Advanced Level
Advanced Level
Casey, James. Exploring Curvature. Braunschweig/Wiesbaden, Germany:
Vieweg, 1996. Coxeter, H. S. M. Introduction to Geometry. New York: John Wiley, 1989. ———. Non-Euclidean Geometry. Washington, D.C.: MAA, 1998. ———. Projective Geometry. New York: Springer-Verlag, 1994. ———. Regular Polytopes. New York: Dover Publications, 1973. Gorini, Catherine A., ed. Geometry at Work: Papers in Applied Geometry.
Washington, D.C.: MAA, 2000. Graver, Jack. Counting on Frameworks. Washington, D.C.: MAA, 2001. Grünbaum, Branko, and G. C. Shephard. Tilings and Patterns. New York: W. H.
Freeman, 1989. Henderson, David W. Differential Geometry: A Geometric Introduction. Upper
Saddle River, N.J.: Prentice Hall, 1998. Hilbert, David, and S. Cohn-Vossen. Geometry and the Imagination. New York:
Chelsea, 1999. Livingston, Charles. Knot Theory. Washington, D.C.: MAA, 1993. Mandelbrot, Benoit. The Fractal Geometry of Nature. New York: W. H.
Freeman, 1983. Okabe, Atsuyuki. Spatial Tessellations: Concepts and Applications of Voronoi
Diagrams. New York: John Wiley, 2000. O’Rourke, Joseph. Computational Geometry in C. Cambridge, U.K.:
Cambridge University Press, 1996. Radin, Charles. Miles of Tiles. Providence, R.I.: AMS, 1999. Senechal, Marjorie. Quasicrystals and Geometry. Cambridge, U.K.: Cambridge
University Press, 1995. Stahl, Saul. The Poincaré Half-Plane. Boston: Jones and Bartlett, 1993.
APPENDIX
Recommended Reading
A’hmosè – dodecahedron INDEX
corresponding angles 37, 117, illustrations or captions.
Italic page numbers indicate
biprism 144
color symmetry 8, 27, 196
bisector
common external tangent 28,
cosine 38, 50, 210, 268–269 A’hmosè 228, 229
angle 6, 63, 256
crystal 39, 40, 188, 202, 225, Academy 196, 215, 229
perpendicular 121
common internal tangent 28,
235, 243 Alexandria 196, 202, 210,
Bolyai, János 187, 237
crystallographic group 40, 212, 215, 230
Bourbaki, Nicholas 187, 188
also compass and
263 Alhambra 231, 241
Brianchon’s theorem 188, 253
straightedge construction
curvature 12, 24, 41, 63, 70, Almagest 215, 231
Brocard angle 18, 189
compass and straightedge
88, 101, 129, 167, 187, 199 alternate exterior angles 4,
132, 137, 172 alternate interior angles 4,
Brouwer fixed point theorem
complete quadrangle 29, 47,
cyclic quadrilateral 188, 231, altitude 5, 14, 113, 249
complex numbers 10, 30,
computer vision 31, 64, 118,
concurrent lines 31, 76, 119,
decapod 11, 15, 25, 43, 158 Apollonius 9, 155, 183, 230
cone 31, 32, 70, 107, 141,
degree 45, 155 problem of 9, 199
variations 19, 185, 233
Delian problem 229. See also apothem 9, 262
Cantor set 20, 189
configuration 18, 32, 64, 115,
duplication of cube arc 9, 85, 87, 264
Carathéodory’s theorem 253
deltahedron 45, 74, 77, 120, Archimedes 10, 24, 62, 156,
catastrophe 21, 180, 221, 226,
conformal mapping 32, 246
congruent triangles 6, 149,
density 41, 45, 95 area 10, 161, 261–265
caustic 21, 213
Desargues, theorem of 46, arithmetic mean 11, 251, 258
Cavalieri’s principle 21,
conic section 23, 32, 95, 183,
