² dr = -2 / (k(1 + kr²)) + C.

📐 Specific measurement formulas

When s = 1/√|k|:

Measurement	Formula	Notes
Distance from 0 to x (0 < x < s)	d_k(0, x) = s ln((s + x)/(s - x))	Derived by evaluating an integral using partial fractions
Circumference of circle with hyperbolic radius r	C = 2πs sinh(r/s)	Greater than 2πr (Euclidean value)
Area of circle with hyperbolic radius r	A = 4πs² sinh²(r/(2s))	Grows exponentially with radius

🔺 Triangle area and angle sum

🔺 Area formula for triangles

Lemma 7.3.1: In hyperbolic geometry with curvature k, the area of a triangle with angles α, β, and γ is A = (1/k)(α + β + γ - π).

• This formula is identical to the elliptic formula.
• However, in hyperbolic geometry, α + β + γ < π (angle sum is less than π).
• Since k < 0 and (α + β + γ - π) < 0, the product (1/k)(α + β + γ - π) is positive, as expected for an area.
• The formula is derived using the area of a 2/3-ideal triangle (a triangle with one or more ideal points), which has area -(1/k)(π - α) for a triangle with interior angle α.

🔺 Angle defect

• The quantity π - (α + β + γ) is called the angle defect in hyperbolic geometry (it is positive).
• The area is proportional to the angle defect: A = -(1/k) × (angle defect).
• Since |k| measures curvature, larger |k| means smaller area for the same angle defect.

👁️ Observing negative curvature: angle of parallelism

👁️ What is the angle of parallelism?

The angle of parallelism θ of a point z to a line L is the angle ∠wzu, where w is the closest point on L to z and u is an ideal point of L.

• Imagine standing at point z and looking at a distant hyperbolic line L.
• The point w on L is closest to you; as you look farther down the line toward a point v, the angle ∠wzv grows.
• As v moves farther and farther away, the angle approaches θ, the angle to the ideal point u.
• Key fact: θ is a function of the distance d from z to L—this relationship does not exist in Euclidean geometry.
• In Euclidean geometry, looking farther down a line always approaches 90°, regardless of distance d.

👁️ Lobatchevsky's formula

Theorem 7.3.3: In hyperbolic geometry with curvature k, the hyperbolic distance d of a point z to a hyperbolic line L is related to the angle of parallelism θ by tan(θ/2) = exp(-√|k| d).

• This formula connects an observable angle θ to the distance d.
• The proof assumes z is at the origin and L is orthogonal to the positive real axis, intersecting it at point x (0 < x < s, where s = 1/√|k|).
• The proof uses the half-angle formula tan(θ/2) = tan(θ)/(sec(θ) + 1) and expresses the Euclidean radius r of the circle containing L in terms of the hyperbolic distance d.
• After substitution and simplification, the result is tan(θ/2) = exp(-d/s) = exp(-√|k| d).
• Example: If you can measure θ, you can deduce d without directly measuring distance.

👁️ Relationship to earlier results

• In the standard Poincaré disk (k = -1), the excerpt references an earlier result: cosh(d) = 1/sin(θ).
• Lobatchevsky's formula provides an alternative relationship between d and θ.

🌌 Application: parallax and curvature bounds

🌌 What is parallax?

Parallax p is the angle by which a relatively close star appears to move relative to distant stars as the Earth moves in its annual orbit around the Sun.

• In an idealized picture, e₁ and e₂ are the Earth's positions at opposite points of its orbit, and star s is orthogonal to the plane of the orbit.
• In a Euclidean universe, the star's distance D from the Sun is given by D = d/tan(p), where d is the Earth-Sun distance (about 8.3 light-minutes).
• Since p is very small, a working formula is D ≈ d/p.
• Example: Friedrich Bessel (1784–1846) measured a parallax of 0.3 arc seconds for star 61 Cygni in 1837, placing it about 10.5 light-years away.

🌌 Parallax in a hyperbolic universe

• If the universe is hyperbolic with curvature k < 0, a detected parallax puts a bound on how curved the universe can be.
• Consider a triangle with vertices e₁, e₂ (Earth's positions), and s (the star), where the distance between e₁ and e₂ is 2d (about 16.6 light-minutes).
• The detected parallax is p, so angle ∠e₂se₁ = 2p.
• The angle α = ∠e₁e₂s is less than the angle of parallelism θ = ∠e₁e₂u, where u is an ideal point.

🌌 Deriving the curvature bound

• Since tan(x) is an increasing function and α < θ, we have tan(α/2) < tan(θ/2).

• By Lobatchevsky's formula, tan(θ/2) = exp(-√|k| 2d).

• Therefore, tan(α/2) < exp(-√|k| 2d).

• Taking logarithms (ln is increasing): ln(tan(α/2)) < -√|k| 2d.

• Squaring (x² is decreasing for x < 0): [ln(tan(α/2))/(2d)]² > |k|.

• Since α ≈ π/2 - 2p (triangles used in stellar parallax have no detectable angular deviation from 180°), we get:

  |k| < [ln(tan(π/4 - p))/(2d)]²

• For values of p near 0, ln(tan(π/4 - p)) has linear approximation -2p, so a working bound (from Schwarzschild's 1900 paper) is:

  |k| < (p/d)²

• Example: Using Bessel's parallax of 0.3 arc seconds for 61 Cygni, one can estimate an upper bound for |k| in units of (light-years)⁻².

• The smaller the detectable parallax, the tighter the bound on |k|; modern telescopes with better resolving power can detect smaller parallaxes and thus provide tighter bounds.

🔗 The unified family (X_k, G_k)

🔗 One family, three geometries

Definition 7.4.1: For each real number k, the geometry (X_k, G_k) has space X_k = D_k if k < 0, C if k = 0, P²_k if k > 0, and transformation group G_k consisting of all Möbius transformations T(z) = (e^(iθ)(z - z₀))/(1 + k z₀ z̄).

• The sign of k dictates the type of geometry: negative → hyperbolic, zero → Euclidean, positive → elliptic.
• The magnitude of |k| dictates the radius of curvature (except when k = 0, in which case the space is the entire complex plane C).
• Euclidean geometry (X₀, G₀) marks the "edge" between hyperbolic and elliptic geometries.

🔗 Identical formulas, opposite behavior

The families (P²_k, S_k) for k > 0 and (D_k, H_k) for k < 0 have:

Feature	Formula/Description	Hyperbolic (k < 0)	Elliptic (k > 0)
Transformation group	T(z) = (e^(iθ)(z - z₀))/(1 + k z₀ z̄)	Identical	Identical
Straight lines	Clines through p, q, and -1/(kp)	Identical	Identical
Arc-length	L(r) = ∫ (2	r'(t)	)/(1 + k
Area	A(R) = ∫∫ (4r)/(1 + kr²)² dr dθ	Identical	Identical
Triangle area	A = (1/k)(α + β + γ - π)	Identical	Identical
Triangle angle sum	α + β + γ < π	α + β + γ > π
Total area of space	Infinite	Finite (2π/k)
Circle circumference (radius r)	C > 2πr (C = 2πs sinh(r/s))	C < 2πr (C = 2πs sin(r/s))

🔗 Euclidean geometry as k → 0

• When k = 0, lines in (X₀, G₀) correspond to Euclidean lines (since -1/(kp) = ∞, the unique line through p and q is the Euclidean line).
• The arc-length formula becomes L(r) = ∫ 2|r'(t)| dt, which is twice the usual Euclidean arc-length (a scaling factor).
• Triangles have angle sum equal to π, and right triangles satisfy the Pythagorean theorem.
• As k → 0⁻ (from the hyperbolic side) or k → 0⁺ (from the elliptic side), the hyperbolic and elliptic formulas for distance, circumference, and area all approach the Euclidean formulas (scaled by a factor of 2 for distance).
• Example: The hyperbolic distance d_k(0, x) approaches 2x as k → 0⁻, and the elliptic distance approaches 2x as k → 0⁺.
• Don't confuse: the limiting case k = 0 is Euclidean, but the formulas are scaled; the geometry is still Euclidean in character (e.g., angle sum = π).

🔗 Lobatchevsky's formula as k → 0

• As k → 0⁻, the formula tan(θ/2) = exp(-√|k| d) shows that exp(-√|k| d) → 1, so tan(θ/2) → 1, which means θ → π/2.
• In the limiting Euclidean case, the angle of parallelism θ is always π/2 (90°), independent of distance d.
• This confirms the Euclidean intuition: looking farther down a line always approaches a right angle, regardless of how far away the line is.

🧭 Overview

🧠 One-sentence thesis

The family of geometries (X_k, G_k) unifies hyperbolic, Euclidean, and elliptic geometry under a single framework where the sign of the curvature constant k determines the geometry type and the magnitude of k determines the scale, revealing that all three geometries share identical transformation groups, arc-length formulas, and triangle area formulas.

📌 Key points (3–5)

• Unified framework: One definition covers all three geometry types—hyperbolic (k < 0), Euclidean (k = 0), and elliptic (k > 0)—with identical formulas for transformations, arc-length, and area.
• Sign of k determines geometry type: k < 0 gives hyperbolic geometry (angle sum < π, infinite area, circumference > 2πr); k = 0 gives Euclidean geometry (angle sum = π); k > 0 gives elliptic geometry (angle sum > π, finite area, circumference < 2πr).
• All geometries are homogeneous and isotropic: every point is congruent to every other point, and every direction at a point is equivalent.
• Common confusion—small vs large triangles: all (X_k, G_k) geometries are locally Euclidean—small triangles have angle sums close to π regardless of k; only large triangles reveal curvature.
• Unified Pythagorean theorem: a single formula A(c) = A(a) + A(b) − (k/2π)A(a)A(b) relates the hypotenuse to the legs of a right triangle across all geometries, where A(r) is the area of a circle with radius r.

