"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 Order from Amazon.com. |
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) |
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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 Brée, 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 Frénicle, 4, 80 De la Hire, Phillipe, 4, 25 De la Hire's method, 4, 45-47, 78 De la Loubère, Simon, 25, 38-42 De la Loubère's method, 38-42, 78, 106, 281 De Méziriac, 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 Dürer, Albrecht, 19-23, 47, 55, 160-161 Dürer'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