Descartes, René 67, 193–194, 202–203, 207, 210, 215, 228,
constructible number 34, 236,
diameter 47, 209, 249 asymptote 11, 33, 266
of rotation 23, 87, 142
Earth 230 attractor 11, 12, 14, 19, 159, 246
centroid 23, 60, 249, 260, 267
construction 4, 5, 12, 99, 101,
dilation 49, 135, 151, 154, 252 axiom 12, 28, 81, 84, 128,
Ceva’s theorem 191, 233, 251
dilative reflection 49, 252 202, 214, 235, 239, 241
cevian 23, 63, 67, 119
compass and straightedge
dilative rotation 49, 252 Pasch’s 118, 235
chaos 23, 197, 209–210, 225,
dimension 49, 52, 68, 146, Playfair 214, 235
201 axiomatic system 12–13, 21,
chord 24, 95, 202, 230, 249,
directed angle 10, 49, 137 29, 71, 85, 88, 104, 143, 173,
convexity 36, 202, 239, 246,
directed magnitude 50, 150, 230
circle 24, 31, 32, 108, 134,
250, 251 axioms, Euclid’s 60, 81, 104,
Conway, John 192, 242, 244,
of the Earth 196
curvilinear 41, 207
division of a segment 52, 63, Barbier’s theorem 253
collinear points 3, 15, 27, 39,
A’hmosè – dodecahedron INDEX
INDEX
double angle formula – linkage
double angle formula 268
instrument 144, 149, 157, drop a perpendicular 14, 54,
four-color
Gergonne
165. See also compass, 114, 119, 136, 151, 164, 255
problem 67, 183, 201,
point 71, 199
straightedge duality 54, 123, 154, 199
triangle 72, 199
interval 26, 75, 88 duplication of the cube 54,
theorem 67, 243, 254
glide reflection 72, 252
inversion 24, 89, 213 183, 233
fourth proportional 67, 258
global analysis 242
involution 75, 104, 252 dynamical system 12, 46, 51,
lines 90, 91 e 55, 270
spiral 73, 258
isogonic center 90, 91 eccentricity 32–33, 266
frustum 68, 209, 228, 263,
graph 16, 29, 48, 73, 109,
isometry 51, 60, 91, 236, Elements 196, 202, 214, 215,
of a function 73, 81, 177
isoperimetric ellipse 12, 32, 56, 250, 264,
theorem 250 ellipsoid 56, 110, 131, 265
commutative 182
isosceles triangle 2, 8, 14, 92, epicycle 58, 215, 230, 231
Galois, Évariste 199, 204,
isotomic point 92, 136 equilateral triangle 59, 77, 78,
Gauss, Carl Friedrich 199,
Jordan curve 93, 204, 214 169, 171, 249, 255, 262
Gauss-Bonnet theorem 187,
Erlanger Program 60, 206,
geodesic 70, 199
Kepler, Johannes 93, 205, Escher, M. C. 196, 213, 217,
geodesic dome 199, 241 geometric mean 71, 251, 258
Hahn-Banach separation
Kepler-Poinsot solid 73, 74, Euclid 159, 196, 230
geometry of numbers 71, 210
theorem 253 Hales, Thomas 201, 246
93, 153, 214 Euclid’s axioms 60, 104, 230,
half angle formula 215, 268 Hamilton, William Rowan
Klein bottle 94, 206, 237 Eudoxus 62, 78, 183, 197, 210
Klein, Felix 60, 116, 197,
Euler formula 60, 208, 234,
knot 65, 70, 94, 113, 128, line 60, 197, 234, 237, 249
polynomial 4, 36, 65, 79, theorem 250
theory 220, 225 exhaustion, method of 62,
elliptic 7, 53, 56, 62, 152,
Helly’s theorem 253
Koch snowflake 94, 206, 240 exotic sphere 62, 210, 242
Euclidean 60, 221, 230, 235
hexiamond 15, 19, 78, 99,
Konigsberg bridges 197, 234 exterior angle 62, 63, 158,
Hilbert, David 202, 204, 211,
207, 237, 239, 246 Fermat prime 65, 235, 250
holyhedron 79, 246
