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concluding remarks This completes our study of the fourth dimension. But in doing so, I wonder why I am personally so compelled to contemplate higher dimensions. It seems to me that it is our nature to dream, to search, and to wonder about our place in a seemingly lonely cosmos. Perhaps for this reason philosophers and even theologians have speculated about the existence of a fourth dimension and what its inhabitants might be like. I agree with Eric Fromm who wrote in The Art of Loving. "The deepest need of man is to overcome his separateness, to leave the prison of his aloneness." The biggest question raised in this book is, "Can humans ever access a fourth dimension?" Or are we more like fish in a pond, near the surface, inches 163
164 surfing through hyperspoce away from a new world, but forever confined, isolated by a seemingly impenetrable boundary: the pond's 2-D surface. If higher spatial dimensions exist, humanity may still have to wait a few hundred years before developing the capacity to explore them, but such a capacity may evolve for our survival—much like fish learned to leave the confines of their pools through evolution. It is possible that humans will some day have proof of higher spatial dimensions, such as those suggested by Kaluza-Klein theories. As discussed in Chapter 4, many cosmological models have been devised in which our universe curves through four-space in a way that could, in theory, be tested. For example, Einstein suggested a universe model in which a spaceship could set out in any direction and return to the starting point. In this model, our 3-D universe is treated as the hypersurface of a huge hypersphere. Going around it would be comparable to an ant's walking around the surface of a sphere. In other universe models, our universe is a hypersurface that twists through four-space like a Klein bottle. These are closed, one-sided, edgeless surfaces that twist on themselves like a Mobius strip. Using various satellites, astronomers now actively search for evidence of the universe's shape by studying temperature fluctuations in deep space. 1 Can we learn to see the fourth dimension? Our inability to clearly visualize hyperspheres and hypercubes may result solely from our lack of training since birth. Our memories are of 3-D worlds, but who knows what might be accomplished with proper early training? This question has been discussed seriously by a number of mathematicians. Also, to say the fourth dimension is beyond imagination may be an exaggeration if we consider how far humans have stretched their imaginations since prehistoric time. From electrons to black holes, the history of science is the history of accepting concepts beyond our imagination. As Edward Kasner and James Newman note in Mathematics and the Imagination, "For primitive man to imagine the wheel, or a pane of glass, must have required even higher powers than for us to conceive of a fourth dimension." Whatever our limitations may be, even today the geometry of four dimensions is an indispensable part of mathematics and physics. Worldview The definitive discovery of 4-D beings would drastically alter our worldview and change our society as profoundly as the Copernican, Darwinian, and Einsteinian revolutions. It would impact religions and spur interest in science as never before.
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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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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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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
Page 178 and 179: THE GODS OF HYPERSPACE 151 Figure 6
Page 180 and 181: THE GODS OF HYPERSPACE 153 You take
Page 182 and 183: THE GODS OF HYPERSPACE 155 "Let's d
Page 184 and 185: Figure 6.4 Cross-sections of higher
Page 186 and 187: THE GODS OF HYPERSPACE 159 The mana
Page 188 and 189: THE GODS OF HYPERSPACE 161 Kindling
Page 192 and 193: CONCLUDING REMARKS 165 If intellige
Page 194 and 195: CONCLUDING REMARKS 167 swered. I ho
Page 196 and 197: appendix a mind-bending four-dimens
Page 198 and 199: MIND-BENDING FOUR-DIMENSIONAL PUZZL
Page 200 and 201: Figure A.3 (right). The capture of
Page 202 and 203: appendix b higher dimensions in sci
Page 204 and 205: HIGHER DIMENSIONS IN SCIENCE FICTIO
Page 212 and 213: appendix c banchoff klein bottle It
Page 214 and 215: BANCHOFF KLEIN BOTTLE 187 Figure C.
Page 216 and 217: QUARTERNIONS 189 Figure D.I author.
Page 218 and 219: FOUR-DIMENSIONAL MAZES 191 graph. T
Page 220 and 221: SMORGASBORD FOR COMPUTER JUNKIES 19
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Page 224 and 225: EVOLUTION OF FOUR-DIMENSIONAL BEING
Page 226 and 227: appendix h challenging questions fo
Page 228 and 229: Figure H.2 An artist's renditions o
Page 230 and 231: 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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