The Zen of Magic Squares, Circles, and Stars:

An Exhibition of Surprising Structures Across Dimensions

Clifford A. Pickover

Princeton University Press, 2002

"A refreshing new look at a timeless topic, brimming over with ideas, littered with surprising twists. Anyone who loves numbers, anyone who enjoys puzzles, will find The Zen of Magic Squares, Circles, and Stars compulsive (and compulsory!) reading."
- Ian Stewart, University of Warwick

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"At first glance magic squares may seem frivolous (Ben Franklin's opinion, even as he spent countless hours studying them!), but I think that is wrong. The great nineteenth-century German mathematician Leopold Kronecker said 'God Himself made the whole numbers--everything else is the work of men,' and Cliff Pickover's stimulating book hints strongly at the possibility that God may have done more with the integers than just create them. I don't believe in magic in the physical world, but magic squares come as close as we will probably ever see to being mathematical magic."
- Paul J. Nahin, University of New Hampshire, author of Duelling Idiots and Other Probability Puzzlers
"Clifford Pickover has compiled such a wonderfully voluminous collection of magic squares and related configurations that the physical book itself threatens to take the shape of a magic cube. Whether or not you achieve arithmetical satori contemplating these engagingly intricate patterns, you will surely come to appreciate their history, beauty, and richness."
- John Allen Paulos, Temple University, author of Innumeracy and A Mathematician Reads the Newspaper
"Who would have thought that the simple numbers we used when learning to count as children could be arranged into so many geometric patterns with interesting properties? If playing with numbers is your thing, this latest work by Cliff Pickover will provide you will countless hours of mystical entertainment and mental challenges."
- Julien Clinton Sprott, University of Wisconsin-Madison
"Pickover carries the mystique of magic squares and their relatives into the twenty-first century with his new book, The Zen of Magic Squares, Circles, and Stars. Whether you're seeking an introduction to magic squares, an in-depth study, some historical information, or just some enjoyable magic figure problems, tricks, properties, or novelties-this book is for you. Pickover does not leave a magic square unturned, and tantalizes us just enough to want to explore further. From the very famous to the less known, Pickover brings them all together in this amazing collection as he points out their roles in science, our lives and the universe."
- Theoni Pappas author of The Joy of Mathematics and Math-A-Day
"Every generation seems to demand its own updated book dedicated to magic squares. Pickover's work meets the needs of the present generation. It is an enthusiastic, whimsical, and idiosyncratic compendium of magic squares and their variants, some of which are centuries old, some as fresh as corn picked this morning."
- Sherman Stein, University of California, Davis
"Well documented and illustrated, this book is essentially a huge catalogue of these arithmetical beasts called magic squares. No book that I know of presents such a large range of 'mysterious, odd, and fun' patterns, as the author puts it. This includes the numerous geometrical diagrams that reveal some hidden symmetries not obvious to the eye contemplating the number arrays. Throughout, Pickover convincingly substantiates his claim that the field of magic square study is still wide open."
- Arturo Sangalli, author of The Importance of Being Fuzzy
"Pickover inspires a new generation of da Vincis to build unknown flying machines and create new Mona Lisas."
- Christian Science Monitor
"Pickover has published nearly a book a year in which he stretches the limits of computers, art, and thought."
- Los Angeles Times "Bucky Fuller thought big, Arthur C. Clarke thinks big, but Cliff Pickover outdoes them both."
- WIRED
"Pickover just seems to exist in more dimensions than the rest of us."
- Ian Stewart, author of Does God Play Dice?
"Strange and beuatiful, stunningly realistic."
- Scientific American
"Pickover is van Leeuwenhoek's 20th century equivalent."
- OMNI

"Clifford Pickover is many things -- scientist, scholar, author, editor, and visionary..." - GAMES

