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A Passion for Mathematics
Numbers, Puzzles, Madness, Religion,and the Quest for Reality
Mathematics is the hammer
Numbers percolate like bubbles in the ocean of mathematics.
Math puzzles, factoids, quotations, trivia, formulas...
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A Smorgasbord of math puzzles, factoids, quotations, trivia, formulas, and much more...
Are you fascinated by numbers and want to learn more? Does the vast world of math problems that humans have solved�and the ones we have yet to begin to comprehend�give you goose bumps? If so, this is the book for you. A Passion for Mathematics is an educational, entertaining trip through the curiosities of the math world, blending an eclectic mix of history, biography, philosophy, number theory, geometry, probability, huge numbers, and mind-bending problems into a delightfully compelling collection that is sure to please math buffs, students, and experienced mathematicians alike.
Ramanujan |
A fragment from the Preface of A Passion for Mathematics: "Readers of my popular mathematics books already know how I feel about numbers. Numbers are portals to other universes. Numbers help us glimpse a greater universe normally shielded from our small brains that have not evolved to comprehend fully the mathematical fabric of the universe. Higher mathematical discussions are a little like poetry. Danish physicist Niels Bohr felt similarly about physics when he said, "We must be clear that, when it comes to atoms, language can be used only as in poetry.'"
"When I think about the vast oasis of numbers that humans have skimmed from the shoreless sea of reality, I get a little shiver. I hope you'll shiver too as you glimpse numbers ranging from integers, fractions, and radicals to stranger beasts like the transcendental numbers, transfinite numbers, hyperreal numbers, surreal numbers, the "nimbers," the quaternions, biquaternions, sedenions, and octonions. Of course, we have a hard time thinking of such queer entities, but from time to time, God places in our midst visionaries who function like the Biblical prophets who touched a universe inches away that most of us can barely perceive." |
Mathematical Marvels from Beyond the Edge
The Ramanujan Code
Blood Dreams and God's Mathematicians
The Mathematical Smorgasbord
Explanation of Symbols
Cultivating Perpetual Mystery
In which we encounter religious mathematicians, mad mathematicians, famous mathematicians, mathematical savants, quirky questions, fun trivia, brief biographies, mathematical gods, historical oddities, numbers and society, gossip, the history of mathematical notation, the genesis of numbers, and "what if?" questions. Do humans invent mathematics or discover mathematics? What is reality? Why does the universe seem to operate according to mathematical laws? What is the most forgettable license plate? Which brilliant mathematicians were mass murderers? What is Ramanujan's tongue? Why was the first woman mathematician murdered? When was the multiply symbol first actually used in a book? Who was the "Number Pope"? Why are there 360 degrees in a circle? Who invented the symbol for infinity? In how many ways can you place Xs and Os on a tic tac toe board? What mathematical genius despised and feared mirrors? What happens when you double the salary of every mathematics teacher? What romantic comedy has the most complicated mathematics ever portrayed in a movie? What number is on Agent Smith's license plate in the movie Matrix Reloaded, and why? What is the earliest evidence we have of humans counting? Can animals count? Which mathematicians claimed that their mathematical breakthroughs came directly from God? What was the largest number ever used in the title of an American movie? Why is there no Nobel Prize for Mathematics? How much mathematics can we know? Does God exist?
In which we encounter fascinating numbers and strange number sequences. We'll explore transcendental numbers, octonions, surreal numbers, obstinate numbers, cyclic numbers, Vibonacci numbers perfect numbers, automorphic numbers, prime numbers, Wilson primes, palindromic primes, Fibonacci primes, Sophie Germain primes, Baxter-Hickerson primes, star-congruent primes, narcissistic numbers, amenable numbers, amicable numbers, p-adic numbers, large palindromes, factorions, hyperfactorials, primorials, palindions and hyperpalindions, exotic-looking formulas for pi, the Golay-Rudin-Shapiro sequence, wonderful Pochhammer notation, famous and curious math constants (like Liouville's constant, the Copeland-Erd�s constant, Brun's constant, Champernowne's number, Euler's gamma, Chaitin's constant, the Landau-Ramanujan constant, Mill's constant, the golden ratio, Ap�ry's constant, and constants even more bizarre)... Who discovered pi? What are "nimbers"? What would happen if everyone's body weight were quantized and came in multiples of pi pounds? Could Jesus of Nazareth ever have worked with a negative number, like -3? Could Jesus multiply two numbers? Is it true that we all live happy lives, coded in the endless digits of pi? Why is the number 13 considered unlucky?