Lemoine circle 66, 145 Feuerbach’s theorem 249
honeycomb 81, 246
Leonardo’s theorem 252 Fields Medal winner 184,
noncommutative 108, 192,
horseshoe map 81, 219, 242
Lie group 97, 209, 238 187, 192, 194, 198, 200, 204,
linear algebra 98, 190, 201 Flatland 66, 98, 154, 182
link 16, 17, 19, 81, 99, 186 focus 32–33, 66, 81, 82, 95,
237. See also mechanism
INDEX
double angle formula – linkage
Lobachevsky, Nicolai – rectangle INDEX
Lobachevsky, Nicolai 115,
trisection of an angle, 209, 236, 237
octahedron 110, 263
estimations of 10, 183,
Plateau’s problem logarithm 99, 107, 232, 238
octonians 111, 190, 236
projection 13, 20, 23, 43, 59, logic 25, 46, 96, 131, 100,
orthic triangle 64, 108, 113
Pick’s theorem 214, 239, 250
130, 138, 168 loran 100, 241, 266
orthocenter 60, 68, 108, 113,
plane, coordinate 36–37, 194
Mercator 102, 110, 168, lune 100, 146, 203
Plateau’s problem 123, 194,
stereographic 159, 232 MacArthur fellow 193, 199,
Plato 124, 183, 214, 221,
projective geometry, 222
packing 93, 115, 201, 205,
fundamental theorem of 253 Mandelbrot set 101, 210, 243
Platonic solid 124, 214. See
projective manifold 48, 101, 140, 239,
Apollonian 9, 246
also regular solid
plane 64, 69, 130, 244 242, 243
pantograph 116, 232
Playfair axiom 214, 235
space 108, 116, 130 matrix 84, 101, 180, 190,
Pappus, theorem of 116,
Poincaré, Henri 214, 238,
proportion 34, 63, 101, 131, mean 10, 11, 71, 75, 101
parabola 32, 50, 116, 194,
Poincaré conjecture 124, 198,
pseudosphere 132, 185, 237 mechanism 76, 101, 178, 223.
parallel lines 117, 196,
polar 125, 171
Ptolemy’s theorem 249 See also linkage
polarity 126, 136
pyramid 25, 132, 152, 193, median 102, 175, 234, 260
parallelogram 22, 72, 117,
pole 126, 151
228, 263 Menelaus, theorem of 251
polygon 6, 36, 123, 127, 138,
Great Pyramid 228 midpoint 2, 9, 16, 25, 38, 60,
parallel postulate 2, 56, 82,
Pythagoras 100, 105, 68, 92, 102, 103, 108, 121,
117, 207. See also postulate,
polyhedron 22, 54, 127, 144,
Euclid’s fifth
Pythagorean identities 132, minimal surface 103, 214
Pascal’s theorem 253. See
polyiamond 106, 120, 127,
215, 268 Minkowski’s convex body
also mystic hexagram
Pythagoreans 106, 165 theorem 253
theorem
polynomial 4, 15, 45, 79, 94,
Pythagorean theorem 18, 196, Miquel circle 104, 236
Pasch’s axiom 118, 235
199, 216, 228, 230–231 Möbius band 39, 104, 211, 237
Peano curve 119, 238
knot 4, 36, 65, 79, 93, 95,
Pythagorean triangle 129, 132 Mohr-Mascheroni
Penrose tiling 17, 21, 24, 36,
Pythagorean triple 132, 228 construction theorem 251
polyomino 106, 111, 120,
vertex neighborhood of 2,
monstrous moonshine
quadrangle, complete 29, 47, theorem 245
polytope 118, 127, 138, 147,
76, 112, 133 Morley’s theorem 239, 249
pentagram 45, 100, 120, 261
quadratrix 133, 203 Museum 230
Perko pair 121, 243
Poncelet, Jean-Victor 215,
quadrilateral 249, 261, 262 mystic hexagram theorem
permutation 5, 42, 61, 111,
quadrilateral cyclic 188, 231, 106, 188, 213
Poncelet-Steiner construction
quasicrystal 133, 187, 218, Nagel point 68, 107
perpendicular bisector 121,
Euclid’s fifth 205, 214,
