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HYPERSPHERES AND TESSERACTS 91 "Sally, ever wonder what a hypersphere would look like projected into our universe?" She smiles. "Sure, every day, every waking hour." You bring out a globe made of glass with all the continents marked. You shine a light on it and look at the projection on the wall. "First, let's consider the projection of an ordinary sphere onto a plane." You point at a spot projected on the wall. "Notice that the two hemispheres will overlap on one another, and that the distance between our FBI headquarters and China seems very short. Of course, that's only because we're looking at a projection. In fact, every point on the projection represents two opposite points on the original globe. China and America don't actually overlap because they are on opposite sides of the globe." Sally studies the projection on the wall. "What we're seeing looks like two flat discs put together and joined along their outer circumferences." "Right. Watch as I rotate the globe. The projection makes it appear that the Earth is rotating both right and left simultaneously. Would a 2-D being go insane trying to picture an object rotating in three dimensions?" "Imagine our difficulty in visualizing a 4-D rotating planet!" You nod. "You can imagine a space-projection of a hypersphere into our world as two spherical bodies put through each other and joined along their outer surfaces. It would be like two apples grown together in the same regions of space and joining at their skins." You place the globe on an old Oriental carpet covering the hardwood floor of your office. "Let's return our attention to hypercubes. Another way to represent a hypercube is to show what it might look like if it was unfolded." You bring out a paper cube that has been taped together and remove some pieces of the tape. "By analogy, you can unfold the faces of a paper cube and make it flat" (Fig. 4.6). You then bring out a paper model of an unfolded hypercube. "Sally, we can cut a hypercube and 'flatten' it to the third dimension in the same way we flattened a cube by unfolding it into the second dimension. In the case of the hypercube, the 'faces' are really cubes" (Fig. 4.7). You point at a poster on the wall. "The hypercube has often been used in art. My favorite is the unfolded hypercube from Salvador Dali's 1954 painting Corpus Hypercubus (Fig. 4.8). By making the cross an unfolded tesseract, Dali represents the orthodox Christian belief that Christ's death was a metahistorical event, taking place in a region outside of our space
92 surfing through hyperspace Figure 4.6 One way to unfold a cube. The arrows show a way of folding the faces to reform the cube—for example, the bottom face connects to the top face. Figure 4.7 One way to unfold a hypercube. Just as with the cube in Figure 4.6, the bottom cubical "face" must join with the top "face" when folding the cubes to reform the hypercube. This folding must be done in the fourth dimension. (The forwardmost cubical face is shaded to help clarify the drawing.)
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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 68 and 69: THE DIVINITY OF HIGHER DIMENSIONS 4
Page 80 and 81: satan and perpendicular worlds Wash
Page 82 and 83: SATAN AND PERPENDICULAR WORLDS 55 a
Page 84 and 85: SATAN AND PERPENDICULAR WORLDS 57 S
Page 86 and 87: SATAN AND PERPENDICULAR WORLDS 59 F
Page 88 and 89: Figure 3.5 A "hyperspace" blood tra
Page 90 and 91: SATAN AND PERPENDICULAR WORLDS 63 t
Page 92 and 93: SATAN AND PERPENDICULAR WORLDS 65 "
Page 94 and 95: SATAN AND PERPENDICULAR WORLDS 67 E
Page 96 and 97: SATAN AND PERPENDICULAR WORLDS 69 T
Page 98 and 99: SATAN AND PERPENDICULAR WORLDS 71 S
Page 104 and 105: Figure 3.13 Chess played on a Mobiu
Page 106 and 107: SATAN AND PERPENDICULAR WORLDS 79 d
Page 108 and 109: hyperspheres and tesseracts FBI Hea
Page 110 and 111: HYPERSPHERES AND TESSERACTS 83 simp
Page 112 and 113: Figure 4.2 Strange universes, (a) T
Page 114 and 115: HYPERSPHERES AND TESSERACTS 87 runn
Page 116 and 117: HYPERSPHERES AND TESSERACTS 89 Poin
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 132 and 133: HYPERSPHERES AND TESSERACTS 105 Fig
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
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the gods of hyperspace The White Ho
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THE GODS OF HYPERSPACE 143 Sally tu
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THE GODS OF HYPERSPACE 145 "What's
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THE GODS OF HYPERSPACE 147 Figure 6
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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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