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SMORGASBORD FOR COMPUTER JUNKIES 195 Code 3. Draw a Klein Bottle (Mathematical Mathematica is a technical software program from Wolfram Research (Champaign, Illinois). With this versatile tool it is possible to draw beautiful Klein bottles as discussed in Chapter 5. The following is a standard recipe for creating Klein bottle shapes using Mathematica. In[l]:= botx = 6 Cos [u] (1 + Sin[u]); boty = 16 Sin[u]; rad =4(1- Cos[u] / 2); In[4]:= X = If[Pi False, ViewPoint-> {1.5, -2.7, -1.6}] Out[7]= -Graphics3D- The following is a fragment of code from a C program that draws little spheres along the surface of a Klein bottle. ALGORITHM: How to Create a Klein Bottle pi = 3.1415; for(u= 0; u pi)&&(u pi)&&(u
appendix g evolution of four-dimensional beings The brain acts as a filter of reality, reducing our perception to what is necessary for survival. Therefore, we have developed sensory organs that only perceive 3-D. As our neural pathways form in the first years of life, our brains are programmed to see reality in accord with these organs, and hence we limit it to 3-D. Given all that cannot be understood in three-dimensional terms (Einstein-Podolsky-Rosen Paradox, or even just the simple wave-particle duality), it seems that we are functioning in a reality composed of more than three dimensions. God may be a being in the infinite dimension, or perhaps a being that can conceive of all dimensions easily. —Lindy Oliver, personal communication The presence of God, the side of this Power which is turned towards us—and indeed with our human thinking we can never penetrate into the essential nature of this Power—is in fact a space. A space, of course, is not a self-contained whole, with definable boundaries separating it in the objective sense from something else. —Karl Heim, Christian Faith and Natural Science Surface Area and Volume It is likely that 4-D beings would possess internal organs with some vague similarities to their Earthly counterparts because 4-D creatures will have to perform functions that are carried out most efficiently by specialized tissues. For example, 4-D beings may have digestive and excretory systems, a transport system to distribute nutrients through the body, and specialized organs to facilitate movement. Evolutionary pressure would probably lead to familiar ecological classes of carnivores, herbivores, parasites, and beneficial symbiotic relationships. Technological 4-D beings will have appendages comparable to hands and feet for manipulating objects. Technological beings must also have senses, such as sight, touch, or hearing, although the precise nature of senses that evolve on another world would depend on the environment. For example, some 4-D aliens may have eyes sensitive 196
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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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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
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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 218 and 219: FOUR-DIMENSIONAL MAZES 191 graph. T
Page 220 and 221: SMORGASBORD FOR COMPUTER JUNKIES 19
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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