🏗️ The unified definition

🏗️ Space X_k

The space depends on the sign of k:

Value of k	Space X_k	Description
k < 0	D_k	Open disk (hyperbolic)
k = 0	C	Complex plane (Euclidean)
k > 0	P²_k	Projective plane (elliptic)

• The magnitude of k determines the radius of the disk: s = 1/√|k| (when k ≠ 0).
• Example: as k approaches 0 from below, the hyperbolic disk grows larger; as k approaches 0 from above, the elliptic projective plane grows larger.

🔄 Transformation group G_k

The group G_k consists of all Möbius transformations of the form T(z) = e^(iθ) (z − z₀) / (1 + kz₀z̄), where θ is a real number and z₀ is a point in X_k.

• This single formula works for all k.
• The transformations preserve arc-length and are the "symmetries" of the geometry.

📏 Arc-length formula

For a smooth curve r : [a,b] → X_k, the arc-length is L(r) = ∫ᵇₐ (2|r'(t)|) / (1 + k|r(t)|²) dt.

• This formula is identical across all three geometry types.
• When k = 0, it reduces to twice the usual Euclidean arc-length (a scaling factor).
• Arc-length is an invariant: transformations in G_k preserve it.

📐 Area formula

The area of a polar region R in (X_k, G_k) is A(R) = ∫∫_R (4r) / (1 + kr²)² dr dθ.

• Again, one formula for all k.
• When k = 0, this reduces to the usual Euclidean area formula (scaled).

🛤️ Lines

• The unique line through points p and q is the unique cline through p, q, and −1/(kp).
• When k = 0, −1/(kp) = ∞, so the line is the Euclidean line through p and q.

🔍 How the sign of k shapes geometry

🔺 Triangle angle sums

The fundamental triangle area formula:

kA = (α + β + γ − π), where A is the area and α, β, γ are the angles.

• k > 0 (elliptic): angle sum > π (angles add up to more than 180°).
• k = 0 (Euclidean): angle sum = π (exactly 180°).
• k < 0 (hyperbolic): angle sum < π (less than 180°).
• Example: a triangle with area 0.01 and k = 0.01 has angle sum π + 0.0001 radians, barely distinguishable from Euclidean.

🌍 Space properties

Property	k < 0 (hyperbolic)	k = 0 (Euclidean)	k > 0 (elliptic)
Total area	Infinite	Infinite	Finite
Circumference vs 2πr	Greater	Equal	Less
Angle sum of triangle	< π	= π	> π

• Don't confuse: the type of geometry is determined by the sign of k, not its magnitude.
• The magnitude of k controls the scale at which non-Euclidean effects become noticeable.

🔬 Local Euclidean nature

• Key insight: small triangles in any (X_k, G_k) have angle sums close to π.
• From kA = (α + β + γ − π): if A is close to 0, then α + β + γ ≈ π regardless of k.
• The closer |k| is to 0, the larger a triangle must be to detect deviation from Euclidean geometry.
• Example: in a universe with very small |k|, explorers need to measure very large triangles to detect curvature.

🧮 Unified theorems

🏠 Homogeneity and isotropy

Theorem: For all real numbers k, (X_k, G_k) is homogeneous and isotropic.

Homogeneous means every point is congruent to every other point:

• For any point p in X_k, the transformation T(z) = (z − p)/(1 + kpz̄) maps p to the origin.
• Since all points are congruent to 0, any two points are congruent to each other.

Isotropic means all directions at a point are equivalent:

• For k < 0: G_k contains elliptic Möbius transformations that rotate points around circles centered at p.
• For k = 0: G_k contains transformations T(z) = e^(iθ)(z − z₀), which include all Euclidean rotations about any point.
• For k > 0: G_k contains rotations around type II clines, corresponding to circles centered at p in the projective plane.
• Don't confuse: isotropy means "looks the same in all directions from a point," not "all points are the same" (that's homogeneity).

🔺 Polygon area formula

For a convex n-sided polygon with interior angles α₁, α₂, ..., αₙ and area A: kA = (Σαᵢ) − (n − 2)π.

• Proof method: divide the n-gon into n − 2 triangles; apply the triangle formula to each; sum the results.
• The total angle sum of the triangles equals the interior angle sum of the polygon.
• When k = 0, this reduces to the Euclidean formula: angle sum = (n − 2)π.

⭕ Circle measurements

For a circle centered at 0 through the point x (where 0 < x < s = 1/√|k|):

Area:

A(x) = (4πx²) / (1 + kx²)

Distance from 0 to x:

• k < 0: d_k(0, x) = s ln((s + x)/(s − x))
• k = 0: d₀(0, x) = 2x
• k > 0: d_k(0, x) = 2s arctan(x/s)

Circumference of a circle with radius r:

• k < 0: C(r) = 2πs sinh(r/s)
• k = 0: C(r) = 2πr
• k > 0: C(r) = 2πs sin(r/s)

Area of a circle with radius r:

• k < 0: A(r) = 4πs² sinh²(r/(2s))

• k = 0: A(r) = πr²

• k > 0: A(r) = 4πs² sin²(r/(2s))

• Note: the derivative of area with respect to r equals circumference: dA/dr = C(r).

🔧 The Unified Pythagorean Theorem

🔧 Statement

For a geodesic right triangle with legs of length a and b and hypotenuse of length c: A(c) = A(a) + A(b) − (k/2π)A(a)A(b), where A(r) denotes the area of a circle with radius r.

• When k = 0, this reduces to c² = a² + b² (the classical Pythagorean theorem).
• The formula works for all k, unifying the three geometries.

🧪 Proof sketch

• Assume the right triangle has vertices at z = 0, p = x, and q = yi (on the real and imaginary axes).
• The legs zp and zq are Euclidean segments with a right angle at z.
• By the lemma, A(a) = (4πx²)/(1 + kx²) and A(b) = (4πy²)/(1 + ky²).
• Use the transformation T(z) = (z − p)/(1 + kpz̄) to find the distance d_k(p, q).
• Let t = |T(q)| = |yi − x|/|1 + kxyi|; then A(c) = (4πt²)/(1 + kt²).
• Direct substitution shows A(c) = A(a) + A(b) − (k/2π)A(a)A(b).

📊 Detecting curvature with right triangles

Example: A bug walks a units along a line, turns left 90°, walks another a units, forming a right triangle with legs of equal length and hypotenuse c.

• k = 0: c² = 2a² (Pythagorean theorem).
• k < 0: cosh(√|k| c) = cosh²(√|k| a) (hyperbolic law of cosines).
• k > 0: cos(√k c) = cos²(√k a) (elliptic law of cosines).

Observations:

• When k < 0, the hypotenuse is longer than the Euclidean prediction.
• When k > 0, the hypotenuse is shorter than the Euclidean prediction.
• By measuring a and c, explorers can determine both the type and the value of k.

🌌 Practical applications: measuring curvature

🧭 Triangle measurements

Example: Explorers measure a triangle with angles 29.2438°, 73.4526°, and 77.2886°, and area 8.81 km².

• Sum the angles: α + β + γ = 179.985° ≈ π radians (convert to radians first).
• Use kA = (α + β + γ − π) to solve for k.
• If the angle sum exceeds π, k > 0 (elliptic); if less than π, k < 0 (hyperbolic); if equal, k = 0 (Euclidean).

📏 Right triangle route

Example: Explorers travel 8 units, turn right 90°, travel 8 more units, and find they are 12 units from the start.

• Use the appropriate law of cosines (hyperbolic, Euclidean, or elliptic) to solve for k.
• If c > √(2a²), the geometry is hyperbolic (k < 0).
• If c < √(2a²), the geometry is elliptic (k > 0).
• If c = √(2a²), the geometry is Euclidean (k = 0).

⭕ Circumference measurements

Example: Explorers swing a volunteer in a circle with radius 18 scrambles; he travels 113.4 scrambles.

• Compare the measured circumference C to 2πr.
• If C > 2πr, k < 0 (hyperbolic).
• If C < 2πr, k > 0 (elliptic).
• If C = 2πr, k = 0 (Euclidean).
• Use the appropriate circumference formula to solve for k.

🔍 Angle measurements

Example: Explorers travel 8 units, turn right 90°, travel 8 more units, and measure the angle at the final vertex as 0.789 radians.

• Use the law of cosines to relate the angle to the side lengths and k.
• Solve for k from the measured angle.
• Don't confuse: you need either two sides and the included angle, or two sides and the opposite angle, or similar combinations to determine k uniquely.

🧭 Overview

🧠 One-sentence thesis

All surfaces can be classified into exactly three families—the sphere, handlebody surfaces (spheres with handles), and cross-cap surfaces (spheres with cross-caps)—and any surface can be characterized by two topological invariants: orientability and Euler characteristic.

📌 Key points (3–5)

• What a surface is: a closed, bounded, connected 2-manifold (compact, connected space where every point has a neighborhood that looks like a flat disk).
• How surfaces are built: via the connected sum operation (removing disks from two surfaces and joining the boundaries with a cylinder) or as polygonal surfaces (polygons with edges identified in pairs).
• The classification theorem: every surface is homeomorphic to either the sphere S², a handlebody surface Hₘ (sphere with g handles), or a cross-cap surface Cₘ (sphere with g cross-caps), and no two in this list are the same.
• Common confusion—topology vs geometry: topology studies features unchanged by stretching (like whether a loop can be contracted to a point); geometry studies features that do change (like volume, curvature, area).
• How to tell surfaces apart: use two topological invariants—orientability status (does it contain a Möbius strip?) and Euler characteristic (an integer computed from a cell division).

🧩 Topology vs geometry

🧩 What topology studies

Topological features: those features of a space that remain unchanged if the space is stretched or otherwise continuously deformed.

• Topology ignores size, curvature, and exact shape.
• Example: as a ball inflates, its volume and surface area change (geometric properties), but a loop drawn on the surface always separates it into two disjoint pieces (topological property).

📐 What geometry studies

Geometric features: features of a space that do change under continuous deformations.

• Volume, curvature, surface area, and distance are geometric.
• Example: inflating a ball changes these properties, but the ball's topological structure stays the same.

🔄 Homeomorphism

Homeomorphism: a continuous bijection between two spaces that has a continuous inverse; two spaces are homeomorphic if one can be continuously deformed to look like the other.