In which we encounter treasure chests of zany and educational math problems involving algebra, fractions, percentages, classic recreational puzzles, and various mathematical manipulation. Some are based on problems over a thousand years old. Others are brand new. Get ready to sharpen your pencils and stretch your brains! Can you show why x^0 = 1? What is a fragile fraction? What is a hyperpower tower? The doomsday formula? The Riemann hypothesis? Are there difficult math problems lurking in the ancient Indian sex manual, The Kama Sutra? Experiment with puzzles involving alien ships, lying robots, sparkling angels, wealthy jewelers, wizards, Martians, alien spheres, the spread of disease, the liquefaction of human flesh, bacterial growth, extraterrestrial slugs, the Bakhshali manuscript, and the Leaning Tower of Pisa.
In which we explore tiles, patterns, position problems, arrays, Venn diagrams, tic tac toe, other games played on boards, K�nigsberg bridges, catenaries, Loyd's and Dudeney's puzzles, Sherck's surface, magic squares, lituuses, inside-out Mandelbrot sets, the Quadratrix of Hippias, hyperspheres, fractal geese, Schmidhuber circles, Pappus' Arbelos, Escher patterns, and chess Knights. Would you like to live in a house shaped like a torus? If everyone on Earth stood on his own 3-by-3 foot piece of ground, how large an area would humanity cover? Is the universe shaped like a dodecahedron? Would it be possible to cram a polar bear into an 11-dimensional sphere with a 6-inch radius? What is a golygon? The mysterious sangaku or the Stomachion of Archimedes? The Lute of Pythagoras? Fractal froth? Osculating circles? Explore The Simpsons and geometry, exotic chess games, strange spirals, caterpillar planets, Jordan curves, alien colonies,M�bius mirrors, Aesculpian mazes, exploding circles, grids of the gods, Procyon maneuvers, Koch curves, robot worms, Ikeda attractors, loxodromes, Borromean rings, chaos machines, Schoenberg curves, Wither's attractors, zenograms, butterfly curves, Lorenz attractors, Johnson functions, and magic squares.
In which we explore casinos, logic, guessing, decisions, combinations, permutations, competition, possibilities, games involving choice, monkeys typing Hamlet, Benford's law, permutations, combinatorics, alien gambits, Rubik's cubes, card shuffles, marble mazes, non-transitive dice, and dangerous movements of air molecules. How can you instantly turn an unfair coin into a fair one for coin tossing games? What is the probability that you will hit Africa with a dart thrown at a world map? In randomly played tic-tac-toe, what is the chance that the first player wins? In how many different ways can we arrange the 52 cards in a playing card deck? What piano note is a blindfolded person most likely to play when randomly striking a key? How does chance and probability affect the way we look at the world? Play with robot jaws, triangles and spiders, apes in barrels, and men that vaporize you with a touch. Play the board game of the gods and enter the tunnels of death and despair.
In which we explore very large numbers, the edges of comprehension, infinity, the funnel of Zeus, the infinite gift, Skewes' number, the Monster group, G�bel's number, the Erd�s-Moser number, the awesome quattuordecillion, Archimedes' famous "Cattle Problem," classic paradoxes, Gauss's "measurable infinity," and Knuth's arrow notation. What is the largest number that a future computer could work with or yield as a result of a computation? How many life experiences can a person have? What is the largest known number described in a work of fiction? What is the number of possible chess positions? What is an infinite gift? How large a number is the Buddhist asankhyeya? Just how enormous is 9^9^9? What is an infinity keyboard? How many numbers have a 7 in them? How big is infinity? How large are our minds?
