Why Beauty is Truth
Page 34
Disney, Thomas, 153
Disquisitiones Arithmeticae (Gauss, C. F.), 67, 134–135, 166
Distance, 141
Diverse Questions and Inventions (Fontana), 55
Division algebra, 157
complex numbers as, 157
real numbers as, 157
Division law, 152
DNA, 205, 241
Donaldson, Simon, 245
Donocrates, 18
Double cones, 39
Drums, vibration patterns, 217
du Motel, Stéphanie-Felicie Poterin, 107, 108
Duchâtelet, Ernest, 107
Dumas, Alexandre, 105
Dynamical theory, 199
Dyson, Frank, 212
e, 132
E = mc2, 191
Earth, rotation of, 187–188
Eclipse, 197
École des Ponts et Chaussées, 127
École Polytechnique, 133
Eddington, Arthur, 198, 212
Eight-squares formula, 265
Einstein, Albert, 173, 174, 181–182, 194–196, 212, 272, 278–279
education of, 183–184
fame of, 198
research papers of, 185–186
theory of relativity of, 189–190
youth of, 182–183
Einstein equations, 197
components of, 225
Einstein, Hermann, 181, 185
Einstein, Maria, 181
Electricity, 179, 187, 222
Electromagnetism, 184, 187, 191, 197, 222, 233. See also Maxwell equations
equations for, 174
as fundamental force, 234–235, 237–239
gravity and, 223–225
induction, 178
models of, 179
radation, 201–202
symmetries in laws of, 192, 232
Electrons, 213, 235, 256
Electroweak theory, 233
Elements of Geometry (Euclid), 19, 22, 25–26, 49
Elements of Geometry (Legendre), 99
Engel, Friedrich, 167
Enlil, 9
Equations, 5–6, 34. See also Quadratic equations
complex numbers and, 68
theory of, 101
Equivalence principle, 194
Eratosthenes, 20
Eridu, 2
ETH (Eidgenössiche Techhnische Hochschule), 183, 184
Euclid, 19–21, 28, 31, 49, 182, 193–194, 270
axioms of, 24–25
innovations of, 22–23
life of, 20–21
omissions of, 26–28
Eudoxus, 21
Euler, Leonhard, 66–67, 68, 69, 76, 85, 132, 148, 264
Euphrates, 1, 2
Exner, Franz, 204
Experimental Researches (Faraday), 179
Fano plane, 262, 270
Fantini, Luigi, 79
Faraday, Michael, 177, 178, 179
Ferdinand, Carl Wilhelm, 66, 67
Fermat, Pierre de, 36, 141, 264
Fermat primes, 135
Fermat’s Last Theorem, 36, 86, 126, 207
Fermi, Enrico, 236
Fermions, 236, 239
Ferrari, Lodovico, 54, 55–56
Fontana and, 55–56
Ferrari Tower, 90–91
Fertile Crescent, 1–2
Feynman diagrams, 249–250, 250
Fibonacci. See Leonardo of Pisa
Fibonacci sequence, 49
Fields Medals, 246, 248
Fields, ordered, 156
Fifth postulate, 22–23
Fifth roots, 68–69, 114
Fior, Antonio Maria, 54
Fitzgerald, Edward, 33
Flatland (Abbott), 227
Fontana, Niccolo, 45, 55
Cardano, Girolamo, and, 53–54
Ferrari and, 55–56
Forces, 233–235. See also
Electromagnetism; Gravity; Nuclear force
characteristics of, 238
Formulas, breeding of, 89
Four Squares Theorem, 76, 77, 264–265
Fourier, Joseph, 103
Fourth dimension, 225, 226
Fractions, 144
in Babylonian times, 12–13
France, Prussia and, 161
Frank, Amelia, 219
French Revolution, 76–77
Freudenthal, Hans, 271
Fridrichsen, Henriette, 95
Fundamental particles, x, 230
Fundamental Theorem of Algebra, 70
G2-manifolds, 254
Galileo, 56, 277
Galois, Évariste, ix, xi, 98–99, 101, 104–105, 107, 162, 275
arrest of, 105–106
duel of, 109–110
education of, 99–100
innovations of, 110–111
proofs of, 106