• Homeomorphic spaces are "topologically equivalent."
• Example: a circle is homeomorphic to a square—map each point on the square to the circle by extending a ray from the center through the point; this mapping is continuous, one-to-one, onto, and has a continuous inverse.
• Don't confuse: homeomorphic does not mean "geometrically identical"; it means "same topological structure."

🌐 Manifolds and surfaces

🌐 What an n-manifold is

Topological n-manifold: a space where each point has a neighborhood homeomorphic to an open n-ball.

• An open n-ball B^n(p, r) is the set of all points in Rⁿ within distance r of p (not including the boundary).
  • Open 1-ball = open interval on the real line.
  • Open 2-ball = interior of a disk in the plane.
  • Open 3-ball = interior of a sphere in 3D space.
• Example: a circle is a 1-manifold (each point has a neighborhood that looks like an open interval); the surface of a sphere is a 2-manifold (each point has a neighborhood that looks like an open disk).

🎯 What a surface is

Surface: a closed, bounded, and connected topological 2-manifold.

• Closed: if a sequence of points in the space converges to a limit, that limit is also in the space.
• Bounded: the space lives entirely within some finite ball.
• Connected: the space has just one piece (not separated into disjoint parts).
• Equivalently: a surface is a compact, connected 2-manifold.

🔍 Examples of 2-manifolds (which are surfaces?)

Space	Bounded?	Connected?	Closed?	Is it a surface?
Parallel planes in R³	No	No (two pieces)	Yes	No
Open unit disk D = {z ∈ C : |z| < 1}	Yes	Yes	No (boundary points missing)	No
Flat torus (rectangle with opposite edges identified)	Yes	Yes	Yes	Yes

• The flat torus is a surface: every point—interior, edge, or corner—has a neighborhood homeomorphic to an open 2-ball (edge points use two half-disks glued together; corner points use four quarter-disks glued together).

🔗 Building surfaces: connected sum

🔗 The connected sum operation

Connected sum X₁ # X₂: remove an open 2-ball from each of two surfaces X₁ and X₂, then connect the boundaries of the removed disks with a cylinder.

• The result is a new surface: still closed, bounded, connected, and locally looks like a 2-manifold.
• Example: T² # T² (torus connected-sum with torus) produces the two-holed torus H₂.

⚪ The sphere as the identity element

• S² # X = X for any surface X.
• Intuition: removing a disk from a sphere and attaching a cylinder to the boundary gives a shape homeomorphic to a closed disk; gluing this to a surface X just "patches the hole" you made in X.
• The sphere plays the role of 0 in the arithmetic of connected sums.

🏺 Handlebody surfaces Hₘ

Handlebody surface of genus g, denoted Hₘ: the connected sum of g copies of the torus T².

• H₀ = S² (the sphere).
• H₁ = T² (the torus).
• H₂ = T² # T² (two-holed torus).
• Hₘ is topologically equivalent to a sphere with g handles attached.

🔀 Cross-cap surfaces Cₘ

Cross-cap surface of genus g, denoted Cₘ: the connected sum of g copies of the projective plane P².

• A cross-cap is the space obtained by removing an open 2-ball from the projective plane P².
• C₁ = P² (the projective plane).
• C₂ = P² # P² is homeomorphic to the Klein bottle K².
• Cₘ is topologically equivalent to a sphere with g open 2-balls removed and replaced with cross-caps.
• Cross-cap surfaces do not embed in R³ the way handlebody surfaces do.

🔲 Polygonal surfaces and boundary labels

🔲 What a polygonal surface is

Polygonal surface: a surface constructed by starting with a finite number of polygons (each having an even number of edges) and identifying edges in pairs.

• If built from a single polygon, the edge identifications can be encoded in a boundary label.
• Each edge gets a letter and an orientation; edges with the same letter are identified.

🏷️ Boundary labels

• Traverse the polygon boundary counter-clockwise and record each edge's letter.
• Use exponent +1 if walking in the direction of the edge's orientation; use exponent −1 if walking opposite.
• Example: the flat torus has boundary label aba⁻¹b⁻¹ (a rectangle with top/bottom edges identified as "a" and left/right edges identified as "b").

📐 Boundary labels for standard surfaces

Surface	Polygon	Boundary label
Torus T²	Rectangle (4-gon)	aba⁻¹b⁻¹
Two-holed torus H₂	Octagon (8-gon)	(a₁b₁a₁⁻¹b₁⁻¹)(a₂b₂a₂⁻¹b₂⁻¹)
Handlebody Hₘ (g ≥ 1)	Regular 4g-gon	(a₁b₁a₁⁻¹b₁⁻¹)(a₂b₂a₂⁻¹b₂⁻¹)⋯(aₘbₘaₘ⁻¹bₘ⁻¹)
Cross-cap C₂ (Klein bottle)	Square (4-gon)	aba⁻¹b
Cross-cap Cₘ (g ≥ 1)	Regular 2g-gon	a₁a₁a₂a₂⋯aₘaₘ

• Any surface can be constructed as a polygonal surface from a 2m-gon with edges identified in pairs, for some m ≥ 1.

🏆 The classification theorem

🏆 Statement of the theorem

Theorem 7.5.12: Any surface is homeomorphic to the sphere S², a handlebody surface Hₘ with g ≥ 1, or a cross-cap surface Cₘ with g ≥ 1. Moreover, no two surfaces in this list are homeomorphic to each other.

• Every surface fits into exactly one of these families.
• The list is complete (covers all surfaces) and non-redundant (no duplicates).

🔁 Two equivalent ways to think about surfaces

1. Geometric picture: any surface is homeomorphic to the sphere, the sphere with some number of handles attached, or the sphere with some number of cross-caps attached.
2. Combinatorial picture: any surface can be constructed from a 2m-gon with its edges identified in pairs.

🧭 Characterizing surfaces: topological invariants

🧭 What topological invariants are

• Features of a surface that do not change under homeomorphism.
• If two surfaces are homeomorphic, they must have the same topological invariants.
• Two key invariants: orientability status and Euler characteristic.

🔄 Orientability status

🔄 The Möbius strip

Möbius strip: obtained from a rectangle by identifying the left and right edges with a twist.

• The Möbius strip has an orientation-reversing path: a clock rotating clockwise, if it travels along the strip, will eventually return to its starting place rotating counterclockwise.
• The Möbius strip itself is not a surface (it has an edge), but it can be embedded in surfaces.

🔄 Orientable vs non-orientable

Non-orientable surface: a surface that contains a Möbius strip.
Orientable surface: a surface that does not contain a Möbius strip.

• All handlebody surfaces Hₘ (g ≥ 0) are orientable.
• All cross-cap surfaces Cₘ (g ≥ 1) are non-orientable (each contains a Möbius strip).
• Example: the Klein bottle K² = C₂ is non-orientable—a bug traveling in the surface can become mirror-reversed; the torus T² is orientable—no such path exists.

🔢 Euler characteristic

🔢 Cell complexes

n-dimensional cell (n-cell): a subset of a space whose interior is homeomorphic to an open n-ball in Rⁿ.

• 0-cell (vertex): a point.
• 1-cell (edge): interior homeomorphic to an open interval.
• 2-cell (face): interior homeomorphic to an open 2-ball (disk).

Cell complex C: a collection of cells in a space such that (1) the interiors of any two cells are disjoint, and (2) the boundary of each cell is the union of lower-dimensional cells in C.

• A 1-complex (graph) consists of vertices and edges.
• A 2-complex consists of vertices, edges, and faces.
• Example: a 1-complex with seven vertices and five edges; a 2-complex with one vertex and one 2-cell (the entire boundary attached to the single vertex) forms the sphere S².

🔢 Computing the Euler characteristic

• The Euler characteristic is an integer calculated from a cell division (a kind of tiling) of the surface.
• (The excerpt ends before giving the formula, but the setup is: count vertices V, edges E, and faces F in a cell complex covering the surface, then compute χ = V − E + F.)

🔍 Simply connected vs multiconnected

🔍 Simply connected spaces

Simply connected space: a space in which any loop drawn on the surface can be continuously contracted (while staying in the space) to a point.

• Example: the sphere S² is simply connected—any loop can shrink to a point.

🔍 Multiconnected spaces

Multiconnected space: a space in which there exist loops that cannot be contracted to a point.

• Example: the torus T² is multiconnected—a loop going around the torus "like an armband" cannot be contracted to a point while staying on the surface.
• This property distinguishes the sphere from the torus topologically.

🧭 Overview

🧠 One-sentence thesis

Every surface admits exactly one type of homogeneous, isotropic, and metric geometry—elliptic, Euclidean, or hyperbolic—determined by its Euler characteristic, and the Gauss-Bonnet theorem elegantly relates a surface's curvature, area, and topology.

📌 Key points (3–5)

• Which geometry a surface admits: The sphere and projective plane admit elliptic geometry; the torus and Klein bottle admit Euclidean geometry; all other surfaces (handlebodies H_g with g ≥ 2 and cross-cap surfaces C_g with g ≥ 3) admit hyperbolic geometry.
• Why corner angles matter: A polygonal surface can only be homogeneous if the corner angles that come together sum to exactly 360°; otherwise, cone points or saddle points create detectable non-uniformity.
• The Gauss-Bonnet formula: For a surface with constant curvature k, area A, and Euler characteristic χ, the relationship kA = 2πχ holds, linking geometry to topology.
• Common confusion: The type of geometry (elliptic/Euclidean/hyperbolic) is fixed by the surface's shape (Euler characteristic), but the magnitude of curvature (when k ≠ 0) can vary by scaling the underlying space.
• Closed geodesic paths: Even when a surface's geometry type and area are determined, there is freedom in the lengths of closed geodesic paths (straight-line loops that return to the starting point).

🎯 Which surfaces admit which geometries

🌐 Elliptic geometry: sphere and projective plane

The sphere (H₀) and projective plane (C₁) admit elliptic geometry.

• The projective plane is the space of elliptic geometry by construction.
• The sphere inherits elliptic geometry via stereographic projection.
• Both have positive Euler characteristic (χ = 2 for the sphere, χ = 1 for the projective plane).

