Académie des Sciences (French Academy of Sciences), 12, 14, 21, 112, 211
Académie Royale des Sciences et Belles Lettres. See Berlin Academy of Sciences
Academy of Sciences in St. Petersburg, 14–17, 23, 103
adjacency graph, 135–137, 140–142
Aleksandrov, Pavel, 263
Alekseyevna, Catherine. See Catherine II of Russia
alternating knot, 194, 195, 197, 199, 200
American Mathematical Society, 142, 158
Anderson, Gary, 163
angle defect. See angle deficit
angle deficit: geodesic polygon, 227, 228, 236
angle excess: geodesic polygon, 227–230, 236–239
polyhedron, 224, 225, 228, 229
angle excess theorem, 219, 223, 229, 230, 238
annulus. See cylinder
Arago, François, 17
Archimedean solid. See semiregular polyhedron
Archimedes, 1, 10, 31, 48–50, 181
Ball, W. W. Rouse, 104
Baltzer, Richard, 133
Barbari, Jacopo de, 51
Barbaro, Daniele, 51
Berlin Academy of Sciences, 14, 18–22
Bernoulli, Jacob, 12
Bernoulli, Nicolaus, 15
Berthollet, Claude Louis, 112
Betti, Enrico, 250–252, 254, 255, 258
Betti group, 263
Betti number, 255–261, 263, 266
Biggs, Henry, 92
Billingsley, Henry, 27
Birkhoff, George D., 141
Blaschke, Wilhelm, 239
Bonnet, Pierre Ossian, 238, 264
boundary. See surface with boundary; manifold with boundary
Boyer, Carl, 25
Boyle, Robert, The Sceptical Chymíst, 43, 58
Brahe, Tycho, 57
bridges of Königsberg problem, 100–111, 135, 156, 187, 253, 269, 284
Brouwer fixed point theorem, 216–218, 285
Brouwer, Luitzen Egbertus Jan, 209, 212, 213, 217, 246, 263
buckminsterfullerene, 7, 8, 78
Burkert, Walter, 36
calculus, ix, 14, 25, 26, 52, 113, 133, 183, 203, 231, 232
Caroll, Lewis, 130
Catherine II of Russia, 11, 14, 22–24
Catherine the Great. See Catherine II of Russia
Cauchy, Augustin-Louis, x, 48, 112–119, 121, 145, 151, 152, 246, 247, 259
Cauchy’s rigidity theorem, 48, 114, 284
Cayley, Arthur, 118, 132, 137, 156
center (zero of a vector field), 205, 206, 208
Challenger, HMS, 33
Chanut, Hector-Pierre, 83
Charles X of France, 113
Christina, Queen of Sweden, 83
Clarke, Arthur C., “The Wall of Darkness,” 163
classification of surfaces, 181, 183–185, 187, 246, 255, 265, 285
Clay Mathematics Institute, 9, 268, 270
Clerselier, Claude, 83
Collegium Carolinium, 234
colorability, knot, 197, 198, 200
complete bipartite graph, 122–124
composite knot, 196
connectivity number, 250–252, 254, 257
Connelly, Robert, 48
convexity, 29, 33, 46–49, 71–73, 95, 97–99, 114–117, 125, 145, 146, 148, 150–152, 155, 174, 220, 224, 225, 246, 247, 249
Conway, John Horton, 127, 128, 184, 270
ZIP proof, 184
Copernican astronomy, 55–57
Copernicus, Nicolaus, 55
Coxeter, Harold Scott MacDonald, 133, 141
Critias, 42
cross cap, 171, 172, 180–182, 184, 242, 246, 258
crossing number, 198–200
Crowell, Richard Henry, 195
cube, 2, 27, 28, 32–34, 36, 40, 42, 46, 47, 55, 58–61, 64, 67–69, 75, 77, 78, 84, 117, 147, 153, 174, 223–225, 241, 245–247
curvature; Gaussian, 233, 235–239
