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A Passion for Mathematics

Numbers, Puzzles, Madness, Religion,
and the Quest for Reality

Clifford A. Pickover
John Wiley & Sons, 2005

Mathematics is the hammer
that shatters the ice of our unconscious.

Numbers percolate like bubbles in the ocean of mathematics.
The mathematician's job is to transport us to new seas,
while deepening the waters and lengthening horizons.

Math puzzles, factoids, quotations, trivia, formulas...

Click here to see the book at Amazon.Com

Read Chapter 1!



Are you Fascinated by Numbers?

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.

In each chapter, Clifford Pickover provides factoids, anecdotes, definitions, quotations, mathematical art, and captivating challenges that range from fun, quirky puzzles to insanely difficult problems. You'll encounter mad mathematicians, strange number sequences, obstinate numbers, curious constants, magic squares, fractal geese, monkeys typing Hamlet, infinity, and much, much more.

If you love all things mathematical, A Passion for Mathematics will feed your fascination while giving your problem-solving skills a great workout!

"A perpetual idea machine, Clifford Pickover is one of the most creative, original thinkers in the world today." —Journal of Recreational Mathematics

Ramanujan and Beyond


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."

Table of Contents

Introduction

Mathematical Marvels from Beyond the Edge
The Ramanujan Code
Blood Dreams and God's Mathematicians
The Mathematical Smorgasbord
Explanation of Symbols
Cultivating Perpetual Mystery

Chapter 1. Numbers, History, Society, and People

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?

Chapter 2. Cool Numbers

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-Erdös constant, Brun's constant, Champernowne's number, Euler's gamma, Chaitin's constant, the Landau-Ramanujan constant, Mill's constant, the golden ratio, Apéry'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?

Chapter 3. Algebra, Percentages, Weird Puzzles, Marvelous Mathematical Manipulation

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.

Chapter 4. Geometry, Games, and Beyond

In which we explore tiles, patterns, position problems, arrays, Venn diagrams, tic tac toe, other games played on boards, Königsberg 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,Möbius 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.

Chapter 5. Probability: Take your Chances

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.

Chapter 6. Big Numbers and Infinity

In which we explore very large numbers, the edges of comprehension, infinity, the funnel of Zeus, the infinite gift, Skewes' number, the Monster group, Göbel's number, the Erdös-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?

Chapter 7. Mathematics and Beauty

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.

Answers and Further Exploring
References
About the Author

Book Index

Note: Many special kinds of numbers, such as rational, prime, or perfect, can be found under the index item "numbers." Special constants, such as the golden ratio, Euler's number, or pi, can be found under the index item "constants."
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, André, 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-Erdös, 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

Dürer, Albrecht, 179
Dase, Johann, 31
decimation, 359
derivatives, 308
Descartes, René, 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
Erdös, 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

Gödel, Kurt, 22, 41, 248, 258, 259, 269
Gödel's theorem, 259
Gabriel's horn, 364
Galois, Évariste, 266
gambling, 211
game shows, 37, 271
Gamm, Rüdiger, 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
Göbel's sequence, 241
Golay-Rudin-Shapiro Sequence, 79
Goldbach conjecture, 59
Golygon, 144, 314, 315
googol, 69
googolplex, 236
Gosper, William, 85
Grünbaum, 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
Hoëné-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
Königsberg bridges, 137
Kronecker, Leopold, 51

L'Hôpital, 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, Gösta, 272
Möbius 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
	 Erdös-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
	 Göbel'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
Schrödinger'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
	 ş, 25, 262
	 Ĺ, 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
Thébault 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