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appendix h challenging questions for further thought The unbelievably small and the unbelievably vast eventually meet—like the closing of a gigantic circle. I looked up, as if somehow I would grasp the heavens. The universe, worlds beyond number, God's silver tapestry spread across the night. And in that moment, I knew the answer to the riddle of the infinite. I had thought in terms of man's own limited dimension. I had presumed upon nature. That existence begins and ends is man's conception, not nature's. And I felt my body dwindling, melting, becoming nothing. My fears melted away. And in their place came acceptance. All this vast majesty of creation, it had to mean something. And then I meant something, too. Yes, smaller than the smallest, I meant something, too. To God, there is no zero. I still exist! —Scott Carey, The Incredible Shrinking Man This appendix includes various unsolved problems, challenges, break<strong>through</strong>s, and recent news items relating to higher spatial dimensions. Photons Is it difficult to imagine a universe with four dimensions in which we are constrained to three dimensions? Could there be some mechanism constraining us— such as being stuck to a surface via adsorption? If we were stuck to some surface in this manner, then the way in which 4-D beings could interact with us would be constrained by that surface. What happens to photons in four dimensions? For example, how do 4-D beings see? Our 3-D universe doesn't seem to be "losing" photons into the fourth dimension, so how would 4-D creatures use photons to see us or themselves? 1 Four-Dimensional Speech, Writing, and Art Would we be able to hear the speech of a 4-D being inches away in the fourth dimension? How would sound work? What would a 4-D being sound like using its 4-D vocal cords? 199
200 appendix I Perhaps a 4-D being's speech will propagate via 4-D sound (pressure) waves and thus its speech will be partially audible to us. Philosopher Greg Weiss suggests that we would hear something, but not likely the whole message, which may have modulations in 4-D space. A 4-D creature could speak "down" to us, but unless we have some way to manipulate four dimensions, we'd have trouble speaking to it in its language. Would a being's voice get louder as the creature grew closer to our 3-D space? Would our houses shield our voices from them? If a 4-D being talked to us, would the sound appear to come from everywhere at once, even from inside us? Of course, we need not use speech to communicate. If we could transmit just a single stream of bits (e.g., ones and zeros) into the fourth dimension, communication would be possible. In fact, the rapid appearance and disappearance of a 4-D object in our world could be used to communicate a binary message. What would the writing of 4-D beings resemble? Our writing is one dimension lower than the space in which we reside. Would the writing of 4-D beings be three-dimensional? What would this writing look like if projected on a piece of paper in our world (Figure H.I)? Figure H.2 and H.3 are an artist's renditions of what a 4-D handwriting might look like as it intersected our world. Wild! As a final artistic piece, consider Figure H.4 by Professor Carlo H. Sequin from the University of California, Berkeley. His representation is a projection of a 4-D 120-cell regular polytope (a 4-D analog of a polygon). This structure consists of twelve copies of the regular dodecahedron — one of the five Platonic solids that exist in 3-D space. This 4-D polytope also has 720 faces, 1200 edges, and 600 vertices, which are shared by two, three, and four adjacent dodecahedra, respectively. Figure H.I What would the intersection of Omegamorph handwriting with a piece of paper look like? Would we recognized it as writing?
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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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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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Page 184 and 185: Figure 6.4 Cross-sections of higher
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Page 188 and 189: THE GODS OF HYPERSPACE 161 Kindling
Page 190 and 191: concluding remarks This completes o
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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
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Page 204 and 205: HIGHER DIMENSIONS IN SCIENCE FICTIO
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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
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Page 230 and 231: CHALLENGING QUESTIONS FOR FURTHER T
Page 240 and 241: HYPERSPACE TITLES 213 tinctly diffe
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Page 244 and 245: notes Only God truly exists; all ot
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Page 252 and 253: notes 225 less mass, the universe e
Page 254 and 255: notes 227 Appendix C 1. For more in
Page 256 and 257: notes 229 Ben Brown suggests that 4
Page 258 and 259: further readings 231 Ehrenfest, P (
Page 260 and 261: about the author Clifford A. Pickov
Page 262 and 263: addendum As this book goes to press
Page 264 and 265: index 10-D universe, 15 Abbott, Edw
Page 266: Index 239 Slade, Henry, 123, 129 So
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