Jacket Flap: Humanity's love affair with mathematics and mysticism reached a critical juncture, legend has it, on the back of a turtle in ancient China. As Clifford Pickover briefly recounts in this enthralling book, the most comprehensive in decades on magic squares, Emperor Yu was supposedly strolling along the Yellow River one day around 2200 B.C. when he spotted the creature: its shell had a series of dots within squares. To Yu's amazement, each row of squares contained fifteen dots, as did the columns and diagonals. When he added any two cells opposite along a line through the center square, like 2 and 8, he always arrived at 10. The turtle, unwitting inspirer of the ''Yu'' square, went on to a life of courtly comfort and fame. (Continued below)

Pickover explains why Chinese emperors, Babylonian astrologer-priests, prehistoric cave people in France, and ancient Mayans of the Yucatan were convinced that magic squares--arrays filled with numbers or letters in certain arrangements--held the secret of the universe. Since the dawn of civilization, he writes, humans have invoked such patterns to ward off evil and bring good fortune. Yet who would have guessed that in the twenty-first century, mathematicians would be studying magic squares so immense and in so many dimensions that the objects defy ordinary human contemplation and visualization?

Readers are treated to a colorful history of magic squares and similar structures, their construction, and classification along with a remarkable variety of newly discovered objects ranging from ornate inlaid magic cubes to hypercubes. Illustrated examples occur throughout, with some patterns from the author's own experiments. The tesseracts, circles, spheres, and stars that he presents perfectly convey the age-old devotion of the math-minded to this Zenlike quest. Number lovers, puzzle aficionados, and math enthusiasts will treasure this rich and lively encyclopedia of one of the few areas of mathematics where the contributions of even nonspecialists count.

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Introduction and History
1 Construction
2 Classification
3 Gallery 1: Squares, Cubes, and Tesseracts
4 Gallery 2: Circles, and Spheres
5 Gallery 3: Stars and Other Beauties
Some Final Thoughts
Notes
For Further Reading

Clifford A. Pickover is the author of over twenty highly-acclaimed books, such as Time: A Traveler's Guide and Computers, Pattern, Chaos, and Beauty, and has written on diverse topics ranging from computers and creativity, art, mathematics, and astronomy-to human behavior and intelligence, time travel, alien life, and science fiction. Dr. Pickover received his Ph.D. from Yale University and is a prolific inventor. His web site pickover.com has received over a half-million visits.

Terms Indexed in The Zen of Magic Squares, Circles, and Stars

(This is meant to give a clearer indication of topics covered.)

Abas, Jan, xx
Abe, Gakuho, 191, 254
Adamson, Gary, 29, 31, 286
Addition-multiplication squares, 133
Agrippa, Cornelius, 12
Alphamagic squares, 133
Amulets, 13-14
Anderson, Arlin, 314, 357-359
Andrews, W. S., xvii, xix, 26, 99, 238
Annihilation squares, 143
Antimagic squares, 108-110, 198
Apocalyptic square, 162
Arens, W., 125-127
Arithmetic series, 7, 197
Arkin, Joseph, 120
Arney, David, 120
Arnoux method, 60
Associated squares, 65-66, 201-203
Associative squares, see Associated squares
Astrology, 12-13

Bachet's method, 42-44
Ball, W., 60
Bent diagonals, 157
Bimagic cube, 103
Bimagic square, 135, 244, 247, 254
Binary numbers, 153, 160-161, 263, 287-289, 383
Bordered magic squares, 79, 159
Br�e, David, 73-74
Brick, integer, 104
Broken diagonals, 17, 69, 162
Brown, Ronald, 93
Bruno, Giordano, 33-34
Buddhism, xiii-xv, 28-30
Butterflies, xi-xii, xx

