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BANCHOFF KLEIN BOTTLE 187 Figure C.2 Cross section of Banchoff Klein bottle, revealing "internal" surfaces. (Computer rendition by author.) our universe may resemble a huge nonorientable space with Klein bottle-like properties, permitting right-handed objects to be transformed into left-handed ones. If you are a teacher, have your students design and program their own patterns by modifying the parameters in these equations. Make a large mural of all the student designs labeled with the relevant generating formulas. ALGORITHM: How to Create a Banchoff Klein Bottle for(u= 0;u
appendix d quarternions The invention of quaternions must be regarded as a most remarkable feat of human ingenuity. —Oliver Heaviside It is as unfair to call a vector a quaternion as to call a man a quadruped. —Oliver Heaviside Some of you may be familiar with the concept of "complex numbers" that have a real and imaginary part of the form a + hi, where z = v — 1. (If you've never heard of complex numbers, feel free to skip this section and simply enjoy the pretty fractal image.) When these 2-D numbers were invented, many people were not sure of their validity. What real-world significance could such imaginary numbers have? However, it didn't take long for scientists to discover many applications for these numbers—from hydrodynamics to electricity. Quaternions are similar to complex numbers but of the form a + hi + cj.+ dk with one real and three imaginary parts. 1 The addition of these 4-D numbers is fairly easy, but the multiplication is more complicated. How could such numbers have practical application? It turns out that quaternions can be used to describe the orbits of pairs of pendulums and to specify rotations in computer graphics. Quaternions are an extension of the complex plane and were discovered in 1843 by William Hamilton while attempting to define 3-D multiplications. Hamilton was a brilliant Irish mathematician whose genius for languages was evident at an early age. He could read at three—by four he had started on Greek, Latin, and Hebrew—and by ten had become familiar with Sanskrit. By age seventeen, his mathematical prowess became evident. In 1843, during a flash of inspiration while walking with his wife, Hamilton realized that it took four (not three) numbers to accomplish a 3-D transformation of one vector into another. In that instant, Hamilton saw that one number was needed to adjust the length, another to specify the amount of rotation, and two more to specify the plane in which rotation takes place. This physical insight led Hamilton to study hypercomplex numbers (or quaternions) with four components, sometimes written with the form: Q = a 0 + a l i + a-j + a 3 k where the as are ordinary real numbers, and i,j, and k are each an imaginary unit vector pointing in 188
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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
Page 164 and 165: 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
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 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
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
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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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