plane curve, 231–233
cylinder, 154, 158–162, 167, 169, 171, 175–177, 179, 181, 182, 184, 235–237, 249, 250, 255, 273
Czartoryski, Prince Adam, 146
D’Alembert, Jean, 21
de Careil, Foucher, 83
De Morgan, Augustus, 132
Dehn, Max Wilhelm, 184
Descartes, René, x, 1, 62, 81–86, 224, 225, 264, 284
Discourse on the Method for Rightly Directing One’s Reason and Searching for Truth in the Sciences, 82
The Elements of Solids, 83, 224
Descartes’ formula, 84, 219, 221, 223, 225, 226, 228–230
Descartes-Euler formula. See Euler’s polyhedron formula
Diderot, Denis, 22
Dieudonné, Jean Alexandre Eugène, 212, 253
differential equation, 10, 113, 183, 202, 203, 212
Diophantus, 50
dipole (zero of a vector field), 205, 206, 208
disk, 158, 159, 169, 171, 178, 179, 181, 182, 184, 185, 189–194, 196, 206, 207, 214, 217, 236, 249, 250, 255
dodecahedron, 32–34, 36, 39, 40, 42, 46, 47, 56, 58–61, 76–78, 260, 266
duality. See Poincaré duality; regular polyhedron, duality
Dürer, Albrecht, 51
Dyck, Walther von, 165, 166, 184, 239, 244
Dyck’s surface, 184
dynamical systems, x, xii, 5, 9, 203, 212, 216, 267
Ehler, Carl, 101
Einstein, Albert, 201
Empedocles, 42
Erdós, Paul, 144
Erlangen University, 262
Escher, Maurits Cornelis, 163
Euclid, x, 1, 27, 31, 32, 44–48, 50, 51, 75, 87, 89, 123
Elements, 27, 32, 40, 41, 44–47, 75, 87, 89
Eudoxus, 44
Euler, Leonhard, x, 1, 10–27, 29, 48, 62, 81, 84–87, 91, 97, 99–107, 112, 113, 116, 118, 145, 146, 152, 156, 180, 211, 225, 233, 234, 250, 269, 270, 283
“Demonstratio nonnullarum insignium proprietatum quibus solida hedris planis inclusa sunt praedita,” 66
“Elementa doctrinae solidorum,” 66
Euler, Marguerite Brucker, 11
Euler characteristic. See Euler number
Euler number, 4–7, 173, 177–181, 184, 185, 187, 189, 193, 201, 205, 208–210, 212, 213, 215, 216, 218, 219, 225, 227, 229, 230, 239–241, 247, 251, 259, 265, 284
Euler-Poincaré characteristic, 259–263, 266
Euler walk, 104–111
Euler’s polyhedron formula, x–xii, 1–4, 8, 9, 26, 28, 29, 32, 46, 58, 75–77, 79, 81, 84–87, 94–101, 112, 114–125, 127, 129, 136, 145–149, 151, 152, 155–157, 174, 179, 241, 246, 247, 249, 250, 269, 270, 283–285
Eulerian polyhedron, 146, 148, 152, 173, 174
exterior angle theorem, 220–223, 225, 232
Ferdinand, Carl Wilhelm, Duke of Brunswick, 234, 235
Fermat’s last theorem, 8, 267, 269
Fibonacci sequence, 9
figure eight knot. See knot, figure eight
five color theorem, 139–141
five neighbors theorem, 136, 137, 140, 141
fixed point, 204, 205, 208, 216–218
four classical elements, 42, 43, 58, 59
four color theorem, 7, 8, 130–144, 267, 284, 285
Francesca, Piero della, 51
Frederick II of Prussia, 11, 18–23
Frederick the Great. See Frederick II of Prussia
Frederick William I of Prussia, 18
Freudenthal, Hans, 113
Fuller, Buckminster, 7
Galilei, Galileo, 56
Gardner, Martin, 127, 133, 135
Gattegno, Caleb, 125