Cameron cube, 120
Carus, Paul, xix, 204-209
Categories, magic square, 37
Chhi-shou, Pao, 102
Chladni patterns, 208-210
Christmas cross, 128
Circles, magic, 95-96, 297-313
Classification, 65-146
	by center number, 107
Codons, 285
Collins, Mark, 160
Collison, David, 249
Complement, magic square, 5
Composite squares, 80, 201, 230, 250
Composite (non-prime) squares, 197
Compound squares, 80, 201
Concentric magic squares, 79
Constant, square, 1, 198
Construction, magic square, 37-64
Conway, John, 56, 129
Cossey, Jay, xx
Coxeter, H., 60
Cross, 229
Crystal magic squares, 143
Cube, magic, 57-60, 97-98, 158-159, 175-180, 268
	perfect, 175-177
	Rubik's, 127
	Latin, 171-172

De Geest, Patrick, 163
De Bessy, Bernard Fr�nicle, 4, 80
De la Hire, Phillipe, 4, 25
De la Hire's method, 4, 45-47, 78
De la Loub�re, Simon, 25, 38-42
De la Loub�re's method, 38-42, 78, 106, 281
De M�ziriac, Claude Gaspar Bachet, 25, 42
De Moivre, Abraham, 210
Decimal expansion, 238
Derksen, Harm, 259
Devedec (Mr.), 389
DeVincentis, Joseph, 348
Diabolique squares, 65, 68
Diagonals
	bent, 157
	broken, 17, 69, 162
	defined, 6 
	method, 53
	rule, 57-60
	short, 77
Diagrams, geometrical, xvii, 26-28, 155-158, 201, 220, 241
Dice, 289-291, 298
DNA, 285
Dominos and magic squares, 141, 289, 332
Doubly even order, 38
Doubly magic squares, see Bimagic
Dudeney, Henry, 25, 212, 214
Durga yantra,  ix, xx, 338-339, 392
D�rer, Albrecht, 19-23, 47, 55, 160-161
D�rer's method, 55

Edwards, Ronald, 191
Emanouilidis, Emanuel, 88
Euler, Leonhard, 90, 211, 235
Even-order squares, 206

Falkener, Edward, 217, 229
Farrell, Jeremiah, 144, 144-145, 295
Faulhaber, Johann, 44
Feisthamel, 91-92
Fibonacci numbers, 16
Fifteen puzzle, 123-127
Files, 118
Fourth dimension, 165
Fractals, xiii-xiv, 351
Fractional squares, 111
Franklin, Benjamin, xvi, 147-155
Franklin's square, 90, 147-155
French contributions, 25-26
Frierson, L., 222-230, 254
Fubine, 359-372
Fults, John Lee, xix, 49, 53

Galilei, Galileo, 15
Ganymede flower, 347
Gardner, Martin, xix, 5, 73-74, 98, 107, 177, 196, 372
Geber, 11-12
Geometrical diagrams, xvii, 26-28, 155-158, 201, 220, 241
Giraffe, 235
Gnomon square, 137
Gonzalez, German, 319
Graphs, supermagic, 141
Gray code, 288
Green, Daniel, 132
Gumdrops, 316
Guzzetta, Cathie, 209
Gwalior square, 220

Hagedorn, Thomas, 146
Harper, Dave, 260-265
Hartsfield, Nora, 142
Hatch, Don, 132
Heath, Royal, 108
Heinz, Harvey, xx, 243, 257, 305, 349
Hendricks, John, xix, 26, 58-60, 62, 103, 114, 119, 121, 165-167, 178, 230, 242, 243, 245-249,
	 267-270, 310
Hermetic geometry, 32
Heterosquares, 108-110, 256
Hexagon, magic, 139, 325-340
Honaker, G., 289, 293
Horner, Walter, 191
Hruska, Frank, 61
Hui, Yang, 96
Hydrogen bonds, 285
Hypercubes, 115, 187

Imperfect squares, 84
India, 16-18, 221
Inlays,
	cubes, 178, 267
	squares 240-241, 242-243, 246, 257
Islam, 14-15
Isomorphic, 5
IXOHOXI squre, 174

Jaenisch, C., 89
Jainism, 16-18, 305
Johnson, Allan, 63, 152, 196, 278

Keith, Mike, 292-293, 312, 314-316, 339
Klein bottle, 217, 219
Knight
	method, 55
	squares, 89-93
	tours, 210-220, 232-235
Koans, xiv
Kordemsky, Boris, 144, 343
Kravitz, Sidney, 111, 199
Kronecker, Leopold, 371
Kurchan square, 164

