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