quaternions 134, 201, 236 Newton, Isaac 66, 212, 233
nine-point circle 108, 188,
radius 24, 134, 135 noncollinear points 29, 169,
ray 57, 136 number
physics 69, 158, 160, 166,
real numbers 44, 136, 193, complex 10, 30, 126, 132,
241. See also relativity,
202, 233, 234, 239. See
general and relativity, special
also duplication of cube,
rectangle 9, 94, 102, 110, 136, constructible 236, 238
pi 122, 207, 210, 223, 233,
four color problem,
squaring of a circle,
Lobachevsky, Nicolai – rectangle INDEX
INDEX
reflection – zonohedron
Steinitz’s theorem 253
137, 144, 169, 170, 261–262 regular decagon 261
semiperimeter 147, 262
straightedge 60, 159. See also
area of 202, 261–262 regular hexagon 78, 127, 138,
semiregular polyhedron 21,
constructions and compass
orthic 64, 108, 113 201, 258
41, 74, 83, 140, 147, 153,
and straightedge construction
Triangle Inequality 169, 248 regular octagon 258
172, 190
string theory 159, 180
trigonometry 170, 182, regular pentagon 120, 250,
set 57, 147, 149
Sulba Sutras 183, 185, 205, 228
202–203, 215, 223, 230, 231, 261
set theory 149, 189, 193
sum, angle 6, 7, 63, 228, 248,
268–269 regular polygon 3, 6, 9, 24,
similar triangles 6, 7, 149,
250, 254, 256
trisection of an angle 171, 30, 135, 138, 250, 262
150, 248
surface 4, 26, 34, 47, 53, 56,
183, 203, 230 regular polyhedron 55, 64,
ultraparallel 14, 119, 173 also Platonic solid
simplex 64, 113, 138, 151, 192
surgery 45, 161, 193
uniform Reidemeister moves 139, 241
Simson line 151, 219, 223,
surveying 123, 165
polyhedron 104, 134, 147, relation 8, 15, 59, 85, 102,
tiling 138, 147, 173 relativity
van Hiele model 175, 242 240, 241
slant height 152, 265
125, 134, 162, 168, 187, 199,
Varignon’s theorem 222, 234, special 103, 209, 210
vector 17, 41, 44, 53, 105, rhombus 47, 100, 119, 140,
space, topological 166,
105, 125, 163, 249, 250, 259,
200, 236 right triangle 11, 21, 83, 141,
space-filling curve 154, 238
tessellation 19. See also tiling
245 rigidity 17, 86, 141, 191, 239,
spacetime geometry 63, 87,
velocity 2, 157, 163, 176, 179 robotics 65, 67, 89, 136, 141
speed 74, 155
theorem 96, 97, 127, 149, 165
vertex 4, 8, 45, 138, 155, 176 rotation 6, 24, 39, 42, 48, 49,
sphere 155, 183, 226, 238,
Thurston, William 222, 243
134, 157–158, 160, 176 center of 23, 87, 142
vertical angles 176, 248
vesica piscis 177, 260 Saccheri quadrilateral 14,
volume 177, 232–233, Samos, tunnel of 228
Voronoi diagram 50, Schlegel diagram 28, 144,
258
transcendental number 168,
School Mathematics Study
wallpaper group 178, 239, Group 242
wavelets 178, 193, 244 sector 145, 264
squaring the circle 133, 157,
121, 137, 154, 168, 172, 252
wind 76, 163, 179 segment 16, 19, 50, 57, 102,
183, 188, 205, 238
transversal 4, 7, 11, 37, 88,
statement, types of 31, 35, 83,
117, 136, 168, 248
145, 255, 256, 257, 259, 264
zonohedron 57, 73, 92, 107, division of 52, 63, 88, 135
84, 85, 89, 107
trapezoid 14, 92, 102, 168
Steiner-Lehmus theorem 248
trefoil knot 4, 169, 193, 240
117, 140, 153, 172, 180
INDEX
reflection – zonohedron