Galois group, 113, 116–117
limitations of, 117
Galois, Nicolas-Gabriel, 99
Galois test, xii
Galois’s Theory of Algebraic Equations (Tignol), 77
Garrone, Lorenzo, 183
Gauss, Carl Friedrich, 63–64, 83, 88, 127, 134–135, 166
death of, 73
education of, 64–65
fame of, 71
theorems of, 70
Gauss, Gebhard Dietrich, 64
Gauss plane, 148
General Theory of Equations (Ruffini), 79
General Theory of Terrestrial Magnetism, 72–73
Geodesics, 196
Geometry
Euclidean, 22–28
noncommutative, 255
of quadratic equation, 15–16
Germain, Sophie, 105
Gibbs, Josiah Willard, 154, 155
Gjerstad, 84
Global Positioning System, 180
Gluons, 237
Göttingen, 71, 72, 160
Gout, 154
Grand Unified Theories (GUTs), 240, 248
Grassmann, Hermann, 154, 155
Graves, John, 261–262, 263, 265, 266
Graviton, 237
Gravity, 193, 195, 196, 197, 225, 233, 234, 251
electromagnetism and, 223–225
loop quantum, 255–256
The Great Art (Cardano, G.), 47, 53–54, 55, 60, 75
Green, Michael, 252
Grossmann, Marcel, 184–185, 194–195, 198
Group disease, 219–220
Group theory, ix, xi, 111–113, 125, 161
cubic equation and, 116
quadratic equation and, 116
quartic equation and, 116
Group Theory and Its Application
to Quantum Mechanics of Atomic Spectra (Wigner, E.), 217
Guigniault, M., 103–104
GUTs. See Grand Unified Theories
Gymnasium, 66
Habilitation thesis, 73, 204
Hahn, Otto, 201
Halley, Edmond, 76
Hamilton, William Rowan, 137–138, 140, 152, 154, 260, 268
on complex numbers, 148–150
education of, 137–138
Hamiltonian systems, 139
Hanging Gardens, 2
Hansteen, Catharine, 95
Hansteen, Christoffer, 85
Hansteen, Johanne, 92
Hardrons, 251
Hardy, Godfrey Harold, 267
Harmony, 21
Harnack, Adolf von, 200–201
Hasenhöhrl, Friedrich, 204
Hashishiyun, 39
Hawkins, Thomas, 167
Heisenberg, August, 206
Heisenberg, Werner, 206–210, 213, 221, 278
Heliocentric theory, 188
Helmholtz, Hermann von, 166, 189, 200
Henry VIII, 178
Hermite, Charles, 133, 134
Hertz, Heinrich, 180, 186
Hexagons, constructing regular, 30–31
Hilbert, David, 134, 165, 197
Hippasus of Metapontum, 47, 130
Hitler, Adolf, 203, 204, 208
Hoesslin, Marga von, 201
Holmboe, Bernt, 85,
93–94
Holten, Florence Hannah, 211
Homer, 19
Hooke, Robert, 175
Horowitz, Gary, 253
Huguenin, Ulrich von, 136
Humboldt, Alexander von, 71
Hunter-gatherers, 1–2
Hurwitz, Adolf, 157, 184, 266
Hutton, Sarah, 137
Huxley, Thomas, 278
Huygens, Christian, 177
Hydrogen, 204
Hyperbola, 40
Hyperspheres, 228
i, 260
Identity, 120–121
Imaginary numbers, 146, 148–149
Inertial frames, 188–189
symmetry and, 189
Infantozzi, Carlos, 107
Inference, 229
Infinite decimals, 145
Inquisition, 56
Institute for Advanced Study, 219, 244, 245
Invariant theory, 191
Inverse square law, 40
Iraq, rubble mounds in, 6
Irrationals, 130, 131
Ishtar Gate, 8
al-Jabr w’al Muqâbala (al-Khwarizmi), 36–37
Jacobi, Carl Gustav Jacob, 67–68
Jean-Louis, 107
Jolly, Philipp von, 199, 200
Jordan, Camille, 122
Jordan, Pascual, 271
Journal de Mathématiques Pures et Appliquées, 127
K(9), 240
Kaiser Wilhelm Institute, 216
Kaluza, Theodor, 224–225, 229, 253
Kaluza-Klein theory, 230
Kant, Immanuel, 182
Karnak, 18
