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THE DIVINITY OF HIGHER DIMENSIONS 45 Figure 2.18 A 4-D creature (Omegamorph) can appear in several different 3-D locations at once. (Note the feet on this creature intersecting our 3-D world represented as a plane. Drawing by Sean Henry.) We can use another analogy to understand our lack of privacy when observed by 4-D beings. Consider an ordinary ant farm in which ants wander a (mostly) 2-D world between panes of glass. Like Flatlanders, the ants would have great difficulty hiding from us, except perhaps by pressing against walls and hoping we would not see them. If there really were a Flatland universe draped along a wall in your home, the Flatlanders could go through their entire lives unaware that you were poised inches above their planar world recording all the events of their lives. Figure 2.20 emphasizes this point by illustrating some fanciful Flatland homes designed from letters of the alphabet. Notice that we can see all parts of the houses simultaneously as well as everything inside their rooms and closets. However, to a Flatlander, these luxurious homes provide perfect privacy from their neighbors. Once a Flatlander closed the door, he or she would be safe inside, even though there is no roof overhead. To climb over the lines would mean getting out of the plane into a third dimension, and no inhabitants of a 2-D world would have any better idea of how to do this than we know how to escape from a room by vaulting into a fourth dimension. Because of your seeming omniscience, Flatlanders would think of you as a God. Your powers would awe them. For example, if you wanted to jail a criminal, you could simply draw a circle around him. However, it might be possible for you to lift the criminal up and deposit him elsewhere in Flatland. This act
46 surfing through hyperspace Figure 2.19 Taking advantage of three-space, a bird can look down at a maze and see its entire structure, while people wandering the 2-D floor are unaware of the maze's structure. would appear miraculous to a Flatlander who would not even have the word "up" in his vocabulary. Could there be even higher-dimensional Gods with greater degrees of omniscience? Could there be universes of five or six or seven dimensions, each one able to look down on its dimensionaHy impoverished predecessor whose inhabitants couldn't hide from the prying eyes of the next-higher beings? I like to imagine a universe where the dimensionaHy impoverished carry picket signs with such words as AFFIRMATIVE ACTION FOR THE DIMENSIONAHY CHALLENGED Who would carry such signs? Three-dimensional creatures parading before the all-seeing eyes of their 4-D brethren? 4-D people in front of 3-D people? Today's government mandates access ramps so that handicapped people can enter public buildings. Similarly, will governments of our far future mandate dimensional conveniences and portals? Perhaps even more weird than these higher dimensions is the 0-D world we might call "Pointland." It possesses neither length nor breadth nor height. This
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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
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
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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
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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
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Page 62 and 63: Figure 2.9 A ball being pushed thro
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
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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
Page 110 and 111: HYPERSPHERES AND TESSERACTS 83 simp
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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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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