In which we explore artistic forms generated from mathematics -- delicate fungi, 22nd-century cityscapes, fractal necklaces, alien devices, and a rich panoply of forms that exhibit a cascade of detail with increasing magnifications.
Al-Battani, 131
Adams, Douglas, 84, 244, 246
Aeon, 244
Agnesi, Maria, 27
AIDS, 219, 349
Al-Samawal, 30
Alhazen, 265
algebra, 113, 131
Al-Khwarizmi, 131, 282
alphamagic squares, 180
amateur math, 182
annihilation magic square, 180
ant puzzle, 149, 160, 320
antimagic square, 180
Antoine, Louis, 275
apocalyptic magic square, 181
Archimedes, 199, 245
arebelos, 170
arithmetic progressions, 108
arrangements, 213, 214
Asimov, Isaac, 83, 90
attractors, 171,173, 189, 332, 333
Babbage, Charles, 24
Bakhshali problem, 128, 309
Barber paradox, 240
Barrow, John, 252
base, 50
Battles, Zachary , 275
Beal reward, 32
Benford's law, 211
Billion, 47, 277
Bloch, Andr�, 254
Boll, David, 87
Boltzman, Ludwig, 271
book towers, 169, 332
Boole, George, 18
Borromean rings, 171
Bourbaki, N., 39, 271
Brahmagupta, 114
Briffault, Robert, 63
Brothers, Harlan, 76
Buddhist, 244
butterfly curves, 188
calculus, 30, 34-35, 267, 308
Cantor, Georg, 24, 216, 237, 238, 260, 327, 362
card arrangements, 217, 229, 348
cardinality, 47, 49
casualty equations, 267
catenary, 142
cattle problem, 245
Chalmers, David, 47
chaos, 190, 333
chess, 143, 144, 145, 185, 195, 196, 197, 198, 199, 217, 247, 340, 347
circles, 16, 146, 154, 159, 164, 165, 170, 172, 193, 203, 206, 254, 317, 324, 329, 330, 333, 334, 350
circular permutations, 213
coin tosses, 211, 227, 228, 342, 355, 358
coincidence, 36, 90, 92, 98, 101
combinatorics, 213, 215
Constantine window puzzle, 203, 342
Constants
0.065988..., 90
0.692200..., 90
1.506592..., 90
1.92878... , 97
Apery's, 90, 359
Brun's, 93
Chaitin, 77
Champernowne, 59, 280
coincidental, 90
Copeland-Erd�s, 60, 280
Euler's gamma, 77, 104, 130
Euler's number, 52, 53, 55, 58, 76, 82, 83, 86, 87, 90, 92, 101, 102, 104, 106, 117, 123, 127, 142, 143, 178, 265, 276, 279, 280, 287, 308, 314, 336
Euler-Mascheroni, 77, 104
fine structure, 86, 87
golden ratio, 80-82, 86, 95, 104, 105, 143, 198, 315
infinite power tower, 90
Landau-Ramanujan, 101
Liouville, 58, 278, 280
Mandelbrot, 90
Mill's (a), 98
pi, 16, 29, 31, 35, 47, 52, 53, 55, 58-60, 62, 72, 75, 76-78, 80-92, 101, 102, 104-106, 109, 143, 147, 182, 256, 265, 270, 276, 277, 279, 280, 283, 286, 287, 292, 294, 314
Thue, 60, 280
continuity, 166, 330, 331
continuum, 48
continuum hypothesis, 237
continuum puzzle, 151, 322
Conway, John, 57, 77, 93, 94, 185, 229
counting, 13, 16, 19, 235, 254, 257
counting and animals, 257
crescent puzzle, 145, 146, 316, 317
D�rer, Albrecht, 179
Dase, Johann, 31
decimation, 359
derivatives, 308
