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MIND-BENDING FOUR-DIMENSIONAL PUZZLES 171 Figure A.l A 3 X 3 X 3 Rubik's cube dissected into three 3X3 layers. cubic in the center of the cube that has no color and plays no role in the puzzle.) Rubik's tesseract has four kinds of pieces that are also distinguished by the number of their colors. Those of you with computers may enjoy "MagicCube4D," a fully functional 4-D analog of Rubik's cube developed by Daniel Green and Don Hatch. The current implementation is for Windows95 and Windows NT. The graphical model is an exact 4-D extension of the original, plastic 3-D puzzle, but with some useful features such as a "reset" button. Employing the same mathematical techniques that are used to project 3-D objects onto 2-D screens, MagicCube software "projects" the 4-D cube into three dimensions. The resulting 3-D objects can then be rendered with conventional graphics software onto the screen. It is very difficult to solve the 4-D Rubik's cube starting from a scrambled initial structure. If you ever do succeed, you will be one of a very elite group of people. You will almost certainly need to have previously mastered the original Rubik's cube before you can hope to solve this one. Luckily, all the skills learned for the original puzzle will help you with this one. Also, you don't need to ever solve the full puzzle to enjoy it. One fun game is to start with a slightly scrambled configuration, just a step or two away from the solved state, and work to back out those few random twists. If you get tired trying to solve the puzzle yourself, it is breathtaking to
172 appendix a Figure A.2 Initial empty Sz'kwa board. watch the computer do it. For more information, visit: http://www.superliminal. com/cube, htm. Hyperdimensiond Sz'kwa An alien challenges you to a simple-looking, competitive game. You hold twentyfive white crystals in your hand; the alien holds twenty-five black crystals. At the start of the game, the circular board would have no crystals (Fig. A.2). You and the alien take turns placing a crystal on the board at the positions with black dots. The rules are as follows. If a player's crystal is completely surrounded by the opponent's crystals, it is captured. Figure A.3 shows the capture of a black piece (left) and the capture of two white pieces (right). When a player has no crystals left to place on the circular board, or no empty sites on which to place a crystal without its being captured, the game ends. The winner is the player who holds the greatest number of crystals. How many different arrangements of crystals on the playing board exist? Is it better to be the first player? Can you write a program that learns strategies by playing hundreds of games and observing its mistakes? Develop a multidimensional Sz'kwa game where the center site on the Sz'kwa board connects center sites on adjacent boards or where all sites connect to the corresponding positions on adjacent boards. First try a game using just two connected boards and then three. Generalize your discoveries to N boards. These versions represent hyperdimensional Sz'kwa (Fig. A.4).
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surfing through hyperspace
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surfing through hyperspace Understa
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This book is dedicated to Gillian A
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An unspeakable horror seized me. Th
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X contents Appendix E Four-Dimensio
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xii preface I first became excited
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XIV preface problems—from a four-
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XVI preface My soul is an entangled
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XVIII preface I've attempted to mak
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The trouble with integers is that w
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XXII introduction cal and national
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xxiv introduction snaps a few photo
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surfing through hyperspace
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degrees of freedom FBI Headquarters
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DEGREES OF FREEDOM 5 Figure 1.1 A f
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DEGREES OF FREEDOM 7 Figure 1.3 A f
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DEGREES OF FREEDOM -9 1800s. Howeve
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DEGREES OF FREEDOM -11 Although phi
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DEGREES OF FREEDOM 13 angles and lo
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DEGREES OF FREEDOM 15 with other my
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DEGREES OF FREEDOM 17 Figure 1.4b K
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DEGREES OF FREEDOM 19 Figure 1.5 An
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DEGREES OF FREEDOM 21 time-like cha
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the divinity of higher dimensions F
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THE DIVINITY OF HIGHER DIMENSIONS 2
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Figure 2.9 A ball being pushed thro
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satan and perpendicular worlds Wash
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SATAN AND PERPENDICULAR WORLDS 55 a
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SATAN AND PERPENDICULAR WORLDS 57 S