Gauss, Carl Friedrich, x, 10, 25, 75, 133, 152, 181, 182, 188, 211, 231, 233–235, 237–239, 248, 264, 270
Gauss, Gerhard, 233
Gauss-Bonnet theorem: global, 219, 239
genus: of a knot, 194–200
of a surface, 169, 170, 181, 183, 184, 193, 194, 239, 251
geocentric astronomy. See Ptolemaic astronomy
geodesic, 89, 227–229, 238. See also geodesic polygon
geodesic polygon, 89–91, 93–96, 156, 227–229, 236–239
geodesic triangle. See geodesic polygon
Gergonne, Joseph Diaz, 115, 117, 147, 148, 152
gingerbread man knot. See knot, gingerbread man
Goldbach, Christian, 1, 63, 66, 101
golden ratio, 39
granny knot. See knot, granny
graph, x, xi, 5, 99–111, 119–129, 134–144, 154, 156, 157, 247, 284
great circle, 87–89, 92, 93, 227
great icosahedron, 149–151
great stellated dodecahedron, 57, 150, 151
Gsell, Georg, 16
Gsell, Salome Abigail, 24
Guthrie, Francis, 131–133
Guthrie, Frederick, 132
Hadamard, Jacques Salomon, 118
Haeckel, Ernst, 33
hairy ball theorem, 5, 209, 210, 214
Hales, Thomas C., 143
Halle University, 133
Hamilton, William Rowan, 132
handle, 169, 170, 172, 179–181, 184, 246, 256
Hardy, Godfrey Harold, 24, 123, 253
Harriot-Girard theorem, 90–96, 227
Heegaard, Pout, 184
heliocentric astronomy. See Copernican astronomy
Hermocrates, 42
Hessel, Johann Friedrich Christian, 146–149, 174, 179, 245
Hierholzer, Carl, 105
Hippasus, 38–40
homeomorphism, 176, 179, 183, 184
homology, 254, 255, 258, 263, 264, 266
homology group, 263
Hopf’s Umlaufsatz. See theorem of turning tangents
Hoppe, R., 154
icosahedron, 27, 32, 34, 36, 40, 42, 43, 46–48, 56, 58–61, 76–78, 260
index (of a zero of a vector field), 206–210, 212, 214, 215
indicatrix, 165
interior angle theorem, 221, 225–227
invariance of dimension theorem, 246
Ivan VI of Russia, 18
Jamnitzer, Wentzel, 51, 54, 149
Jonquières, Ernest de, 84, 85, 145
Jordan, Marie Ennemond Camille, 118, 183
Jordan curve theorem, 166, 169, 231
Kant, Immanuel, 101
Kauffman, Louis, 199
Kelvin, Lord. See Thomson, William
Kempe, Alfred Bray, 137–142
Kepler, Johannes, x, 1, 8, 32, 49, 51–61, 92, 98, 143, 150, 260, 270
Cosmic Mystery, 55–57
The Harmony of the World, 57–59
Kepler conjecture, 143
Kepler-Poinsot polyhedron, 57, 58, 98, 149–151
Klein, Felix, 156, 159, 162, 165–169, 253, 262
Klein bottle, 167–169, 171, 178, 182, 209, 242, 245, 251, 257, 258, 273
figure eight, 186, 187, 193–197, 199, 200
gingerbread man, 186, 187, 193–200
granny, 196
trefoil, 185–200
unknot, 186, 187, 189, 190, 194–198, 200
knot invariant, 188, 189, 194, 197–200
Koch, John, 142
Koestler, Arthur, 53
Kuratowski’s reduction theorem, 124
Lagrange, Joseph-Louis, 21, 112, 243
Lakatos, Imre, 115, 117, 148, 284
Laplace, Pierre-Simon, 10, 112
Lefschetz, Solomon, 157–158, 213
Legendre, Adrien-Marie, 65, 67, 87–89, 92, 94–99, 145, 149, 150, 152, 156, 234
Elements of Geometry, 87
Leibniz, Gottfried, 14, 83, 91, 102, 103, 113
Leipzig University, 133