Kelvin, Lord, 199
Kemp, Christine, 86, 87, 93, 94, 95
Kepler, Johannes, 40, 276, 277
Khayyám, Omar, 33, 37, 38–39, 41–44
al-Khwarizmi, Mohamed ibn Musa, 36–37
Killing, Wilhelm Karl Joseph, 165, 166, 167, 168, 169, 170–171, 272
proofs of, 168–169
Killing’s list, 239, 268
Kinetic theory, 186
Kirchhoff, Gustav, 200
Klein, Felix, 133, 160
Klein, Oskar, 230
Klein’s Elangen program, 270
Kline, Morris, 50
Kollros, Louis, 108
Königsberg, 224
Kragh, Helge, 278
Kronecker, Leopold, 160
Kummer, Ernst, 160, 166
Küssner, Elizabeth, 134
Lacroix, Sylvestre-François, 103
Lagash, 2
Lagrange, Joseph-Louis, 75–77, 80, 81, 83, 111, 264–265
Lambert, Johann, 130
Lambert’s proof, 131, 133
Laplace, Pierre Simon, 77
Lavoisier, Antoine, 77
Law, 3–4
Laws of nature, x
Le Monnier, Renée-Françoise-Adélaide, 77
Legendre, Adrien-Marie, 93, 99–100, 103
Lehrbuch der Algebra (Weber, H.), 216
Leibniz, 148
Leonardo of Pisa, 49–50, 264
Levi-Civita, Tullio, 195
Liber Abbaci (Fibonacci), 49
Lie algebras, 166, 168, 169, 245, 268
simple, 167
Lie groups, xii, 163, 165, 214, 263, 267, 271, 272
exceptional, 169, 170, 253, 254, 263, 269, 270, 271–272
features of, 164
in physics, 253–254
Lie, Marius Sophus, 159, 160, 162, 170, 213
Liebniz, Gottfried Wilhelm, 5
Light, 174–175, 197–198
change of phase in, 232, 233
speed of, 190, 191
Limitations, 28
Lindemann, Carl Louis
Ferdinand von, 133–134
Lindemann’s proof, 134
Linear equations, 14
Lines, 28, 122, 270
Liouville, Joseph Louis, 111, 127, 131
Logarithms, 132, 148
Loop quantum gravity, 255–256
Lorentz group, 192, 232
Lorentz, Hendrik, 192, 212
Lorentz invariance, 194
Louis XVIII, 99, 103
Louis-Le-Grand, 100, 133
Louis-Philippe, 103, 104
Ludus Algebrae et Almucgrabalaeque, 37
Magic square, 271
Magnetism, 179, 187, 222
Manhattan Project, 220
Manifolds, 73
Map making, 48
Marconi, Guglielmo, 180
Marduk, 8
Maric, Mileva, 185
Marie, Anne, 86
Maskerade (Pratchett), 205
Mathematical Principles of Natural Philosophy (Newton), 174
Mathematical Thought from Ancient to Modern Times, 50
Mathematics
ancient study of, 2
applied, 22
in Babylon, 13–14
beauty and, 278, 280
competitions, 54
role of, 5
symmetry and, 279–280
truth and, 275
universe and, xiii
Mathias, Hans, 85
Matrix, 122, 208, 276
Matter, 186, 197
Maurice of Nassau, 144
Maxwell equations, 180, 187–189, 222
symmetry of, 191
Maxwell, James Clark, 174, 177, 178–179, 184
Maxwell-Boltzmann doctrine, 199
McGovern, George, 244
Mechanico-optical analogy, 142
Mechanics (Euler), 132
Meitner, Lise, 201
Merck, Marie, 200
Mesopotamia, 1–2
history of, 7
Metric, 195–196
Michelson, Albert, 189, 191
Michelson-Morley experiment, 189
Micheria, Chiara, 50
Mills, Robert, 238
Minkowski, Hermann, 184, 192
Minkowski space-time, 192
geometry of, 193
Moon, movements of, 23–24
Morley, Edward, 189, 191
Morse, Marston, 247–248
Motzfeldt, Ernst, 159
Mowaffak, Imam, 37, 39
M-theory, 254
al-Mulk, Nizam, 37
Müller, Hermann, 200
Multidimensional spaces, 73
Multiplication, 150
Nabopolassar, 8
Nabu, 8
Naishapur, 37
Nambu, Yoichiro, 249
Napoleon, 77, 79, 161
Nature, beauty and, 278–279
Nazis, 203, 210