Descartes, Ren�, 24, 132
dice, 222, 352
Diderot, Denis, 30
Diopantus, 258
doomsday equation, 127
Dudeney, Henry, 176, 177, 178, 186, 336
dynamical systems, 332
Dyson, Freeman, 40, 41, 114, 248
Earls, Jason, 107
Education, 43
Edwards, Anthony, 140
Ehrenfest, Paul, 5
Einstein, Albert, 24, 26, 27, 40, 42, 271
Elkies, Noam, 120
ellipse game, 174, 335
entropy, 106
epitaphs, 39
Erd�s, Paul, 18, 26, 33, 51, 65, 69, 72, 246, 256, 290
Eschergram, 335
Euclid, 187
Euler, Leonhard, 18, 30, 33, 52, 77, 80, 83, 85, 96, 98, 103, 105, 120, 137, 185, 265, 274, 359
even number, 47, 77
exponential growth, 126, 308
factors, 100, 229
Fermat numbers, 59
Fermat spiral, 169
Fermat, Pierre de, 32, 187, 254
Fermat's Last Theorem, 120
Feynman, Richard, 87
Fibonacci, 49, 50, 66, 71, 73, 95, 98, 107, 212, 287, 291
fifteen puzzle, 147, 317
fine structure constant (a), 86, 87
Ford froth, 163
Fourier, Jean, 263
fractals, 85, 87, 90, 155, 163, 175, 188, 190, 191, 194, 251, 325, 331
caves, 175
drums, 194
geese, 190
swords, 194
The Fractal Murders (novel), 42
Funnel of Zeus, 364
G�del, Kurt, 22, 41, 248, 258, 259, 269
G�del's theorem, 259
Gabriel's horn, 364
Galois, �variste, 266
gambling, 211
game shows, 37, 271
Gamm, R�diger, 21
Ganson, Mark, 89, 107, 349
Gardner, John, 275
Gardner, Martin, 94, 162, 183, 219, 251, 274, 275
Gauss, Carl Friedrich, 17, 237, 284
gears, 164, 329
Gell-Mann, Murray, 8
geometry, 135-209
Germain, Sophie, 27
Go, 244
G�bel's sequence, 241
Golay-Rudin-Shapiro Sequence, 79
Goldbach conjecture, 59
Golygon, 144, 314, 315
googol, 69
googolplex, 236
Gosper, William, 85
Gr�nbaum, Branko139
graph theory, 137
grasshopper sequences, 68, 289
greatest common divisor, 319
Greek notation, 31
Grim, Patrick, 155, 325
group theory, 266
Guberlet, Constantin, 49
Guy, Richard, 94, 229
Hales, Thomas, 4
Hamburger, Peter, 139
Hankel, Hermann, 33
Harborth configuration, 192
Hardy, G. H., 2, 23, 142, 143, 241, 247, 251, 257, 275, 290
harmonic series, 103, 119, 130, 132, 310, 311
Heisenberg Uncertainty Principle, 82
Heisenberg, Werner, 4
helical curves, 188
Heron's problem, 173, 334
heterosquares, 150, 322
hexgonal cutting, 165
hex-congruent primes, 100
Hilbert, David, 33, 101
hoax, 90, 294
Ho�n�-Wronski, Josef, 32
Honaker, G. L., 99, 108, 109
Hypatia, 259
hypercubes, 192, 194
hyperpower towers, 123
hyperpyramids, 195
hyperspheres, 193, 359
Ikeda attractor, 332
infinite series, 102
infinity symbol, 25
infinity keyboard, 365
infinity, 26, 233, 237, 239, 240, 241, 244, 363, 365
infinite product, 103
James, William, 275
Jeans, James, 17
Jefferson, Thomas, 204
Jerusalem crystal puzzle, 205
Jesus, 61, 282
jeweler puzzle, 116
Jews, 35, 270
Jacobs, Bobby, 34
Johnson functions, 198
Johnson's theorem, 317