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SATAN AND PERPENDICULAR WORLDS 59 F
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Figure 3.5 A "hyperspace" blood tra
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SATAN AND PERPENDICULAR WORLDS 63 t
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SATAN AND PERPENDICULAR WORLDS 65 "
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SATAN AND PERPENDICULAR WORLDS 67 E
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SATAN AND PERPENDICULAR WORLDS 69 T
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SATAN AND PERPENDICULAR WORLDS 71 S
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Figure 3.13 Chess played on a Mobiu
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SATAN AND PERPENDICULAR WORLDS 79 d
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hyperspheres and tesseracts FBI Hea
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HYPERSPHERES AND TESSERACTS 83 simp
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Figure 4.2 Strange universes, (a) T
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HYPERSPHERES AND TESSERACTS 87 runn
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HYPERSPHERES AND TESSERACTS 89 Poin
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HYPERSPHERES AND TESSERACTS 91 "Sal
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Figure 4.8 The Crucifixion {Corpus
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HYPERSPHERES AND TESSERACTS 95 Figu
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HYPERSPHERES AND TESSERACTS 97 Hint
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HYPERSPHERES AND TESSERACTS 99 Next
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Figure 4.10 Slices of a hypercube a
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Figure 4.13 Embryonic 5-D cube in F
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HYPERSPHERES AND TESSERACTS 105 Fig
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HYPERSPHERES AND TESSERACTS 107 Fig
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HYPERSPHERES AND TESSERACTS 109 4-D
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Figure 4.21 Volume of a radius 2 sp
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HYPERSPHERES AND TESSERACTS 113 How
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HYPERSPHERES AND TESSERACTS 115 ans
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HYPERSPHERESANDTESSERACTS 117 Figur
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mirror worlds FBI Headquarters, Was
Page 148 and 149: MIRROR WORLDS 121 "I've designed a
Page 150 and 151: MIRROR WORLDS 123 Sally puts her he
Page 152 and 153: MIRROR WORLDS 125 Figure 5.4 A stri
Page 154 and 155: MIRROR WORLDS 127 "Me too." She pla
Page 156 and 157: MIRROR WORLDS 129 Zollner Experimen
Page 158 and 159: MIRROR WORLDS 131 not possible; per
Page 160 and 161: MIRROR WORLDS 133 Rotate Figure 5.6
Page 162 and 163: MIRROR WORLDS 135 To get a better u
Page 164 and 165: MIRROR WORLDS 137 Figure 5.9 Comput
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Page 168 and 169: the gods of hyperspace The White Ho
Page 170 and 171: THE GODS OF HYPERSPACE 143 Sally tu
Page 172 and 173: THE GODS OF HYPERSPACE 145 "What's
Page 174 and 175: THE GODS OF HYPERSPACE 147 Figure 6
Page 176 and 177: Figure 6.2 The cross section of 4-D
Page 178 and 179: THE GODS OF HYPERSPACE 151 Figure 6
Page 180 and 181: THE GODS OF HYPERSPACE 153 You take
Page 182 and 183: THE GODS OF HYPERSPACE 155 "Let's d
Page 184 and 185: Figure 6.4 Cross-sections of higher
Page 186 and 187: THE GODS OF HYPERSPACE 159 The mana
Page 188 and 189: THE GODS OF HYPERSPACE 161 Kindling
Page 190 and 191: concluding remarks This completes o
Page 192 and 193: CONCLUDING REMARKS 165 If intellige
Page 194 and 195: CONCLUDING REMARKS 167 swered. I ho
Page 196 and 197: appendix a mind-bending four-dimens
Page 200 and 201: Figure A.3 (right). The capture of
Page 202 and 203: appendix b higher dimensions in sci
Page 204 and 205: HIGHER DIMENSIONS IN SCIENCE FICTIO
Page 212 and 213: appendix c banchoff klein bottle It
Page 214 and 215: BANCHOFF KLEIN BOTTLE 187 Figure C.
Page 216 and 217: QUARTERNIONS 189 Figure D.I author.
Page 218 and 219: FOUR-DIMENSIONAL MAZES 191 graph. T
Page 220 and 221: SMORGASBORD FOR COMPUTER JUNKIES 19
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Page 224 and 225: EVOLUTION OF FOUR-DIMENSIONAL BEING
Page 226 and 227: appendix h challenging questions fo
Page 228 and 229: Figure H.2 An artist's renditions o
Page 230 and 231: CHALLENGING QUESTIONS FOR FURTHER T
Page 240 and 241: HYPERSPACE TITLES 213 tinctly diffe
Page 242 and 243: HYPERSPACE TITLES 215 meaningfully
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notes 221 tiny discrete units. To q
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notes 223 less universes could be v
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notes 225 less mass, the universe e
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notes 227 Appendix C 1. For more in
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notes 229 Ben Brown suggests that 4
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further readings 231 Ehrenfest, P (
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about the author Clifford A. Pickov
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addendum As this book goes to press
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index 10-D universe, 15 Abbott, Edw
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Index 239 Slade, Henry, 123, 129 So
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