📐 Euclidean geometry: torus and Klein bottle

The torus (H₁) and Klein bottle (C₂) admit Euclidean geometry.

• Both are built from regular 4-gons (squares) whose edges are identified so that all 4 corners come together at a single point.
• In the Euclidean plane, each corner angle is π/2 radians, so the sum is 2π radians—exactly what is needed for homogeneity.
• Both have Euler characteristic χ = 0.

🌀 Hyperbolic geometry: almost everything else

All handlebody surfaces H_g with g ≥ 2 and all cross-cap surfaces C_g with g ≥ 3 admit hyperbolic geometry.

• These surfaces are built from regular n-gons where n ≥ 6, with all n corners coming together at a single point.
• In the Euclidean plane, a regular n-gon has interior angle (n − 2)π / n radians, so the corner angle sum is (n − 2)π radians, which exceeds 2π when n ≥ 6.
• Placing the polygon in the hyperbolic plane and adjusting its size allows the corner angles to shrink until their sum equals exactly 2π radians.
• Example: C₃ is built from a hexagon in the hyperbolic plane with each corner angle equal to 60° (so six corners sum to 360°).

🔧 Fixing corner angles to achieve homogeneity

🎪 The cone-point and saddle-point problem

• A polygonal surface can fail to be homogeneous at the point where corners come together.
• If the corner angles sum to less than 360°, the surface has a cone point (like the tip of a cone).
• If the corner angles sum to more than 360°, the surface has a saddle point (like a mountain pass).
• A two-dimensional bug could detect these special points by measuring triangles or circles, violating homogeneity.

🛠️ How to fix the angles

• To shrink corner angles: place the polygon in the hyperbolic plane, where polygons can have smaller interior angles than in Euclidean space.
• To expand corner angles: place the polygon in the projective plane (elliptic geometry), where polygons can have larger interior angles.
• By choosing the right size and space, we can make the corner angles sum to exactly 360°.

Example: In Example 7.6.3, a hexagon with boundary label abcabc has corners that come together in groups of two. In the Euclidean plane, each corner is 120°, so two corners sum to 240° < 360°, creating cone points. Placing the hexagon in the projective plane and expanding it until each corner angle is 180° (so two corners sum to 360°) eliminates the cone points; the resulting surface is the projective plane itself, admitting elliptic geometry.

🧮 The Gauss-Bonnet theorem

📏 The formula

Gauss-Bonnet theorem: For a surface with constant curvature k, area A, and Euler characteristic χ, the relationship kA = 2πχ holds.

• This formula elegantly connects:
  • Geometry (curvature k and area A)
  • Topology (Euler characteristic χ)
• It applies to all surfaces with constant curvature.

🔍 What the formula tells us

Surface	Euler characteristic χ	Curvature k	Area A
Sphere (H₀)	2	positive	4π / k
Projective plane (C₁)	1	positive	2π / k
Torus (H₁)	0	0	any
Klein bottle (C₂)	0	0	any
H_g (g ≥ 2)	2 − 2g	negative	2π(2 − 2g) / k
C_g (g ≥ 3)	2 − g	negative	2π(2 − g) / k

• The sign of the curvature matches the sign of the Euler characteristic.
• For surfaces with k ≠ 0, the area is determined by χ and k.
• For surfaces with k = 0 (torus and Klein bottle), the formula reduces to 0 = 0, and the area is unrestricted.

🌌 Application: deducing the shape of a universe

• A two-dimensional cosmologist who measures the curvature k and total area A of her universe can compute χ = kA / (2π).
• If χ is 2 or an odd integer, the shape is uniquely determined.
• If χ < 2 and even, there are two possible shapes: one orientable (handlebody H_g) and one non-orientable (cross-cap C_g).

Don't confuse: The Gauss-Bonnet formula determines the product kA, not k and A separately. The magnitude of curvature can vary (by scaling the underlying space), but once k is fixed, A is determined by χ.

🧵 Flexibility in geometric measurements

🎨 Curvature scale can vary (when k ≠ 0)

• The type of geometry (elliptic, Euclidean, or hyperbolic) is fixed by the surface's Euler characteristic.
• However, the magnitude of curvature can vary by placing the polygonal surface in a scaled version of the projective plane or hyperbolic disk.
• Example: The surface C₃ has constant curvature −1 if placed in the standard hyperbolic plane, but it can have curvature k = −8 if placed in the hyperbolic plane with radius 1 / √8.

🔁 Closed geodesic paths

A closed geodesic path in a surface is a path that follows a straight line (in the underlying geometry) and starts and ends at the same point.

• Even when a surface's geometry type and area are determined, there is freedom in the lengths of closed geodesic paths.
• Example: A torus can be built from any rectangle in the Euclidean plane. The width and length of the rectangle correspond to the lengths of two closed geodesic paths. These dimensions can vary, giving different tori with the same geometry type but different "sizes" in different directions.
• For hyperbolic surfaces, even though the area is fixed (by Gauss-Bonnet), the lengths of closed geodesic paths can still vary.

🩳 Building surfaces from pairs of pants

• A "pair of pants" is a surface with three boundary circles.
• Any pair of pants can be cut into two hexagons.
• Example: A two-holed torus (H₂) can be constructed from two pairs of pants, which decompose into four hexagons.
• In the hyperbolic plane, there exists a right-angled hexagon for each triple of alternate side lengths (a, b, c) (by Theorem 5.4.19).
• This means there is a family of two-holed tori, one for each choice of six seam lengths (a₁, a₂, a₃, b₁, b₂, b₃), all with the same geometry type but different closed geodesic path lengths.

🌍 Surfaces in three-dimensional space

🏔️ Positive curvature is unavoidable in R³

• If a surface X lives in three-dimensional space R³, it must have at least one point with positive curvature.
• Why: The surface is bounded, so there exists a sphere centered at the origin that contains the entire surface. Shrink this sphere until it just touches the surface at some point. At that point, the curvature of the surface matches the curvature of the sphere, which is positive.
• Only the sphere can be embedded in R³ with constant positive curvature everywhere.
• Most constant-curvature surfaces (especially hyperbolic ones) cannot be embedded in R³.

Don't confuse: The handlebody surfaces in Figure 7.5.10 live in R³, but they do not have constant curvature—they have points of varying curvature, including at least one point of positive curvature.

🧭 Overview

🧠 One-sentence thesis

Quotient spaces provide a systematic way to construct surfaces and higher-dimensional manifolds by identifying points in a space according to an equivalence relation, allowing us to build complex geometric objects like tori and Klein bottles from simpler spaces through group actions.

📌 Key points (3–5)

• What quotient spaces do: they "divide out" redundancies in a mapping by grouping points that should be identified together into equivalence classes.
• How they're built: start with an equivalence relation on a set (reflexive, symmetric, transitive), then form a new set whose elements are the equivalence classes.
• Orbit spaces from groups: when a group of transformations acts on a space, points that can be mapped to each other form natural equivalence classes called orbits.
• Common confusion: not every quotient inherits geometry—the group must be fixed-point free and properly discontinuous for the quotient to inherit a metric from the parent space.
• Why it matters: all surfaces can be viewed as quotients of the Euclidean plane, hyperbolic plane, or sphere by appropriate groups of isometries.

🎯 Motivation and basic idea

🎯 The cylinder example

The section opens with a concrete motivation: bending a square sheet of paper into a cylinder.

• The map p from the unit square I² to a cylinder C is continuous, onto, and almost one-to-one.
• It fails to be one-to-one only where edges are glued: points (0, y) and (1, y) both map to (1, 0, y).
• The quotient space construction formalizes "dividing out" these redundancies so the result becomes genuinely one-to-one.

Example: When you roll paper into a cylinder, the left and right edges meet—quotient spaces formalize this identification process mathematically.

🧩 Why surfaces are quotient spaces

Every surface can be viewed as a quotient space M/G where:

• M is the Euclidean plane C, hyperbolic plane D, or sphere S²
• G is a group of isometries in the corresponding geometry
• M is called the universal covering space of M/G

🔗 Equivalence relations

🔗 What a relation is

A relation on a set S is a subset R of S × S.

• In other words, a collection of ordered pairs (a, b) where both elements are in S.
• We write aRb or a ∼ b to mean (a, b) is in the relation.

✅ Equivalence relations

An equivalence relation on a set A is a relation ∼ satisfying three conditions:

1. Reflexivity: x ∼ x for all x ∈ A
2. Symmetry: if x ∼ y then y ∼ x
3. Transitivity: if x ∼ y and y ∼ z then x ∼ z

Example from the text: Define z ∼ w in the complex numbers if Re(z) − Re(w) is an integer and Im(z) = Im(w). This groups together all complex numbers that differ by an integer in their real part but have the same imaginary part.

📦 Equivalence classes

The equivalence class of a, denoted [a], is the subset of all elements in A related to a: [a] = {x ∈ A | x ∼ a}.

Key property (Lemma 7.7.4): Two equivalence classes either have no elements in common or are completely equal—they cannot partially overlap.

Why this matters: Equivalence classes partition the set into non-overlapping pieces, each representing a "single point" in the quotient.

🗂️ Partitions

A partition of a set A consists of non-empty subsets that are mutually disjoint and have union equal to A.

An equivalence relation naturally creates a partition via its equivalence classes.

🏗️ Building quotient sets

🏗️ The quotient set definition

If ∼ is an equivalence relation on a set A, the quotient set of A by ∼ is A/∼ = {[a] | a ∈ A}.

• The elements of A/∼ are the distinct equivalence classes.
• This is a new set whose "points" are entire equivalence classes from the original set.

🎨 Fundamental domains

A fundamental domain of a quotient set contains a representative of each equivalence class and at most one representative in its interior.

• Think of it as a "sample region" that shows one copy of each point type.
• Boundary points may represent multiple equivalence classes (they're where identifications happen).

Example: For the cylinder built from a square, the square itself is a fundamental domain—every point on the cylinder appears once in the interior, and left/right edges are identified.

🌀 Orbit spaces from group actions

🌀 The natural equivalence relation from a group

Given a geometry (X, G), define x ∼_G y if and only if T(x) = y for some T ∈ G.