Jordan curves, 160, 328
Kaczynski, Ted, 261
Kalman, Rudolf, 106
Kama Sutra puzzle, 125, 307
Kant, Immanuel, 14
knight placement, 197, 198, 340
knights tours, 185, 186, 195, 196, 197
Knuth, Donald, 20, 94, 236, 361
Koch curve, 166, 331
K�nigsberg bridges, 137
Kronecker, Leopold, 51
L'H�pital, Gillaume de, 35
Lagrange, Joseph, 35, 98
Lanchester, F., 267
large law, 235
Latin squares, 320
lattice numbers, 196, 339
least common multiple, 302
Lego sequence, 65, 287
Leibniz, Gottfried, 22, 34, 36, 267, 277
license plates, 16, 36, 255
limit, 167, 332
Liouville, Joseph, 58
lituus, 187
Lobachevsky, Nikolai, 36, 144
Lorenz attractor, 189
lost in hyperspace, 150, 321
Lotka-Volterra equations, 267
love formulas, 129
Lovelace, Ada, 264
loxodrome, 170
Loyd, Sam, 7, 113, 114, 115, 145, 146, 147, 295, 296, 297, 317
lute of Pythagoras, 204, 315
lyrics, song, 270
MacArthur awards, 41, 48
magic squares, 56, 57, 150, 153, 154, 179-184, 195, 278, 321, 324
magic circles, 165, 329
magic spheres, 166, 330
Mahavira problem, 127, 308
Mandelbrot set, 85, 87, 90, 188, 191, 251, 252
map coloring, 207
map projections, 345
Mathematicians
blind, 42, 274
epitaphs, 39, 271
first, 37
homosexual, 26, 257
madness and creativity, 14, 23, 26, 39, 260, 263
mirror phobic, 257
murderers, 25, 254, 261
murdered, 259
number Pope, 23
prolific, 28, 33
religious, 14,15, 17, 18, 20-24, 37, 38, 265
savants, 21, 31, 34
starving, 258
women, 22, 26, 27, 28, 259, 263, 264
Mathematics
beauty, 13, 239
computers, 4
dreams, 15
education, 43, 263, 275
Jews, 35, 73, 270
game shows, 37
God and religion, 3, 14, 15, 17, 18, 23-25, 27, 28, 30, 31, 33, 34, 50, 74, 80, 83, 143, 235, 238, 254, 260, 261, 269, 272
proofs, 4
mind, 4
movies, 22, 26, 35, 36, 93, 260, 270, 271
novels, 42
nature and cosmos, 8, 13, 30, 42, 50, 52, 84, 162, 183, 198, 223, 247-249, 268, 273, 340
plays, 40
predictive power, 8
real world, 8
reality, 13, 30, 40, 41, 43, 220, 226, 229, 249, 251, 268, 273, 275
simultaneity, 36
The Matrix, 2, 36, 271
maze, 161, 214, 215, 328, 345
Mersenne numbers, 19
Mersenne, Marin, 18
mirror phobia, 19, 257
Mirror magic squares, 182
Mittag-Leffler, G�sta, 272
M�bius strip, 199
molecule puzzle, 202, 341
The Modulor, 2
Mondrian puzzle, 142
monkeys typing Hamlet, 211, 343
moon puzzle, 145, 146, 316, 317
Morin, Bernard, 274
Morse code, 285
Morse-Thue sequence, 286
Moskowitz, Paul, 255
multiplication principle, 213
murder, 254
Newton, Isaac, 14, 17, 29, 30, 34, 35, 36, 265, 267, 271
Nobel Prize, 40, 272
Nobl, George, 119, 301
Noether, Emmy, 27
non-transitive dice, 222, 352
Notation
good notation, 169
Greek, 31
bases, 50
Knuth, 236, 361
Pochhammer, 106
See also symbols
number theory, 49
number Pope, 23
Numbers
153, 68-69, 289
216, 270