Leonard, K. C. von, 148
Lhuilier, Simon-Antoine-Jean, 146–149, 152, 154, 170, 174, 179, 180
Listing, Johann Benedict, 108, 118, 157, 164, 188, 247–250, 253
Little, Charles Newton, 188
London, England, 14, 18. See Royal Society of London
London Mathematical Society, 132
manifold, 214, 243–246, 252, 258–262, 265–268
with boundary, 244, 259, 262, 265
Marinoni, Giovanni, 102
Marquis de Condorcet, 25
Mathematical Association of America, 142
May, Kenneth O., 131
Menelaus of Alexandria, 89
Sphaerica, 89
minimal criminal, 137, 138, 140–142
Möbius, August, 133–135, 164, 165, 176, 183, 184, 247, 250, 253
Möbius band, 3, 156, 162–165, 167, 169, 171, 176, 177, 180–182, 184, 188–190, 247, 250, 273
music of the spheres, 38
Nash, John Forbes Jr., 270
Newton, Isaac, 10, 14, 16, 24, 57, 113
Noether, Emmy Amalie, 262, 263
nonorientable, 165–172, 177, 181, 183, 184, 190, 244, 246, 250, 256, 261
normal vector, 166, 231, 232, 235
octahedron, 32–34, 36, 40, 42, 46, 47, 56, 58–61, 75, 77, 78, 228, 229, 246, 247
one-sided surface, 164–166, 190
optics, 91
orientable, 169, 170, 172, 177, 181–185, 190, 191, 193, 238, 239, 244, 245, 250, 251, 255, 256, 260, 261, 265
Pacioli, Fra Luca, De Divina Proportione, 51, 54
Paterson, Michael, 127
pentafoil knot. See knot, pentafoil
pentakis dodecahedron, 53
perspective in artwork, 51
Peter I of Russia, 11, 13–16, 18, 22, 23
Peter III of Russia, 22
Peter the Great. See Peter I of Russia
phase space, 202–204
Pick, Georg Alexander, 124, 125
Pick’s theorem, 124–126
plane angle, 46, 47, 63, 84, 85, 225, 226
Plato, 1, 8, 31, 32, 40–43, 51, 58
Epínomís, 31
The Sophist, 40
Theaetetus, 40
Timaeus, 42–43
Platonic solid. See regular polyhedron
Poincaré, Henri, x, 28, 75, 85, 155, 157, 158, 173, 211, 212, 241, 244, 245, 252–262, 264, 266, 270
Analysis Situs, 173, 253, 254, 258, 261
Poincaré conjecture, 8, 9, 265–270, 285
Poincaré’s dodecahedral space, 266
Poincaré-Hopf theorem, 205, 208–211, 214–217
Poincaré, Raymond, 211
Poinsot, Louis, 97–99, 145, 149, 150, 152
Pólya, George, 220
Pont, Jean-Claude, 147
prime knot, 195, 196, 198, 200
Princeton University, 158, 213
problem of the five princes, 133–135
projective plane, 168, 169, 171, 172, 180, 182, 242, 251, 257, 258, 273
pseudosphere, 236
Ptolemaic astronomy, 55
Ptolemy, 50
pyramid, 27, 32, 34, 40, 67–72, 116, 152, 245, 259. See tetrahedron
Pythagoras, x, 1, 31, 36–39, 45, 51, 270
Pythagorean theorem, 8, 37, 125
quadratic formula, 8
quadratic reciprocity law, 234
Raleigh, Sir Walter, 91
Raphael, School of Athens, 51, 52
reducible configuration, 141, 142, 144
regular polyhedron, xi, 8, 31–35, 39–43, 45–48, 51, 55–62, 75–78, 84, 86, 151, 174, 181, 227, 260, 273, 285
Riemann, Georg Friedrich Bernhard, x, 118, 181–183, 244, 248, 250–254, 258, 264, 270
Riemann hypothesis, 267
Riemann surface, 183
Royal Society of London, 14, 21, 87, 139
