# Index Index

¶2-sphere, Definition

3-manifold

3-sphere, Example

3-torus, Example

half turn manifold, Example

Hantschze-Wendt, Example

one-sixth turn manifold, Example

one-third turn manifold, Example

Poincaré dodecahedral space, Example

quarter turn manifold, Example

Seifert-Weber space, Example

3-sphere, Example

3-torus, Example

8, the greatness of, Paragraph

\((\mathbb{D}_k,{\cal H}_k)\) with \(k \lt
0\), Paragraph

\((\mathbb{P}^2_k,{\cal S}_k)\) with \(k \gt
0\), Paragraph

\((X_k,G_k)\), Definition

\(\infty\), Paragraph

\(\mathbb{P}^2\), Definition

\(\mathbb{S}^1\), Example

\(C_g\), Paragraph

\(H_g\), Paragraph

\(i\), Paragraph

all-star equation, Paragraph

angle

between curves, Paragraph

between rays, Paragraphs

determined by three points, Paragraphs

transformation preserves, Definition

transformation preserves magnitudes, Definition

angle of parallelism, Paragraph

antipodal points of \(\mathbb{C}^+\), Definition

Apollonian Circles Theorem, Theorem

Beltrami, Enrico, Paragraph

Bessel, Friedrich, Paragraphs

block, Example

Bolya, János, Paragraph

Bolyai, Farkas, Paragraph

bounded set in \(\mathbb{R}^n\), Paragraph

cell complex, Definition

\(n\)-dimensional, Definition

edge, Paragraph

face, Paragraph

vertex, Paragraph

cell division, Definition

Clifford translation, Paragraph

cline, Definition

closed geodesic path, Paragraph

closed set in \(\mathbb{R}^n\), Paragraph

compact space, Paragraph

complex number, Paragraph

addition of, Paragraph

argument, Paragraph

Cartesian form, Paragraph

conjugate, Paragraph

division of, Paragraph

imaginary part, Paragraph

modulus, Paragraph

multiplication, Paragraph

polar form, Paragraph

real part, Paragraph

scalar multiplication, Paragraph

Coneland, Example

connected sum, Paragraphs

construction

antipodal point, Exercise

cline through three distinct points, Theorem

elliptic line through two points, Theorem

hyperbolic circle centered at \(p\) through \(q\), Exercise

hyperbolic line through two points, Exercise

hyperbolic reflection sending a point to the origin, Section

octagon in \((\mathbb{D},{\cal H})\) with 45\(\circ\) interior angles, Exercise

right-angled hexagon in \((\mathbb{D},{\cal H})\), Theorem

symmetric point \(i_C(z)\)