Labeling, magic, 138
Langman, H., 101
Latin 
	cubes, 171-172
	squares, 136, 169-171
Laue, Ralf, 267
Leapers, 235
Lebesgue, Henri, 216
Lee, Siu-Leung, viii, xix
Legendre, Adrien, 210
Lindon, J., 110
Lloyd, Sam, 123-127
Lobeck, T., 172
Lo-Shu, 8-11, 369
Lozenge method, 56

Madachy, Joseph, xix, 110, 201
Magic circles, 297-310
MagicCube4D, 132
Magic cubes, 57-60, 97-98, 158-159, 175-180, 268
	bimagic, 103
	perfect, 97, 101
	semiperfect, 98
Magic graphs, 141
Magic hexagons, 139, 325-340
Magic labeling, 138
Magic rectangles, 146
Magic squares
	addition-multiplication, 133
	annihilation, 143
	associated, 65-66, 201-203
	associative, see associated 
	bimagic, 135, 244, 247, 254
	bordered, 79, 159
	categories, 37
	complement, 5
	compound, 80, 201
	composite, 80, 201, 230, 250
	composite (non-prime), 197
	concentric, 79
	constant, 1, 198
	construction, 37-64
	definition of, 1
	Diabolique, 65, 68
	diagonals, 6
	dividing, 88
	doubly magic, 135
	domino, 141
	fractional, 111
	Franklin, 90
	gnomon, 137
	imperfect, 7, 84
	isomorphic, 5
	knight's move, 89-93
	most perfect 72-73, 217
	multimagic, 136
	multiplying, 88, 140
	Nasik, 65, 68, 92, 260-262, see also Pandiagonal
	negative, 111
	order, 1, 38, 198
	overlapping, 83, 245, 251-252
	Pandiagonal, 65, 68, 174, 196, 254, 257, 268, 278
	Perfect, 65, 68, 217
	physical, 267
	prime, 196-197, 256-257
	pure, 7
	Pythagorean, 104
	semidiabolique, 77
	semi-Nasik, 65, 77
	serrated, 120
	simple, 65
	subtracting, 88
	sum, 1, 198
	trimagic, 135

Magic series, 141
Magic spheres, 297-310
Magic spider, 349
Magic stars, 325-340
Magic tesseracts, 114-120, 165-170, 187
	perfect, 120
Magic tetrahedrons, 343
Magic word squares, 144-145
Mandelbrot, Benoit, xiii
Margossian method, 60
Matrix multiplication, 61
McCranie, Judson, 292-293
Meyers, Richard, 175
Miguel, Gomez, 342
Mirror square, 164, 174, 205
Misfits, 84
Moebius band, 217
Moran, Jim, 155
Moschopoulus, Emanuel, 11
Most perfect squares, 72-73, 217
Multimagic squares, 136
Multplication squares, 140
Music, 94
Myer, Lewis, 98

Nagle, Nancy, xx, 359-360
Nasik squares, 65, 68, 92, 260-262, see also Pandiagonal
Negative squares, 111
Nelson, Harry, 196
Nesi, James, xx, 359, 361-368
Number of the beast, 162-163

Odd order, 38
Ollerenshaw Dame Kathleen, 73-74
Order, magic square, 1, 38
Overlapping squares, 83, 245, 251-252
Overlay patterns, 193