This is an equivalence relation because:

• Reflexive: the identity transformation is in G
• Symmetric: if T(x) = y then T⁻¹(y) = x, and inverses are in G
• Transitive: compositions of transformations in G are in G

🔄 Orbits

The orbit of x consists of all points in X to which x can be mapped under transformations of G: [x] = {y ∈ X | T(x) = y for some T ∈ G}.

• The orbit is the set of all points congruent to x in the geometry (X, G).
• If the geometry is homogeneous, every orbit is the entire space X (not interesting for quotients).

🎯 Groups of homeomorphisms and isometries

A group of homeomorphisms of X means each transformation in G is continuous.

A group of isometries means each transformation preserves distance.

The quotient X/G is called an orbit space when G is a group of homeomorphisms.

📐 Examples of orbit spaces

📐 The topological cylinder

Consider the horizontal translation T₁(z) = z + 1 of the complex plane C.

The group generated by T₁ is:

• ⟨T₁⟩ = {Tₙ(z) = z + n | n ∈ ℤ}
• The orbit of point p is [p] = {p + n | n ∈ ℤ}
• Fundamental domain: the vertical strip 0 ≤ Re(z) ≤ 1

Geometric picture: We're "rolling up" the plane into an infinitely tall cylinder by identifying points that differ by integer horizontal translations.

🔲 The torus from two translations

Let ⟨T_a, T_bi⟩ be the group generated by horizontal translation T_a(z) = z + a and vertical translation T_bi(z) = z + bi.

• An arbitrary transformation has the form T(z) = z + (ma + nbi) where m, n are integers.
• Fundamental domain: rectangle with corners 0, a, a + bi, bi.
• The quotient C/⟨T_a, T_bi⟩ is homeomorphic to the torus.

Key insight: Boundary points are identified in pairs, matching the polygonal surface representation of the torus from earlier sections.

🎡 A finite rotation group (not a good quotient)

The rotation R_π/2 by π/2 about the origin generates a finite group:

• ⟨R_π/2⟩ = {1, R_π/2, R_π, R_3π/2}
• The orbit of 0 is just {0}; the orbit of any other point has four elements.

Why this fails: The group is not fixed-point free (all rotations fix the origin), so the quotient cannot inherit Euclidean geometry properly.

🔐 When quotients inherit geometry

🔐 Fixed-point free condition

A group G is fixed-point free if each isometry in G (other than the identity) has no fixed points.

🔐 Properly discontinuous condition

A group G is properly discontinuous if every x in X has an open ball U_x whose images under all isometries in G are pairwise disjoint.

📏 Distance in quotient spaces

When G is a fixed-point free, properly discontinuous group of isometries, the orbit space inherits a metric:

d([u], [v]) = min{distance(z, w) | z ∈ [u], w ∈ [v]}

Interpretation: The distance between two equivalence classes is the shortest distance between any representative from the first class and any representative from the second class.

Example: In the torus, to find the distance between two points, consider all possible paths including those that "wrap around" the identified edges—the shortest such path gives the distance.

🌐 Building surfaces from hyperbolic quotients

🌐 The projective plane

Let T_a: S² → S² be the antipodal map T_a(P) = −P.

• This is a fixed-point free isometry of the sphere.
• The group ⟨T_a⟩ has just two elements: T_a and the identity.
• The quotient S²/⟨T_a⟩ is the projective plane P².

🔷 The 2-holed torus from an octagon

Build a regular octagon in the hyperbolic plane D with interior angles π/4.

For each pair of identified edges (a, b, c, d):

• Find a hyperbolic isometry that maps one edge to its partner (respecting orientation).
• These isometries are constructed by composing two hyperbolic reflections.
• Since the reflection lines don't intersect, the result is a hyperbolic translation with no fixed points in D.

The group generated by T_a, T_b, T_c, T_d creates a quotient D/G that:

• Is homeomorphic to H₂ (the 2-holed torus)
• Inherits hyperbolic geometry
• Has the octagon as a fundamental domain

Tiling property: Moving copies of the octagon by isometries in the group tiles all of D without gaps or overlaps.

🎯 General construction for surfaces

All surfaces H_g (g ≥ 2) and C_g (g ≥ 3) can be built as quotients of D by:

1. Starting with a properly sized polygon in D (corner angles sum to 2π)
2. Making identified edges have equal length
3. Finding hyperbolic isometries that map paired edges to each other
4. Generating a group from these isometries

The resulting quotient is homeomorphic to the surface and inherits hyperbolic geometry.

🎨 Dirichlet domains

🎨 Definition and construction

The Dirichlet domain with basepoint x consists of all points y in M such that d(x, y) ≤ d(x, T(y)) for all T in G.

Interpretation: The Dirichlet domain at x contains all points that are at least as close to x as they are to any image of themselves under the group.

Visualization method:

1. Start with a small circle centered at x
2. Construct equal circles at all points in the orbit of x
3. Inflate all circles simultaneously
4. When circles touch, let them press into each other forming geodesic boundary edges
5. When the circle fills the entire surface, you have the Dirichlet domain

🎨 Properties

• The Dirichlet domain is itself a fundamental domain for M/G.
• It represents the fundamental domain from a local inhabitant's perspective.
• It's a polygon whose edges are geodesics in the local geometry.

Don't confuse: The Dirichlet domain may have a different shape than the original polygon used to construct the surface, and its shape can vary from basepoint to basepoint.

🎨 Klein bottle example

The text describes a Klein bottle built from a hexagon in C with corners at 0, 1, 2, 2+i, 1+i, i.

• The group is generated by vertical translation T(z) = z + 2i and a reflection-translation r(z) = z̄ + (1 + 2i).
• The quotient C/Γ is a Klein bottle with Euclidean geometry.
• The Dirichlet domain shape varies: at points on Im(z) = 1/2 it's a square; at points on Im(z) = 0 it's a rectangle.

Key insight: Even though the fundamental domain is a hexagon, the Dirichlet domain at different basepoints can be squares or rectangles—the "natural" fundamental domain depends on where you stand.

🧭 Overview

🧠 One-sentence thesis

The universe at any fixed time can be modeled as a 3-manifold admitting one of three geometries—Euclidean, hyperbolic, or elliptic—and many candidate shapes can be constructed by identifying faces of polyhedra (like cubes or dodecahedra) in pairs, with the geometry determined by whether corner angles must shrink, expand, or remain unchanged for the corners to fit together perfectly.

📌 Key points (3–5)

• Three geometries for 3-manifolds: Just as in two dimensions, a homogeneous and isotropic 3-manifold admits Euclidean, hyperbolic, or elliptic geometry, distinguished by triangle angle sums (equal to π, less than π, or greater than π, respectively).
• Construction via face identifications: Many 3-manifolds are built by taking a 3-dimensional solid (a cube, dodecahedron, or prism) and identifying opposite faces in pairs; the geometry depends on how corner angles must adjust to form a perfect patch of space.
• Euclidean 3-manifolds are rare: Only ten compact, connected Euclidean 3-manifolds exist (six orientable, four non-orientable), including the 3-torus and the Hantzsche-Wendt manifold.
• Common confusion—geometry vs topology: The same topological shape (e.g., a dodecahedron with face identifications) can admit different geometries depending on the twist angle used in the identification; a one-tenth twist yields elliptic geometry (Poincaré dodecahedral space), while a three-tenths twist yields hyperbolic geometry (Seifert-Weber space).
• Hyperbolic uniqueness: Unlike surfaces, a connected orientable 3-manifold supports at most one hyperbolic structure, so geometry and shape are rigidly linked in the hyperbolic case.

🌐 The three geometries in three dimensions

🌐 Euclidean geometry in ℝ³

Euclidean geometry is the geometry of our experience in three dimensions; planes look like infinite tabletops, and lines in space are Euclidean straight lines.

• The space is ℝ³, the set of ordered triples (x, y, z).
• Distance between points v = (x₁, y₁, z₁) and w = (x₂, y₂, z₂) is given by the Euclidean distance formula: square root of (x₁ − x₂)² + (y₁ − y₂)² + (z₁ − z₂)².
• Transformations: rotations about lines and translations by vectors; any transformation can be expressed as a screw motion (a translation along a line followed by a rotation about that same line).
• Example: A rotation about a line or a pure translation is a special case of a screw motion.

🌀 Hyperbolic geometry in H³

Hyperbolic three-space H³ consists of all points inside the unit ball in ℝ³, bounded by the "sphere at infinity" (the unit 2-sphere), which is not part of H³.

• The space: H³ = {(x, y, z) ∈ ℝ³ | x² + y² + z² < 1}.
• Lines (geodesics): arcs of circles or lines in H³ that meet the sphere at infinity orthogonally; Euclidean lines through the origin are also hyperbolic lines.
• Planes: portions of spheres or Euclidean planes inside H³ that meet the sphere at infinity at right angles.
• Transformations: generated by inversions about spheres orthogonal to the sphere at infinity; composing two such inversions gives an orientation-preserving transformation.
• Inversion about a sphere: For a sphere S centered at v₀ with radius r, the point v∗ symmetric to v satisfies |v − v₀| · |v∗ − v₀| = r².
• Any plane in H³ inherits two-dimensional hyperbolic geometry.

🔵 Elliptic geometry on S³

Three-dimensional elliptic geometry is derived from the 3-sphere S³, the set of all points in 4-dimensional space one unit from the origin.

• The space: S³ = {(x, y, z, w) ∈ ℝ⁴ | x² + y² + z² + w² = 1}.
• Geodesics and planes: great circles (circles of maximum diameter) are geodesics; great 2-spheres (2-spheres of maximum diameter) inherit two-dimensional elliptic geometry.
• Quaternions: The transformation group is conveniently described using quaternions.
  • A quaternion has the form a + bi + cj + dk, where a, b, c, d are real and i² = j² = k² = −1, with ijk = −1.
  • The modulus is |q| = √(a² + b² + c² + d²); a unit quaternion has modulus 1.
  • The conjugate q∗ = a − bi − cj − dk satisfies q · q∗ = |q|².
• Transformations: For unit quaternions u and v, the map T(q) = uqv is a typical transformation; these preserve antipodal points (distinct points on the same Euclidean line through the origin in ℝ⁴).
• Model: S³ can be visualized as two solid balls in ℝ³ whose boundary 2-spheres are identified point for point.

