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QUARTERNIONS 189 Figure D.I author.) A 2-D slice of a 4-D quaternion Julia set. (Computer rendering by the three mutually perpendicular directions of space, in a simple extension of ordinary complex numbers of 2-D space. Although it was difficult to visualize quaternions, Hamilton found a way to use them in electrical circuit theory. Oliver Heaviside, a great Victorian-age genius, remarked: "It is impossible to think in quaternions— you can only pretend to do it." Today, quaternions are everywhere in science. They are used to describe the dynamics of motion in three-space. The space shuttle's flight software uses quaternions in its computations for guidance, navigation, and flight control for reasons of compactness, speed, and avoidance of singularities. Quaternions are used by protein chemists for spatially manipulating models of protein structure. Ted Kaczynski, the Unabomber, spoke of quaternions fondly <strong>through</strong>out his highly theoretical mathematical journal articles. Quaternion representations are so complicated that it is useful to develop methodologies to aid in their display. Such methods reveal an exotic visual universe of forms. In particular, I enjoy image processing of the beautiful and intricate structures resulting from the iteration (repeated application) of quaternion equations. Figure D.I is actually a 2-D slice of a resultant 4-D object called a quaternion Julia set. This slice has a fractional dimension. For details on these 4-D fractal shapes, see note 1 to Appendix D.
appendix e four-dimensional mazes So long as we have not become aware that the presence of God is a space, encompassing the whole of reality just as the three-dimensional space does, so long as we conceive the world of God only as the upper story of the cosmic space, so long will God's activity, too, always be a force which affects earthly events only from above. —Karl Heim, Christian Faith and Natural Science Mazes are difficult to solve in two and three dimensions, but can you imagine how difficult it would be to solve a 4-D maze? Chris Okasaki, from Carnegie Mellon University's School of Computer Science, is one of the world's leading experts on 4-D mazes. When I asked him to describe his 4-D mazes, he replied: My 4-D mazes are two-dimensional grids of two-dimensional grids. Each of the subgrids looks like a set of rooms with some of the walls missing, allowing the maze-solver to travel directly between certain rooms. In addition, each room may have a set of arrows in it, pointing North, South, East, and West. The arrows mean that you can travel directly between this room and the corresponding room in the next subgrid in that direction. For example, ina2X2X2X2 maze, if you are in the upper left corner of the upper left subgrid, an arrow pointing south means that you can travel to the upper left corner of the lower left subgrid. Mathematically, the mazes I generate are based on "random spanning trees" of some graph representing all the possible connections between rooms. Contrary to what you might expect, however, random spanning trees do not make very good mazes. The problem is that they have far too many obvious dead-ends, which do not lure the person solving the maze into exploring them. Therefore, I post-process each random spanning tree to convert a tree with many short dead-ends into one with fewer, longer dead-ends. A 4-D grid is no harder to model as a graph than a 2-D grid, so my software can generate 4-D mazes just by starting with the appropriate 190
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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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Page 168 and 169: the gods of hyperspace The White Ho
Page 170 and 171: THE GODS OF HYPERSPACE 143 Sally tu
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Page 176 and 177: Figure 6.2 The cross section of 4-D
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Page 180 and 181: THE GODS OF HYPERSPACE 153 You take
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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
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
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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.
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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 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
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Index 239 Slade, Henry, 123, 129 So
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