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FOUR-DIMENSIONAL MAZES 191 graph. The only difference is in how the resulting spanning tree is displayed. I've also thought about how to do this for 6-D (or even higher) mazes. My visual representation of a 6-D maze is a 2-D grid of 2-D grids of 2-D grids. I use arrows just as in the 4-D maze, except that now arrows can be short or long. A short arrow indicates connections within a 4-D subgrid. A long arrow indicates connections between adjacent 4-D subgrids. You can find a gallery of mazes, including a random 4-D maze, at http://www.es.columbia.edu/-cdo/maze/maze.html. Note, however, that this particular 4-D maze does not include the post-processing I mentioned earlier. Those of you who wish to learn more about 4-D geometry will be interested in HyperSpace, a fascinating journal on all subjects relating to higher-dimensional geometries, complex mazes, geometry, and art, and unusual patterns. The journal has articles in English and Japanese. Contact: Japan Institute of Hyperspace Science, c/o K. Miyazaki, Graduate School of Human and Environmental Studies, Kyoto University, Sakyo-ky, Kyoto 606 Japan.
appendix f sm<strong>org</strong>asbord for computer junkies The intersection of a 4-D object with a 3-space does not need to be connected, just like a continuous coral formation can appear as multiple disjoint islands where they protrude above the ocean's surface. A collection of creatures such as a hive of bees may be different parts of a single 4-D animal. Similarly, all people may be part of a single 4-D entity. The aging process can be represented as the slow motion of an intersecting hyperplane <strong>through</strong> a 4-D entity. —Daniel Green, Superliminal Software The presence of God is not an upper story of the one cosmic space, but a separate, all-embracing space by itself, so that the polar and the suprapolar worlds do not stand with respect to one another in the same relation as two floors of the same house but in the relation of two spaces. —Karl Heim, Christian Faith and Natural Science Code 1. Hyper-hypercube Program The following is C computer code I wrote to compute the attractive models of higher-dimensional cubes for Figures 4.12 to 4.17. Cubes of dimension N may be generalized to higher dimensions N + 1 by translating the TV-cube and interconnecting the appropriate vertices—just as a graphical, representation of a cube can be generated by drawing two squares and interconnecting the vertices. At higher dimensions, the cubes become so complex that they may be difficult to graphically represent. In the program, n = 4 generates a hypercube; n - 5 generates a hyperhypercube, and so on. The idea for this approach comes from a BASIC program written by Jonathan Bowen based on a Fortran program by C. S. Kuta. [See, for example, "Hypercubes" in Practical Computing, 5(4): 97-99, April, 1982.] For more details, see "On the Trail of the Tesseract," a section in Chapter 4. 192
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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
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 190 and 191: concluding remarks This completes o
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 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
Page 240 and 241: HYPERSPACE TITLES 213 tinctly diffe
Page 242 and 243: HYPERSPACE TITLES 215 meaningfully
Page 244 and 245: notes Only God truly exists; all ot
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Page 248 and 249: notes 221 tiny discrete units. To q
Page 250 and 251: notes 223 less universes could be v
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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