🔺 Triangle angle sums distinguish geometries

Geometry	Triangle angle sum	Implication for polyhedra
Euclidean (ℝ³)	Equal to π radians	Corner angles remain unchanged
Elliptic (S³)	Exceeds π radians	Corner angles must increase (inflate)
Hyperbolic (H³)	Less than π radians	Corner angles must decrease (shrink)

• Placing a dodecahedron in H³ shrinks the corner angles; placing it in S³ increases them.
• This flexibility is useful: many 3-manifolds are constructed by identifying faces of a polyhedron in pairs, and the geometry is determined by how corner angles must adjust for corners to form a perfect patch of space.

🧊 Euclidean 3-manifolds

🧊 The 3-torus T³

The 3-torus can be thought of as a cubical room where flying through the ceiling brings you back through the floor, through the left wall brings you back through the right wall, and through the front wall brings you back through the back wall.

• Face identifications: Each point on the top face is identified with the corresponding point on the bottom face; similarly for left/right and front/back.
• Corners: All eight corners come together at a single point, and the angles form a perfect patch of three-dimensional space without adjustment.
• Orbit space construction: T³ is the quotient ℝ³/Γ, where Γ is the group generated by unit translations Tₓ, Tᵧ, Tᵤ in the x, y, z directions.
  • Any transformation in Γ has the form T(x, y, z) = (x + a, y + b, z + c) for integers a, b, c.
  • The fundamental domain is the unit cube I³ = {(x, y, z) | 0 ≤ x, y, z ≤ 1}.
• Geometry: The 3-torus inherits Euclidean geometry.
• A slice of the 3-torus parallel to a face is a (two-dimensional) torus.

🔲 The Hantzsche-Wendt manifold

This manifold can be constructed from two identically sized cubes sharing a face, forming a box with 10 faces identified in pairs.

• Face identifications:
  • Top and bottom faces are identified as in the 3-torus.
  • Top back and bottom back faces: each point is identified with its reflection across the vertical bisector of the face; similarly for top front and bottom front.
  • Top left and bottom right faces are identified with a 180° rotation; similarly for bottom left and top right.
• Orbit space construction: The manifold is the quotient of ℝ³ by the group generated by three screw motions T₁, T₂, T₃.
  • Each screw motion consists of a 180° rotation about a segment followed by a unit translation along that segment.
  • The ten-faced solid is a fundamental domain.
• Geometry: Inherits Euclidean geometry.
• Repeated applications of the transformations produce image cubes covering half of ℝ³ in a checkerboard pattern.

🔄 Four more Euclidean 3-manifolds

Manifold	Fundamental domain	Face identification
Quarter turn	Cube	Front/back with 90° rotation; other pairs direct
Half turn	Cube	Front/back with 180° rotation; other pairs direct
One-third turn	Hexagonal prism	Opposite parallelograms direct; hexagons with 120° rotation
One-sixth turn	Hexagonal prism	Opposite parallelograms direct; hexagons with 60° rotation

• Classification: There are exactly ten compact, connected Euclidean 3-manifolds (six orientable, four non-orientable).
• The 3-torus, Hantzsche-Wendt manifold, and the four above are the six orientable ones.

🌌 Elliptic 3-manifolds

🌌 The 3-sphere S³

The 3-sphere is the simplest elliptic 3-manifold; Einstein assumed the universe had this shape when he first solved his equations for general relativity.

• Why Einstein chose it: A static, finite, simply connected universe without boundary was aesthetically appealing and cleared up paradoxes arising in an infinite universe.
• Limitation: General relativity fixes only the local nature of space, not the global shape.
• Alternative shape: In 1917, Willem De Sitter noticed that Einstein's solutions admitted three-dimensional elliptic space obtained from S³ by identifying antipodal points.
• Slicing S³: Slicing the 2-sphere with a plane in ℝ³ gives a circle (or point); slicing S³ with ℝ³ gives a 2-sphere (or point).
  • A great 2-sphere (maximum diameter) is a geodesic surface.
• Stereographic projection: S³ can be mapped to ℝ³ with a point at infinity (R̂³) via stereographic projection from the north pole N = (0, 0, 0, 1).
  • For P = (a, b, c, d) on S³, φ(P) = (a/(1−d))i + (b/(1−d))j + (c/(1−d))k if d ≠ 1; φ(N) = ∞.
  • Antipodal points P and −P on S³ map to points q and u in R̂³ satisfying q · u∗ = −1.

🔷 The Poincaré dodecahedral space

Start with a dodecahedron in ℝ³ and identify opposite faces with a one-tenth clockwise twist (rotation by 2π/10 radians).

• Face identification: Each pair of opposite faces is identified with a 36° rotation (one-tenth of a full turn).
• Corners: The twenty corners come together in five groups of four.
• Geometry: The corner angles are too small in Euclidean space; placing the dodecahedron in elliptic space inflates the angles to form a perfect patch.
• Orbit space: The Poincaré dodecahedral space is a quotient of S³ by a group of isometries; the dodecahedron is a fundamental domain.
• Tiling: S³ is tiled by 120 copies of this dodecahedron.
• Cosmology: This manifold has been considered as a model for the shape of our universe.

🔶 Lens spaces L(p, q)

The lens spaces form an infinite family of elliptic 3-manifolds, constructed from a solid ball by identifying points on its boundary 2-sphere.

• Parameters: p > q are positive integers with no common factor.
• Construction: Each point (z, t) on the boundary 2-sphere (where z is complex, t is real, and |z|² + t² = 1) is identified with (e^(2πqi/p) z, −t).
  • The north pole (0, 1) is identified with the south pole (0, −1).
  • A point in the northern hemisphere is identified with a single point in the southern hemisphere (reflected across the xy-plane, then rotated around the z-axis by 2πq/p radians).
  • Each point on the equator is identified with p − 1 other points.
• Cell division: L(p, q) can be obtained from a cell division of the solid ball with p + 2 vertices (north pole, south pole, and p equally spaced vertices on the equator), 3p edges, and 2p triangular faces.
• Face identification: Face Nᵢ (in the northern hemisphere) is identified with face Sᵢ₊ᵩ (in the southern hemisphere), where the sum is modulo p.
• Example: In L(5, 2), face N₀ is identified with S₂, N₁ with S₃, N₂ with S₄, N₃ with S₀, and N₄ with S₁.

🔢 Classification of elliptic 3-manifolds

• There are infinitely many different types of elliptic 3-manifolds.
• All elliptic 3-manifolds are orientable.

🌀 Hyperbolic 3-manifolds

🌀 The Seifert-Weber space

Identify opposite faces of a dodecahedron with a three-tenths clockwise turn (rotation by 3 · 2π/10 radians).

• Face identification: Each pair of opposite faces is identified with a 108° rotation (three-tenths of a full turn).
• Corners: All twenty corners come together at a single point.
• Geometry: The corner angles must be drastically shrunk; placing the dodecahedron in hyperbolic space H³ achieves this.
• Result: The Seifert-Weber space is a hyperbolic 3-manifold.

🔒 Uniqueness of hyperbolic structure

A connected orientable 3-manifold supports at most one hyperbolic structure.

• Contrast with surfaces: In two dimensions, a two-holed torus can have varying geometric properties (e.g., length of minimal closed geodesics) even though the area is fixed by the Gauss-Bonnet formula.
• No such freedom in 3D: If a 3-manifold admits hyperbolic geometry, that structure is unique.
• Don't confuse: This is a major difference from the two-dimensional case, where the same topological surface can have different hyperbolic structures.

🔀 Common confusion: twist angle determines geometry

• The same polyhedron (e.g., a dodecahedron) with different face-identification twists yields different geometries:
  • One-tenth twist (Poincaré dodecahedral space): corners need inflated angles → elliptic geometry.
  • Three-tenths twist (Seifert-Weber space): corners need shrunk angles → hyperbolic geometry.
• The twist angle determines whether corners fit together in Euclidean, elliptic, or hyperbolic space.

🔭 Cosmic topology context

🔭 Why study 3-manifolds

• Assumption: The shape of the universe at any fixed time is a 3-manifold.
• Evidence: The universe appears isotropic and homogeneous on the largest scales.
• Implication: Just as in two dimensions, the universe admits one of three geometries: hyperbolic, elliptic, or Euclidean.
• Goal: Cosmic topology seeks to determine the global shape of the universe through observational techniques (discussed in later sections).

🔭 Historical note

• Einstein's choice: The 3-sphere S³ was Einstein's initial model for a static, finite universe.
• De Sitter's alternative: Three-dimensional elliptic space (S³ with antipodal points identified) is another solution to Einstein's equations.
• Limitation: General relativity constrains local geometry but does not fix the global shape.

🧭 Overview

🧠 One-sentence thesis

Cosmic crystallography detects whether the universe is finite by searching for spikes in pair separation histograms that reveal repeated images of the same objects caused by the universe wrapping around itself.

📌 Key points (3–5)

• What the method does: analyzes catalogs of astronomical objects by computing distances between all pairs and looking for statistically significant spikes in the histogram.
• Why spikes appear: in a finite universe with certain geometries, we see multiple images of the same object at characteristic distances determined by the universe's dimensions.
• Clifford translations: transformations that move every point the same distance; only universes built from these produce detectable spikes.
• Common confusion: not all finite universes reveal themselves this way—hyperbolic manifolds don't produce spikes because hyperbolic isometries are not Clifford translations.
• Current status: no statistically significant spikes have been found in real catalogs to date, suggesting our observable universe is either simply connected or too large to detect topology.

🔭 The basic idea: seeing multiple images

🌌 How a finite universe creates repeated views

• If we live in a finite universe (e.g., a torus), light can travel in different directions and reach the same object by wrapping around the space.
• Example: In a two-dimensional torus universe, an observer at point E can see galaxy G in three different directions because light follows different paths that wrap around the torus.

