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HYPERSPHERES AND TESSERACTS 105 Figure 4.17 An 8-D cube. We can think of these representations as "shadows" of hypercubes on 2-D pieces of paper. Luckily, we don't have to build the object to compute what its shadow would look like. (The computer code I used to create these forms is listed in Appendix I.) Projections of higher-dimensional worlds have stimulated many traditional artists to produce geometrical representations with startling symmetries and complexities (Figs. 4.18 to 4.20). Although computer graphic devices produce projections of higher dimensions on mere 2-D screens, the computer can store the location of points in higher dimensions for manipulations such as rotation and magnification. The computer can then display projections of these higher-dimensional forms from various viewpoints. In fact, the computer is frequently used to represent higher dimen-
106 <strong>surfing</strong> <strong>through</strong> hyperspace Figure 4.18 Symmetrical, hyperbolically warped city, by Peter Raedschelders. sions in all kinds of practical scientific problems where temperature or electric charge are the additional "dimensions" represented as colors on 3-D objects. We can prepare ourselves for any invasions of 4-D creatures entering our world. Although it may be difficult for us to fully see higher dimensions, we can use computers to develop ways of responding to the intersections of higher-dimensional phenomena in our world. The computer also makes us gaze in awe at the beauty and complexity of higher dimensions. On this theme, Professor Thomas Banchoff of Brown University writes The challenge of modern computer graphics fits right in with one of the chief aims of Edwin Abbott Abbott in the introduction of his time-
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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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Page 88 and 89: Figure 3.5 A "hyperspace" blood tra
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Page 96 and 97: SATAN AND PERPENDICULAR WORLDS 69 T
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Page 104 and 105: Figure 3.13 Chess played on a Mobiu
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Page 108 and 109: hyperspheres and tesseracts FBI Hea
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Page 112 and 113: Figure 4.2 Strange universes, (a) T
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Page 116 and 117: HYPERSPHERES AND TESSERACTS 89 Poin
Page 118 and 119: HYPERSPHERES AND TESSERACTS 91 "Sal
Page 120 and 121: Figure 4.8 The Crucifixion {Corpus
Page 122 and 123: HYPERSPHERES AND TESSERACTS 95 Figu
Page 124 and 125: HYPERSPHERES AND TESSERACTS 97 Hint
Page 126 and 127: HYPERSPHERES AND TESSERACTS 99 Next
Page 128 and 129: Figure 4.10 Slices of a hypercube a
Page 130 and 131: Figure 4.13 Embryonic 5-D cube in F
Page 134 and 135: HYPERSPHERES AND TESSERACTS 107 Fig
Page 136 and 137: HYPERSPHERES AND TESSERACTS 109 4-D
Page 138 and 139: Figure 4.21 Volume of a radius 2 sp
Page 140 and 141: HYPERSPHERES AND TESSERACTS 113 How
Page 142 and 143: HYPERSPHERES AND TESSERACTS 115 ans
Page 144 and 145: HYPERSPHERESANDTESSERACTS 117 Figur
Page 146 and 147: 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
Page 166 and 167: MIRROR WORLDS 139 Figure 5.11 A mor
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
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Page 180 and 181: THE GODS OF HYPERSPACE 153 You take
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THE GODS OF HYPERSPACE 155 "Let's d
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Figure 6.4 Cross-sections of higher
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THE GODS OF HYPERSPACE 159 The mana
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THE GODS OF HYPERSPACE 161 Kindling
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concluding remarks This completes o
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CONCLUDING REMARKS 165 If intellige
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CONCLUDING REMARKS 167 swered. I ho
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appendix a mind-bending four-dimens
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MIND-BENDING FOUR-DIMENSIONAL PUZZL
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Figure A.3 (right). The capture of
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appendix b higher dimensions in sci
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HIGHER DIMENSIONS IN SCIENCE FICTIO
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appendix c banchoff klein bottle It
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BANCHOFF KLEIN BOTTLE 187 Figure C.
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QUARTERNIONS 189 Figure D.I author.
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FOUR-DIMENSIONAL MAZES 191 graph. T
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SMORGASBORD FOR COMPUTER JUNKIES 19
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SMORGASBORD FOR COMPUTER JUNKIES 19
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EVOLUTION OF FOUR-DIMENSIONAL BEING
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appendix h challenging questions fo
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Figure H.2 An artist's renditions o
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CHALLENGING QUESTIONS FOR FURTHER T
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HYPERSPACE TITLES 213 tinctly diffe
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HYPERSPACE TITLES 215 meaningfully
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notes Only God truly exists; all ot
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notes 219 4. Unfortunately, there a
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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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