666, 74, 75, 90, 91, 92, 109
abundant, 280
algebraic, 48, 54
almost integers, 85, 87, 90, 294
amenable, 88
ammicable, 93
apocalypic Fibonacci, 73
Armstrong, 289
asankhyeya, 243
augmented primes, 61, 281
automorphic, 264, 287
Baxter-Hickerson prime, 109
billion, 47, 277
biquaternions, 48, 57
cardinal, 47, 49, 260
catalan, 71
complex, 48, 53, 54, 95
composite, 51, 96
congruent primes, 99, 100
continued fractions, 75, 314
coprime, 75, 91
cube, 91-93, 119, 294, 295, 302
cyclic, 89
decillion, 236
diagram, 48, 54
emirp, 73
Erd�s-Moser, 246
even, 47, 277
exclusionary squares, 60, 281
factorial, 92, 104, 247
factorions, 68, 289
father prime, 89
Fibonacci, 49, 50, 66, 71, 73, 95, 98, 107, 212, 287, 296
Fibonacci prime, 98
fractions, 48, 115, 297, 314
fragile fractions, 115, 297
G�bel's, 241
googol, 69
googolplex, 236
Graham's, 361
Hamiltonian integers, 48, 55
hex-congruent primes, 100
hypercomplex, 48, 54, 55
hyperfactorial, 92
hyperimaginary, 201
hyperpalindion, 107
hyperpower towers, 123
hyperreal, 48, 95
integers, 47, 48, 49, 51, 58
irrational, 48, 54, 70, 72, 291, 293, 327
isoprimes, 60, 97, 281
kinky, 242
large, 234
legion's, 109
leviathan, 243
Mersenne, 19
narcissistic, 289
natural, 48, 54
nimbers, 48, 57
novemdecillion, 236
obstinate, 62, 284
octonions, 48, 55, 57
ordinal, 260
p-adic, 95
palindions, 107
palindromes, 70, 71, 92, 107, 291
palindromic primes, 97, 108, 109
palinpoints, 66, 288
perfect, 73, 74, 240
prime, 19, 49, 50, 51, 60, 61, 74, 75, 77, 78, 79, 80, 83, 86, 88, 89, 92, 93, 96, 97, 98, 99, 100, 103, 104, 108, 109, 110, 183, 184, 281, 287, 288, 295
primorial prime, 92
pyramidal, 67
quaternions, 48, 54, 55
quattuordecillion, 236
radicals, 48, 96, 103, 106, 143, 266
rational, 19, 47, 53, 104, 120, 129, 237, 268, 302, 327, 362
real, 47, 48, 363
relatively prime, 75, 91
repunit primes, 281
Roman, 40
round, 99
scalar, 54
sedenions, 48, 55, 56
sextillion, 236
skewes, 241
Sopie Germaine prime, 98
square, 68, 70, 92, 98, 281, 291, 294
star-congruent primes, 99
superfactorial, 247
surreal, 48, 93, 94
taxicab, 58, 279
transcendental, 48, 53-55, 58, 279, 280
transfinite, 47, 260
transionic, 248
triangular, 78, 91, 92, 288
twin primes, 93
ultrahex, 235, 360
undulating, 50, 62, 80, 284, 289
untouchable, 65, 287
undulating primes, 80
vampire, 287
vibonacci, 107
vigintillion, 236
Wilson prime, 97
zebra, 102
zero-free, 109
osculatory packing, 334
Pagels, Heinz, 113
Pappus' Arbelos, 170
parabolic spiral, 169
paradoxes, 58, 240, 248, 279
parallel universes, 31, 268
paraskevidekatriaphobia, 293
Pascal, Blaise, 14, 15, 36, 71, 254
Pascal's triangle, 71
Pe, Joseph, 66, 86, 255, 310
pendulums, 190
Penrose, Roger, 251
pentagonal numbers, 286
pentagram, 198
permutations, 213, 215