when \(z\) is inside \(C\), Exercise

when \(z\) is outside \(C\), Exercise

cosmic background radiation (CMB), Paragraph

cosmic crystallography, Paragraph

collecting correlated pairs, Example

pair separation histogram, Paragraph

cosmic topology, Paragraph

circles-in-the-sky method, Paragraph

cosmic crystallography method, Paragraph

cross ratio, Definition

invariance of, Theorem

cross-cap, Paragraph

curvature of a space at a point, Definition

curve, Paragraph

smooth, Paragraph Definition

general theory of relativity, Paragraph

special theory of relativity, Paragraph

elliptic geometry, Definition

arc-length, Definition

area, Definition

circle, Definition

distance formula, Paragraph

line, Definition

lune, Paragraphs

elliptic geometry with curvature \(k\), Paragraph

equivalence relation, Definition

Euclidean distance between points, Example

Euler characteristic, Definition

extended plane, Paragraph

fractional linear transformation, Paragraph

Friedmann Equation, Paragraph

fundamental domain, Paragraph

Gauss-Bonnet Formula, Theorem

general linear transformation, Paragraph

general theory of relativity, Paragraph

geodesic, Paragraph

geometry, Definition

\((X_k,G_k)\), Definition

congruent figures in, Definition

Euclidean, Example

figure in, Definition

homogeneous, Definition

hyperbolic geometry, Poincaré disk model, Definition

invariant function, Definition

invariant set, Definition

isotropic, Definition

metric, Definition

minimal invariant set, Definition

point, Definition

rotational, Exercise

translational, Example

geometry with curvature \(k = 0\), Paragraph

geometry with curvature \(k \gt 0\), Paragraph

geometry with curvature \(k \lt
0\), Paragraph

golden ratio, Example

great circle, Paragraph

great circle in \(\mathbb{C}^+\), Paragraph

group of homeomorphisms, Paragraph

group of isometries, Paragraph

group of transformations, Definition

Guth, Alan, Paragraph

half turn manifold, Example

handlebody surface of genus \(g\), Paragraph

Hantschze-Wendt manifold, Example

Hogan, Craig, Paragraph

homeomorphic spaces, Paragraph

homeomorphism, Paragraph

homogeneous geometry, Definition

horocycle, Paragraph

Hubble constant, Paragraph

Hubble, Edwin, Paragraph

hyperbolic cosine function, Definition

hyperbolic geometry

\(\cal H\text{,}\) the transformation group in hyperbolic geometry, Definition

\(\frac{2}{3}\)-ideal triangle, Paragraph

\(\mathbb{D}\text{,}\) the hyperbolic plane, Definition

arc-length, Definition

area, Definition

circle, Definition

circle at infinity, Paragraph

circle centered at \(p\) through \(q\text{,}\) construction of, Exercise

distance between points, Paragraph

ideal point, Definition

ideal triangle, Paragraph

line, Definition

line through \(p\) and \(q\text{,}\) construction of, Exercise

parallel lines, Definition

Poincaré disk model, Definition

upper half-plane model, Definition

hyperbolic geometry with curvature \(k\), Paragraph

hyperbolic plane, Definition

reflection of, Paragraph

rotation of, Paragraph

translation of, Paragraph

hyperbolic refelction, Paragraph

hyperbolic sine function, Definition

hyperbolic transformation group, Definition

parallel displacement, Paragraph

rotation, Paragraph

translation, Paragraph

image of a set, Definition

inflationary universe theory, Paragraph

inverse pole of a Möbius transformation, Paragraph

inversion in a circle, Paragraph

involution transformation, Exercise

isometry

Euclidean, Definition

fixed-point free, Paragraph

hyperbolic, Paragraph

properly discontinuous, Paragraph

isotropic geometry, Definition

last scattering surface (LSS), Paragraph

lens space, \(L(p,q)\), Example

Lobachevsky, Nikolai, Paragraph

Lobatchevsky's formula, Theorem

manifold, Paragraph

metric, Definition

Euclidean, Example

hyperbolic, Corollary

multiconnected space, Paragraph

Möbius geometry, Definition

Möbius strip, Paragraph

Möbius transformation, Paragraph

determinant of, Paragraph

elliptic, Paragraph

fundamental theorem of, Theorem

hyperbolic, Paragraph

inverse pole of a, Paragraph

loxodromic, Paragraph

normal form!one fixed point, Paragraph

normal form!two fixed points, Paragraph

parabolic, Paragraph

pole of a, Paragraph

one-sixth turn manifold, Example

one-third turn manifold, Example

one-to-one, Definition

onto, Definition

open \(n\)-ball, Paragraph

orbit of a point, Paragraph

orbit space, Paragraph

orthogonal clines, Paragraph

parallax, Paragraphs

of star 61 Cygni, Paragraphs

Parallel Postulate, Paragraph

partition, Paragraph

Perlmutter, Saul, Paragraph

planar curve, Paragraph

Planck Satellite, Paragraph

Platonic solids, Example

Playfair's Axiom, Paragraph

Playfair, John, Paragraph

Poincaré, Henri, Paragraph

point at \(\infty\), Paragraph

pole of a Möbius transformation, Paragraph

polygonal surface

boundary label, Paragraph

power of a point, Paragraph

projective plane, Definition

Pythagorean triples, Exercise

quarter turn manifold, Example

quaternion, Paragraphs

conjugate, Paragraphs

modulus, Paragraphs

pure, Example

scalar term, Paragraphs

unit, Paragraphs

quotient set, Definition

\(X/G\) built from geometry \((X,G)\), Paragraph

radius of curvature, Paragraph

real \(n\)-dimensional space, Paragraph

reflection, Example

relation on a set, Paragraph

Riess, Adam, Paragraph

rotation, Example

saddle point, Paragraph

Schmidt, Brian, Paragraph

screw motion, Paragraphs

Seifert-Weber space, Example

simply connected space, Paragraph

Slipher, Vesto, Paragraph

smooth curve, Paragraph

sphere

diametrically opposed points, Paragraph

spherical geometry, Paragraph

square root, Example

stereographic projection, Paragraphs

surface, Definition

cross-cap surface of genus \(g\) (\(C_g\)), Paragraph

handlebody surface a genus \(g\) (\(H_g\)), Paragraph

non-orientable, Paragraph

orientable, Paragraph

polygonal, Paragraph

symmetric points

with respect to a circle, Paragraph

with respect to a sphere, Paragraphs

Theorem Egregium, Paragraph

three-dimensional geometry

elliptic, Paragraphs

Euclidean, Paragraphs

hyperbolic, Paragraphs

Thurston, William, Paragraph

transformation, Definition

dilation, Example

fixed point, Definition

fractional linear, Paragraph

general linear, Paragraph

inverse, Paragraph

involution, Exercise

Möbius, Paragraph

preserves angle magnitudes, Definition

preserves angles, Definition

reflection, Example

rotation, Example

screw motion, Paragraphs

translation, Example

translation, Example

type I cline of \(p\) and \(q\), Paragraph

type II cline of \(p\) and \(q\), Paragraph