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Figure 4.10 Slices of a hypercube as it moves corner-first through our world. Figure 4.11 Central projections of a rotating hypercube in four-space. Starting from the left and going clockwise, the shade cube opens toward the bottom; if the rotation is continued, the small cube will flatten out and come back to become the large exterior cube. (After Banchoff.)
102 surfing through hyperspace with eight corners. Even though we cannot easily visualize the next step in the process, we can predict that if we were able to move a cube perpendicular to all its edges, we would generate a 4-D object: a hypercube. It would have sixteen corners. The trend in the number of corners is a geometric progression (2, 4, 8, 16 . . .), and we can therefore calculate the number of corners in any dimension by using the formula 2" where n is the number of dimensions. We can also consider the number of boundaries for objects in different dimensions. A line segment has two boundary points. A square is bounded by four line segments. A cube is bounded by six squares. Following this trend, we would expect a hypercube to be bounded by eight cubes. This sequence follows an arithmetic progression (2, 4, 6, 8 . . .). The area of a square of edge length a is a 2 . The volume of a cube of edge length a is c?. The hypervolume of an w-cube is a". Past books typically provide wire-frame diagrams for tesseracts produced by the "trail" of a cube as it moves in a perpendicular direction, similar to the one in Figure 4.4. Of course, we can't really move in a perpendicular direction, but we can move the cube diagonally, in the same way a square is moved diagonally to represent a cube. Now prepare yourself for some wild trails of higher-dimensional objects rarely, if ever, seen in popular books. To give you an idea of the beauty and complexity of higher-dimensional objects, I produced Figures 4.12 to 4.17 using a computer program. Modern graphics computers are ideal tools for visualizing structures in higher dimensions. Figure 4.12 A 5-D cube produced by moving a hypercube along the fifth dimension.
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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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Page 78 and 79: THE DIVINITY OF HIGHER DIMENSIONS 5
Page 80 and 81: satan and perpendicular worlds Wash
Page 82 and 83: SATAN AND PERPENDICULAR WORLDS 55 a
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Page 86 and 87: SATAN AND PERPENDICULAR WORLDS 59 F
Page 88 and 89: Figure 3.5 A "hyperspace" blood tra
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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 120 and 121: Figure 4.8 The Crucifixion {Corpus
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Page 130 and 131: Figure 4.13 Embryonic 5-D cube in F
Page 132 and 133: HYPERSPHERES AND TESSERACTS 105 Fig
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Page 138 and 139: Figure 4.21 Volume of a radius 2 sp
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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
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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
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Page 176 and 177: Figure 6.2 The cross section of 4-D
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THE GODS OF HYPERSPACE 151 Figure 6
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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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