📏 The observable universe constraint

Observable radius (r_obs): the distance to which we can see, which defines the radius of our observable universe.

• To see multiple images of the same object, the diameter 2r_obs must exceed some length dimension of the universe.
• Practical complication: different lines of sight have different lengths, so we see the same object at different times in its evolution—galaxies evolve dramatically, making recognition difficult.

🗺️ The tiling visualization technique

• To find all possible directions to view an object, tile the plane (or space) with identical copies of the fundamental domain.
• Place Earth and other objects at the same relative positions in each copy.
• The visual boundary (circle of radius r_obs) shows which images are visible.

📊 The pair separation histogram method

📐 Building the histogram

• Start with a catalog of N similar objects that don't evolve too rapidly (e.g., galaxy superclusters).
• Compute the distance between each pair of sources: N(N−1)/2 total distances.
• Must assume a geometry (Euclidean, hyperbolic, or elliptic) to compute these distances.
• Plot a histogram of all these distances—this is the pair separation histogram (PSH).

📈 What to expect in different universes

Universe type	PSH appearance	Reason
Simply connected (infinite)	Smooth Poisson distribution	No repeated images; distances are random
Finite with Clifford translations	Spikes at characteristic distances	Multiple images of same objects at fixed separations

🎯 Interpreting the spikes

Example from torus simulation:

• Spike at distance 10: horizontal dimension of the torus—many objects have images displaced horizontally by this amount.
• Spike at distance 14: vertical dimension—objects have images displaced vertically.
• Spike at distance ~17.2: diagonal displacement, calculated as square root of (10² + 14²).
• The largest spike corresponds to the smallest dimension (more pairs visible at that separation).

⚠️ Critical assumption dependency

• The PSH depends on which geometry you assume when computing distances.
• If you assume Euclidean geometry but the universe is hyperbolic, the spikes vanish.
• Don't confuse: finding spikes doesn't uniquely determine the shape—different manifolds might produce identical spike patterns.

🔄 Clifford translations and method limitations

🔄 What makes a Clifford translation

Clifford translation: a transformation T of a metric space where d(p, T(p)) = d(q, T(q)) for all points p and q—every point gets moved the same distance.

• Example of Clifford translation: any translation in Euclidean space moves every point by the same vector length.
• Example of non-Clifford: rotation about the origin—points farther from the origin move greater distances.

🚫 Which universes cosmic crystallography can detect

Geometry	Detectable by cosmic crystallography?	Reason
Euclidean (all 10 manifolds)	Yes	Built from translations (Clifford)
Some elliptic manifolds	Yes	Some isometries are Clifford translations
Hyperbolic manifolds	No	Hyperbolic isometries are not Clifford translations

• This is a fundamental limitation: the method cannot detect any hyperbolic 3-manifold.
• Even for detectable geometries, we must see far enough (large enough r_obs) to observe the finite dimensions.

🔬 Advanced methods: Type I vs Type II pairs

🔢 Two types of recurring distances

Type II pairs: {p, T(p)} where T is an isometry from the group generating the manifold.

• These are what basic cosmic crystallography detects.
• Produce spikes only if T is a Clifford translation.
• Example: an object and its image displaced by one fundamental domain width.

Type I pairs: any pair {p, q} of distinct points.

• If we see images T(p) and T(q) in another copy of the fundamental domain, then d(p, q) = d(T(p), T(q)) because transformations preserve distance.
• This distance appears multiple times in the PSH but doesn't produce statistically significant spikes.

📦 Collecting correlated pairs (CCP) method

A more sensitive approach that attempts to detect Type I pairs:

1. Compute all P = N(N−1)/2 distances and order them from smallest to largest.
2. Calculate Δᵢ = (i+1)st distance minus ith distance for i = 1 to P−1.
3. Count Z = number of Δᵢ values equal to zero (or less than some small threshold ε).
4. Compute ratio R = Z/(P−1), which measures the proportion of repeated distances.

• R serves as a single-number measure of the likelihood of living in a multi-connected (finite) universe.
• Practical adjustment: in real catalogs with measurement errors, use Δᵢ < ε instead of Δᵢ = 0.
• This method might detect any 3-manifold regardless of geometry, overcoming the Clifford translation limitation.

🌐 Current observational status

📡 Results to date

• No statistically significant spikes have been found in pair separation histograms computed from real astronomical catalogs.
• This suggests either:
  • Our universe is simply connected (infinite), or
  • The finite dimensions are larger than our observable universe, or
  • Our universe has hyperbolic geometry (undetectable by this method).

🔍 Ongoing refinements

• More sensitive methods beyond basic cosmic crystallography have been proposed.
• The collecting correlated pairs method is one example that might detect topologies regardless of underlying geometry.
• Accurate measurement of astronomical distances remains a fundamental challenge for all these methods.

🧭 Overview

🧠 One-sentence thesis

If the universe is finite and the last scattering surface is large enough relative to the fundamental domain, matching circles with identical temperature distributions should appear in the cosmic microwave background, revealing the universe's topology.

📌 Key points (3–5)

• What the last scattering surface (LSS) is: a giant 2-sphere centered on us, from which cosmic microwave background radiation reaches us after traveling the same distance from every direction since ~350,000 years after the big bang.
• How matching circles arise: if the LSS is large relative to the fundamental domain of a finite universe, it intersects copies of itself, creating circles with matching temperature distributions.
• Detection method: the circles-in-the-sky method searches for pairs of matching circles on the LSS; it can detect any compact manifold regardless of geometry and does not require assumptions about the metric.
• Common confusion: the injectivity radius vs observable radius—matching circles appear only when the observable radius r_obs exceeds the injectivity radius r_inj (half the shortest closed geodesic path).
• Current results: no matching circles have been found, placing a lower bound on the universe's size (injectivity radius greater than ~78 billion light-years).

🌌 The cosmic microwave background and the last scattering surface

🔥 Origin of the CMB radiation

• Immediately after the big bang, the universe was too hot for normal matter to form; photons constantly collided with free electrons and could not move freely.
• About 350,000 years after the big bang, the universe expanded and cooled enough for light to travel unimpeded—this free radiation is the cosmic microwave background (CMB).
• The universe has since expanded and cooled further, stretching this radiation to microwave wavelengths (~1–2 millimeters).

🌐 What the last scattering surface is

Last scattering surface (LSS): the surface of a giant 2-sphere centered on the observer, from which all detectable CMB radiation at a given instant has traveled the same distance (because it has all traveled at the same speed for the same amount of time).

• Everyone in the universe has their own LSS with the same radius, and this radius grows over time.
• The CMB temperature is remarkably uniform across the LSS—constant to a few parts in 100,000, smoother than a billiard ball.
• Slight temperature variations exist due to imbalances in the early universe's matter distribution; these variations might reveal the universe's shape.

🔄 How matching circles appear in a finite universe

🧩 Tiling space with fundamental domains

• Imagine the universe is a finite 3-manifold (e.g., a 3-torus) with a fundamental domain (e.g., a rectangular box) that is our Dirichlet domain, with us at the center.
• We can tile all of space (e.g., ℝ³) with copies of this fundamental domain, placing ourselves in the same position in each copy.
• Each copy of the fundamental domain contains a copy of our LSS surrounding a copy of us.

⭕ When the LSS intersects itself

• Small LSS (relative to fundamental domain): the LSS does not intersect any of its copies; no matching circles appear (Figure 8.3.1(a)).
• Large LSS (relative to fundamental domain): the LSS intersects one or more of its copies; adjacent copies of the LSS intersect in circles (Figure 8.3.1(b)).
• Key property: these circles are copies of the same circle, so the temperature distribution around them will match.
• From our viewpoint at the center of the LSS, the two images of a circle will be directly opposite one another in the sky.

Example: In a 3-torus universe with a rectangular box fundamental domain (dimensions a × b × c, where a < b < c), if the LSS radius r_obs is greater than a/2 but less than b/2, we would see one pair of matching circles diametrically opposed on the LSS, with one circle traced counterclockwise matching the other traced clockwise, with no phase shift.

🎯 The detection strategy

• Scan the temperature distribution in the LSS for matching circles.
• If found, these circles reveal that the universe is finite and provide clues about its topology.

🔍 The circles-in-the-sky method

🛠️ How the method works

Circles-in-the-sky method: a strategy for detecting a finite universe by searching for pairs of circles on the LSS with matching temperature distributions.

• The search involves analyzing a six-parameter space:
  • Centers of the two circles: (θ₁, φ₁) and (θ₂, φ₂)
  • Common angular radius α (the circles are copies, so they have the same radius)
  • Relative phase β (the angular offset between the temperature patterns; β ≠ 0 if face identifications in the 3-manifold involve rotations)
• The goal is to determine whether a statistically significant correlation exists between temperatures as we proceed around the circles.

✅ Advantages over other methods

• Universality: can detect any compact manifold, regardless of the geometry it admits.
• Metric-independence: the search for matching circles does not depend on a metric; no need to claim a specific geometry to detect a finite universe.
• Contrast with cosmic crystallography: the circles-in-the-sky method has these advantages in cosmic topology.

⚠️ Computational intensity

• The method is computationally very intensive due to the six-parameter search space.
• Most searches to date have focused on circles that are diametrically opposed (or nearly so), reducing the search space from six parameters to four.

📏 Injectivity radius and detection conditions

📐 What the injectivity radius is

Injectivity radius (r_inj) at a point: half the distance of the shortest closed geodesic path that starts and ends at that point.

• This is a convenient length dimension for comparing the size of the LSS to the size of space.
• Don't confuse: the injectivity radius is a property of the manifold's topology and the observer's location, not the LSS itself.

🔑 Necessary condition for detection

• For matching circles to appear in the LSS at our location, the observable radius r_obs must exceed the injectivity radius r_inj.
• If r_obs < r_inj, the LSS is too small relative to the fundamental domain, and no matching circles will appear.

Example: In a 3-torus with dimensions a × b × c (a < b < c), the injectivity radius is a/2. If r_obs > a/2, the LSS will intersect itself and produce matching circles.