Perrin sequence, 290
persistence, 288
Pi,
pi, living in digits, 62, 283
pi-magic squares, 182
pi-palindromic prime, 109
See also constants, pi
Pincus, Steve, 106
Pisa puzzle, 113, 295
Platonic patent, 200
Platonic solids, 154, 325
plays, 40
Pochhammer notation, 106, 107
polyhedreal universe, 340
Pontryagin, Lev, 274
Pope Sylvester II, 23
Poussin proof, 78
prime numbers, See numbers, prime
prime magic square, 183, 184
probability, 209-232
proofs, 4
Pythagoras, 24, 37-39, 261
Pythagoras, lute, 204, 315
Pythagorean triangle, 187, 287
quadratix of Hippias, 192
radicals, 48, 96, 103, 106, 143, 266
Ramanujan, Srinivasa
formulas, v, 76, 82, 85, 100-106, 279, 294
life and philosophy, 2-3, 15, 16, 42, 99-101, 108
plays, 40
random chords, 350
random selection, 277-278
reality, 13, 30, 40, 41, 43, 220, 226, 229, 249, 251, 268, 273, 275
Ribenboim sequence, 68
Rice, Marjorie, 182
Riemann zeta function, 103, 359
Riemann hypothesis, 130
Riemann, Georg, 21, 130
Rivera, Carlos, 74, 84, 109
Roman numerals, 40, 273
rope puzzle, 153
Rubik's cube, 216, 246
Rupert puzzle, 178, 337
Russell, Bertrand, 18, 141, 169, 248
Safford, Truman, 34
Sallows, Lee, 180
sangaku geometry, 201
Saunderson, Nicholas,274
savants, 21, 31, 34
Scherk's surface, 174
Schmidhuber circles, 206
Schoenberg curves, 173, 175
Schr�dinger's wave equation, 40
series
indefinitely divergent, 131
geometric, 296
slow-growing, 120
See also harmonic series
set theory, 51
seven, 74
Shandy paradox, 248
Sierpinski, Waclaw, 57
sign language, 62, 284
simultaneity, 36. See also coincidence
Soler, Cyril, 90
space-filling curves, 175
spirals, 81, 169, 171, 187, 188
St. Denis, Paul, 155, 325
star, 198
star-congruent primes, 99
Stirling's formula, 104
stomachion, 199
Streisand, Barbra, 93
string theory, 70
students, 134
stylometry, 229, 359
Sumerians, 50, 254
symbols (history of)
+, 13, 29
-, 13
�, 20
> and <, 22
m, 22
! (factorial), 40
�, 25, 262
�, 25
e, 265
�, 47, 260
pi, 265, 276
calculus, 35
numbers, 32, 34
power notation, 24
variables, 132
synchronicity. See also coincidence
talisman square, 154, 324
tea puzzle, 115, 297
teacher puzzle, 114, 296
Tegmark, Mark, 247, 268
Th�bault construction, 173
tic-tac-toe, 25, 141-143, 145, 155, 211, 223, 262, 325, 352
Tinkertoy computer, 24
toilet paper, 144, 162
Tolstoy, Leo, 92
tori, 145, 146, 316
Torricelli's trumpet, 364
transfinite numbers, 47
triangle of Gods, 61, 282
Tritar, 201
Unabomber, 261
Urantia, 34
Vampire numbers, 287
Venn diagrams, 138- 140, 312
Viswanath, Divakar, 107
Vos Savant, Marilyn, 273
Wafa, Abu'l, 130
Wallis, John, 262
war, 185
Wigner, Eugene, 42
Wilson's theorem, 96
Wither's attractor, 173
Yates, Samuel, 100
Yuki Pomo, 50
zenogram, 177
zero, 47
zeta function, 103, 359