saddle (on a surface), 206, 227, 235–237
saddle (zero of a vector field), 205–209
San Marco Basilica, 52
Schläfii, Ludwig, 168, 247, 259
Seifert circle, 191
Seifert surface, 190–196
semiregular polyhedron, 49, 57, 174, 181
simple closed curve, 166, 169, 219, 222, 223, 231, 238
simplex, 244, 245, 259, 260, 262
simplicial complex, 244, 245, 258, 259, 262
simply connected, 255, 256, 266, 268
sink (zero of a vector field), 205–209
Smale, Stephen, 267–269
small stellated dodecahedron, 57, 150, 151
Sommerville, D. M. Y., 145
source (zero of a vector field), 205, 206, 208, 209
sphere, 3, 4, 30, 38, 39, 46, 55–57, 61, 67, 80, 87–98, 118, 120, 143, 155–159, 166, 169–172, 174, 177–184, 192, 194, 204–206, 208–210, 212, 214, 217, 219, 225, 227, 236–239, 244, 246, 247, 249, 250, 254, 255, 258, 260
n-sphere (Sn), 212, 213, 217, 244, 247, 260, 266–268
sprouts, 127–129
square knot. See knot, square
star polyhedron. See Kepler-Poinsot polyhedron
star-convex polyhedron, 97–99, 145, 150
Staudt, Karl Georg Christian von, 152–155
Geometrie der Lage, 152
Steiner, Jakob, 152
Steinitz, Ernst, 117
stereoisomer, 48
surface, xi, xii, 5, 158–186, 189, 191, 204, 206, 208, 209, 212, 214–216, 219, 223, 227–230, 232, 233, 235–239, 241–246, 250, 251, 254–260, 265
with boundary, 159, 162, 167–169, 171, 177, 180, 181, 183–185, 189–196, 214, 217, 244, 250
Tait, Peter Guthrie, 157, 188, 189, 199, 249
Terquem, Orly, 109
tetrahedron, 27, 31–34, 36, 40, 42, 46, 47, 56, 58–61, 64, 67, 68, 75, 77, 78, 224, 225
Thales, 31
Theaetetus, 1, 40–42, 44, 49, 86, 151, 181
Theodorus, 40
theorem of turning tangents, 222, 223, 232
Thistlethwaite, Morwen, 199
Thomson, William, 188, 189, 201
vortex model of atomic theory, 188, 201
Thurston, William, 214, 268, 269
Thurston’s geometrization conjecture, 268, 269, 285
topological invariant, 4, 177–179, 189, 225, 240, 241, 246, 250–252, 258–260
torsion coefficient, 257–261, 263, 266
torus, 3, 4, 148, 154, 156, 158–162, 167, 169, 170, 174–176, 178, 180, 182, 183, 185, 192, 193, 204–206, 208, 210, 225, 235, 239, 244, 245, 249–251, 255, 256, 258–260, 273
g-holed, 169, 170, 181, 182, 184, 192, 219, 258, 260
double, 161, 162, 169, 170, 174–176, 182, 183, 193, 208, 209, 251, 256, 258, 260, 266,
trefoil knot. See not, trefoil triangulate, 116, 184, 262
truncated icosahedron, 2, 27, 53
Tucker, Albert, 158
unavoidable set, 141, 142, 144
University of Berlin, 181
University of Breslau, 213
University of Cambridge, 127
University of Erlangen, 152
University of Göttingen, 133, 152, 181, 234, 235, 251, 262
University of Pisa, 251
University of Würzburg, 152
unknot. See knot, unknot
Vandermonde, Alexandre-Théophile, 187
vector field, 203–210, 212, 214–217
Verona, Fra Giovanni da, 52, 53
Waterhouse, William, 34, 35, 40, 86
Weber, Wilhelm Eduard, 182
Whitney, Hassler, 141, 171, 245
Whitney embedding theorem, 246
Whitney umbrella, 171
Zeno, 31