Nebuchadnezzar, 6, 7, 8
Negative numbers, 144
square roots of, 59–60, 260
Negatives, 29
Negentropy, 205
Neugebauer, Otto, 13–14
Neusis constructions, 27
Neutrinos, 235–236
Neveu, André, 251
Newton, Isaac, 5, 174, 175–176, 277
physics of, 190
Nichomachus, 35
Nineveh, 2
Nippur, 2
Nixon, Richard, 243
Nobel Prize, 185, 204, 209, 220, 244
Noether, Emmy, 165
Noncommutative geometry, 255
Notation systems, 10–11, 144–145
Babylonian, 11–12
Nuclear force, 233, 235
Nuclear weapons, 210
Number theory, 76
Numbers. See also Complex numbers;
Imaginary numbers;
Negative numbers;
Prime numbers;
Real numbers;
Square numbers;
Transcendental numbers;
Triangular numbers
notation systems for, 10–11
O(1), 240
Observation, 23
Octonions, xii, 259–260, 268, 273
units of, 262–263
Octooctonions, 272
On the Conditions of Solvability of Equations by Radicals (Galois), 104–105
On the Revolutions of the Heavenly Spheres (Copernicus), 56
&n
bsp; Opticks (Newton), 175
Order, 156
Ørsted, Hans, 178
Osiander, Andreas, 56
Oslo Fjord, 84
Osthoff, Johanna, 71
Pacioli, Luca, 50
Pais, Abraham, 174, 220
Pappus, 21
Parabola, 40
Paris, 87
Particle physics, 231
Pauli principle, 236
Pauli, Wolfgang, 208–209
Pausanias, 18
Peirce, Benjamin, 154, 155
Perihelion position, 196
Peritonitis, ix
Permutations, 81, 111–112
group property of, 121
multiplying, 81–83, 111
subgroups of, 114
symmetries and, 113
Pfaff, Johann, 67, 71
Phase, 232
shift, 232
Philip II, 18
Philosophy, 221
Photoelectric effect, 186
Photons, 237
Physics, 149, 173, 194, 199
lie groups in, 253–254
Pi, 132
Piazzi, Giuseppe, 70–71
Planck, Max, 200, 202, 203, 204, 218–219
Planes, 147
Plasma, 242
Plato, 20, 21, 44, 269, 277
Plücker, Julius, 159–160, 165–166
Poetry, 139–140
Poincaré, Henri, 133, 192
Poinsot, Louis, 103
Poisson brackets, 213
Polchinski, Joseph, 257
Polygons, 29
regular, 29, 30
Polynomials, 57, 81
Poncelet, Jean-Victor, 159
Positron, 236
Powers, 35
Practica Geometriae (Leonardo of Pisa), 49
Pratchett, Terry, 176, 205
Preserve, 118–119
defining, 119
Prime numbers, 264
Principle of Least Time, 141
Problems of antiquity, 125–126
Projections, 269, 270
Proofs, 22–23
necessity of, 23–24
Proportion, 118
Proposed Draft of an Attempt to
Treat the Results of a Cone Intersecting a Plane (Desargues), 271
Protons
stability of, 240
structure of, 238–239
symmetry groups in, 239
Protractors, 128
Prussia, France and, 161
Ptolemy, 20
Pythagoras, 130, 144
cult surrounding, 21
Pythagorean theorem, 25, 144, 193, 264
Pythagoreanism, 21–22
Quadratic equations, 13–14, 42, 57, 78, 280
geometric picture of, 15–16
groups and, 116
solving, 90
Quadratic reciprocity, 66–67
Quadratrix, 27
Quadruples, 152
Quanta, 236
Quantum chromodynamics, 233, 238
Quantum field theory, 245, 248
Quantum mechanics, 192, 210, 213, 230, 259
Quantum physics, x
Quantum theory, xiii, 202, 209, 230
Copenhagen interpretation of, 205
relativity and, 221–222
Quarks, 237, 238, 256
Quartic equation, 61
groups and, 116
solving, 90
Quaternions, xii, 142–143, 153, 154, 156, 214, 261, 263
Quateroctonions, 272
Quintic equations, xi, 61, 78, 83, 97, 114, 117, 280