🌟 Examples of matching circles in different universes

📦 Euclidean 3-manifolds

• All six compact orientable Euclidean 3-manifolds would have matching circles diametrically opposed to one another on the LSS (if the size is right).
• The phase shift on these matching circles will be non-zero if the faces are identified with a rotation.
• Only the 3-torus has a Dirichlet domain that is location-independent; for other Euclidean 3-manifolds, the Dirichlet domain can vary from point to point.

🔷 Poincaré dodecahedral space

• If we live in a Poincaré dodecahedral space and the LSS is large enough, we might see six pairs of matching circles.
• Each pair consists of diametrically opposed circles with matching temperature distributions after a relative phase shift of 36°.
• Example: the identification of the front and rear faces of the dodecahedron produces one such pair, where circle C₁ traced counterclockwise matches circle C₂ traced clockwise, with a 36° phase shift.

🗺️ Location dependence

• In general, the matching circles we might see depend not only on the shape of the universe but also on where we are in the universe.
• This is because the Dirichlet domain can vary from point to point (except in the 3-torus and some elliptic 3-manifolds).
• In any hyperbolic 3-manifold, the Dirichlet domain depends on location.
• If we observe matching circles, it can reveal both the topology and the Earth's location in the universe.

📊 Current search results and implications

❌ No matching circles found

• At the time of writing, no matching circles have been found in the LSS.
• This negative result places bounds on the size of our universe.

📏 Lower bound on universe size

• The absence of matching circles implies the universe has an injectivity radius r_inj greater than 24 gigaparsecs.
• Converting units: 24 × 3.26 × 10⁹ ≈ 78 billion light-years.
• A geodesic closed path trip in the universe would be at least 156 billion light-years long (twice the injectivity radius).
• This distance is so large that we can likely abandon the possibility of seeing a distant image of our own Milky Way Galaxy.

📚 Further resources

• The excerpt mentions accessible papers on the circles-in-the-sky method and cosmic crystallography method.
• Jeff Weeks discusses both research programs in The Shape of Space.

🧭 Overview

🧠 One-sentence thesis

Current observations suggest our universe is homogeneous, isotropic, nearly flat (Euclidean), and dominated by dark energy, with its geometry and fate tied to the ratio of actual mass-energy density to the critical density required for flatness.

📌 Key points (3–5)

• What determines geometry: Einstein's general relativity ties the universe's geometry to its mass-energy content via the density parameter Ω (ratio of actual density to critical density).
• Three geometry possibilities: Ω < 1 → hyperbolic; Ω = 1 → Euclidean (flat); Ω > 1 → elliptic, corresponding to the three geometries studied in the text.
• The flatness puzzle: Inflationary theory predicts Ω = 1, but early estimates showed Ω ≈ 1/3; the discovery of dark energy (≈72% of total) reconciles this by adding a second density component.
• Common confusion: "flat" from CMB measurements vs "shape"—CMB analysis detects near-zero curvature (Ω_k ≈ 0), not the specific topology; circles-in-the-sky would reveal shape but none have been detected.
• Why it matters: the geometry and dark energy content determine the universe's ultimate fate (big crunch, asymptotic expansion, or accelerated expansion).

🌌 Evidence for homogeneity and flatness

🌡️ Cosmic microwave background radiation

• The cosmic microwave background (CMB) arrives from every direction with nearly constant temperature.
• This uniformity is evidence that the universe is homogeneous (same everywhere) and isotropic (same in all directions).

💥 Inflationary universe theory

• Pioneered in the 1980s by Alan Guth and others.
• States that during the first 10^−30 seconds after the big bang, the universe expanded at a stupendous rate.
• This rapid expansion caused the universe to appear homogeneous, isotropic, and flat (Euclidean geometry).
• The theory predicts Ω = 1 (critical density).

📡 WMAP and Planck satellite measurements

• The Wilkinson Microwave Anisotropy Probe (launched 2001) carefully plotted CMB temperature.
• Five-year WMAP data estimated Ω = 1.0045 ± 0.013, supporting near-flatness.
• The Planck satellite (launched 2009, orbiting 1.5 million km from Earth) gave improved CMB measurements.
• Planck team concluded the universe appears flat to 0.25% accuracy (one standard deviation).
• Nine-year WMAP + supernova data suggest −0.0066 < Ω_k < 0.0011 (Ω_k measures deviation from flatness).

Don't confuse: CMB analysis detects curvature (whether Ω_k ≈ 0), not the specific shape (topology). The circles-in-the-sky method would reveal shape by finding matching circles, but none have been detected.

🔗 Geometry tied to mass-energy content

⚖️ The Friedmann equation

The Friedmann equation relates the mass-energy density ρ of the universe to its curvature k: H² = (8πG/3)ρ − k/a²

• G: Newton's gravitational constant
• H: Hubble constant, measuring the expansion rate (current estimates: 68–70 km/s per megaparsec, where 1 megaparsec = 3,260,000 light-years)
• k: curvature constant (−1, 0, or 1)
• a: scale factor (both a and H change over time but may be viewed as constant in the present period)

📏 Critical density and the density parameter Ω

• In a Euclidean universe, k = 0. Solving the Friedmann equation gives the critical density:

  ρ_c = 3H²/(8πG) ≈ 1.7 × 10^−29 grams per cubic centimeter

• This is the precise density required for a Euclidean universe.

• The density parameter Ω is the ratio of actual density to critical density:

  Ω = ρ/ρ_c

🔺 Three geometry cases

Condition	Geometry	Implication
Ω < 1	Hyperbolic	Universe has low mass-energy content
Ω = 1	Euclidean (flat)	Universe has exactly the critical mass density
Ω > 1	Elliptic	Universe has high mass-energy content

Why Euclidean seems unlikely from a naive view: If the mass-energy content deviates by the mass of just one hydrogen atom from the critical amount, the universe fails to be Euclidean. Yet observations suggest Ω ≈ 1.

🌑 The dark energy discovery

🔭 The 1/3 problem

• Until the late 1990s, all estimates of mass-energy content put Ω ≈ 1/3, suggesting a hyperbolic universe.
• This contradicted the inflationary model's prediction of Ω = 1.
• If mass-energy density is only about 1/3 of what is required for a Euclidean universe, but CMB data show the universe is flat, some other form of energy must exist.

💫 Type Ia supernovae observations

• In 1999, observations of distant exploding stars (Type Ia supernovae) showed they are fainter than expected for a universe whose expansion is slowing down.
• This suggests the universe is accelerating its expansion.
• Saul Perlmutter, Brian Schmidt, and Adam Riess won the 2011 Nobel Prize in Physics for this discovery.

⚡ The cosmological constant revived

• When Einstein first proposed the 3-sphere as the universe's shape, his theory predicted it should be expanding or collapsing.
• He added a cosmological constant to his field equations to counteract gravity and keep the universe static.
• Edwin Hubble, Vesto Slipher, and others discovered galaxies are receding (the universe is expanding), so Einstein withdrew the constant.
• Now the constant has new life: it can represent the repulsive dark energy driving accelerated expansion.

🧩 Two-component density

The density parameter now has two components:

• ρ_M: mass-energy density from ordinary and dark matter (what cosmologists estimate by observation)
• ρ_Λ: dark energy from the cosmological constant

The Friedmann equation becomes: H² = (8πG/3)(ρ_M + ρ_Λ) − k/a²

Dividing by H² and defining:

• Ω_M = ρ_M/ρ_c
• Ω_Λ = ρ_Λ/ρ_c
• Ω_k = −k/(aH)²

We get the fundamental equation:

1 = Ω_M + Ω_Λ + Ω_k

Current estimates (nine-year WMAP + supernovae):

• Ω_Λ ≈ 0.72 (dark energy dominates)
• Ω_M ≈ 0.28 (ordinary + dark matter)
• −0.0066 < Ω_k < 0.0011 (nearly flat)

As the universe evolves, the density parameters may change, but their sum always equals one.

🔮 Fate of the universe

🎯 Simple case: zero cosmological constant

If the cosmological constant is zero, the relationship is simple:

Condition	Geometry	Fate
Ω > 1	Elliptic 3-manifold	Universe will eventually fall back on itself → "big crunch"
Ω = 1	Euclidean (one of 10 possible shapes, 6 if orientable)	Expansion rate asymptotically approaches 0 but never collapses
Ω < 1	Hyperbolic	Universe continues expanding until everything spreads out → "big chill"

🌀 With nontrivial dark energy

• Gravitational force from usual mass-energy tends to slow expansion.
• Dark energy causes acceleration.
• If dark energy wins the tug of war with gravity:
  • The curvature of the universe would approach 0 as expansion accelerates (regardless of whether the universe is hyperbolic, elliptic, or Euclidean).
  • The density of matter would approach 0.

Don't confuse: The fate depends on both the geometry (determined by Ω) and the nature of dark energy. With dark energy, even a hyperbolic or elliptic universe could have curvature approaching zero over time.

🔍 Open questions and future prospects

🎲 Geometry still barely open

• Current estimates for Ω_k leave the question of geometry open, though just barely.
• It is still possible the universe is a hyperbolic or elliptic manifold, but the curvature would have to be very close to 0.

🔄 Six Euclidean possibilities

• If the universe is a compact, orientable Euclidean manifold, there are six different possibilities for its shape.
• Euclidean manifold volumes aren't fixed by curvature, so there is no reason to expect the dimensions to be close to the radius of the observable universe.
• If the size is right, the circles-in-the-sky method would reveal the shape through matching circles.

🌠 No circles detected yet

• The circles-in-the-sky method searches for shape (topology), different from CMB analysis that detects curvature.
• Planck satellite data enabled more refined circles-in-the-sky tests.
• Alas, no circles have been detected.
• Perhaps the universe is just too big.

🌟 A remarkable feat

The excerpt concludes: pursuing the question of the shape of the universe is a remarkable feat of the human intellect. It is inspiring to think, especially looking up at a clear, star-filled night sky, that we might be able to determine the shape of our universe, all without leaving our tiny planet.