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contents Preface xi 1. 2. 3. 4. 5. 6. Introduction Degrees of Freedom The Divinity of Higher Dimensions Satan and Perpendicular Worlds Hyperspheres and Tesseracts Mirror Worlds The Gods of Hyperspace Concluding Remarks Appendix A Mind-Bending Four-Dimensional Puzzles xxi 3 23 53 81 119 141 163 169 Appendix B Higher Dimensions in Science Fiction Appendix C Banchoff Klein Bottle Appendix D Quaternions ix 175 185 188
X contents Appendix E Four-Dimensional Mazes Appendix F Sm<strong>org</strong>asbord for Computer Junkies Appendix G Evolution of Four-Dimensional Beings Appendix H Challenging Questions for Further Thought Appendix I Hyperspace Titles Notes Further Readings About the Author Addendum Index 190 192 196 199 212 217 230 233 235 237
Page 2 and 3: surfing through hyperspace
Page 4 and 5: surfing through hyperspace Understa
Page 7 and 8: This book is dedicated to Gillian A
Page 9: An unspeakable horror seized me. Th
Page 13 and 14: xii preface I first became excited
Page 15 and 16: XIV preface problems—from a four-
Page 17 and 18: XVI preface My soul is an entangled
Page 19 and 20: XVIII preface I've attempted to mak
Page 21 and 22: The trouble with integers is that w
Page 23 and 24: XXII introduction cal and national
Page 25 and 26: xxiv introduction snaps a few photo
Page 28 and 29: surfing through hyperspace
Page 30 and 31: degrees of freedom FBI Headquarters
Page 32 and 33: DEGREES OF FREEDOM 5 Figure 1.1 A f
Page 34 and 35: DEGREES OF FREEDOM 7 Figure 1.3 A f
Page 36 and 37: DEGREES OF FREEDOM -9 1800s. Howeve
Page 38 and 39: DEGREES OF FREEDOM -11 Although phi
Page 40 and 41: DEGREES OF FREEDOM 13 angles and lo
Page 42 and 43: DEGREES OF FREEDOM 15 with other my
Page 44 and 45: DEGREES OF FREEDOM 17 Figure 1.4b K
Page 46 and 47: DEGREES OF FREEDOM 19 Figure 1.5 An
Page 48 and 49: DEGREES OF FREEDOM 21 time-like cha
Page 50 and 51: the divinity of higher dimensions F
Page 52 and 53: THE DIVINITY OF HIGHER DIMENSIONS 2
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THE DIVINITY OF HIGHER DIMENSIONS 3
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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
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MIRROR WORLDS 121 "I've designed a
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MIRROR WORLDS 123 Sally puts her he
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MIRROR WORLDS 125 Figure 5.4 A stri
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MIRROR WORLDS 127 "Me too." She pla
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MIRROR WORLDS 129 Zollner Experimen
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MIRROR WORLDS 131 not possible; per
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MIRROR WORLDS 133 Rotate Figure 5.6
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MIRROR WORLDS 135 To get a better u
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MIRROR WORLDS 137 Figure 5.9 Comput
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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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