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2-sphere, Definition
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
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
circles of Apollonius, Paragraph Paragraph
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
cone point, Example Paragraph
Coneland, Exercise
connected sum, Paragraphs
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
De Sitter, Willem, Paragraph
Dirichlet domain, Paragraph
Dostoevsky, Fyodor, Paragraph
Einstein, Albert, Paragraph Paragraph Paragraph
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
Erlangen Program, Paragraph Quotation
Euclidean distance between points, Example
Euler characteristic, Definition
extended plane, Paragraph
Friedmann Equation, Paragraph
fundamental domain, Paragraph
Gauss, Carl Friedrich, Paragraph 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
Euclidean, Definition
fixed-point free, Paragraph
hyperbolic, Paragraph
properly discontinuous, Paragraph
isotropic geometry, Definition
Klein bottle, Example Exercise
Klein, Felix, Paragraph Paragraph
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
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
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
unit 2-sphere, Definition
unit circle, Example
universal covering space, Paragraph
Weeks, Jeff, Paragraph
Wilkinson Microwave Anisotropy Probe (WMAP), Paragraph