Palindromes, 283
Pandiagonal squares, 65, 68, 174, 196, 254, 257, 268, 278, see also Nasik
Pandiagonal torus, 349
Pandigital, 164
Pantriagonal, 178
Patel, Lalbhai, 155
Patents, 129
Pearl Harbor, 256
Pegg, Ed, 235
Penrose, Roger, xiv
Perfect
	cube, 97, 101, 175-177
	squares, 65, 68, 217
	tesseract, 120-121
Perlman, Y, 125-127
Petersen, Ivars, 135
Physical squares, 267
Pi square, 172
Pi rings, 272
Pillars, 118
Pirsig, Robert, 373
Planck, C., 178
Planetariums, 343
Porter, Bruce, 120
Poyo, F. 179-187
Prime 
	squares, 196-197, 256-257
	stars, 352
Prison squares, 194-195
Prometheus, 323
Pythagorean squares, 104

Quadragonals, 118, 119
Queens solutions, 279
Quincunx, 299

Rectangles, magic, 146
Reentrant, 211-213, 237
Reid, Michael
Reversible square, 75
Rhombus, 228, 230
Ringel, Gerhard, 142
Rivera, Carols, 256
Rook's tour, 237
Rosser, J., 177
Rubik's cube, 127
Rubik's tesseract, 130
Runes, 134

Sallows, Lee, 134
Sator Arepo, 23-24
Sayles, Harry, 238
Schimmel, Annemarie, 14
Schroeppel, Richard, 4, 26, 99
Seferovich, PennLea, xx, 339
Semidiabolique squares, 65, 77
Semi-Nasik squares, 65, 77
Semiperfect magic cube, 98
Serrated squares, 120
Series, magic, 141
Shelley, Percy, 324
Shineman, E., 256-257
Short diagonals, 77
Shri-yantra, 31-32
Shuldham, Chas, 197
Simple squares, 65
Singly even order, 38
Sisolak, Marsha, 319, 341
Skew cells, definition, 3
Spheres, magic, 95-96, 297-313
Spiders, magic, 349
Squares, Latin, 136
Stairstep method, 48
Stars, magic, 94, 325-340
Stars, prime, 352
Stewart, Ian, 74, 146, 214
Strachey, Ralph, 51
Sum, square, 1, 198
Supermagic graphs, 141
Swastika, 284

Talisman squares, 111, 199, 332
Ta-Wei, Cheng, 6
Templar magic square, 23-24
Tesseract, Rubik's, 130
Tesseracts, 114, 120-121, 165-170, 187
	perfect, 120-121
Tetrahedrons, 343
Thai, Minh, 129
Tilegrams, 260
Torus, pandiagonal, 349
Transversal, 171, 384
Trapezoids, 246
Trees, 138
Triagonals, 98, 121, 175, 268
Triangles, 290-292, 345
Triangular-parallelogram method, 49
Trigg, Charles, 108
Trimagic squares, 135
Turtle, 9

Upside-down square, 173

Vandermonde, Alexandre, 214
Velleman, Dan, 130

Violle,  Jean-Yves , 25

Walker, Robert, 177
Wenzelides, 92
Winder, Dik, 128
Word squares, 144, 144-145
World records, 267

Yantra, Shri, 31, 32
Yantra, Durga, ix, xx, 338-339, 392
Yin and yang, 10
Yu (emperor), 8-9

Zen, xiii-xv, 28-30, 372-373

A note on pentamagic squares. As described in my book, a magic square is p-MULTIMAGIC if the square formed by replacing each element by its kth power (for k=1, 2, ?, p) is also magic. (If this definition seems unclear, the book give numerous examples for you to study.) Bimagic squares (2-multimagic) are presented in the book, and references for trimagic squares (3-multimagic) are given. The new multimagic record is a pentamagic square (5-multimagic) built in 2001 by Andr� Viricel and Christian Boyer. Look at Pentamagic, where you can study and download this pentamagic square. This website is in English, French and German.

Return to Cliff Pickover's home page which includes computer art, educational puzzles, higher dimensions, fractals, virtual caverns, JAVA/VRML, alien creatures, black hole artwork, and animations.

The Zen of Magic Squares, Circles, and Stars:

An Exhibition of Surprising Structures Across Dimensions

Clifford A. Pickover

Table of Contents

Terms Indexed in The Zen of Magic Squares, Circles, and Stars