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acknowledgments and disclaimers Dost thou reckon thyself only a puny form When within thee the universe is folded? —Baha'u'llah quoting Imam Ali, the first Shia Imam I owe a special debt of gratitude to mathematician Dr. Rudy Rucker for his wonderful books and papers from which I have drawn many facts regarding the fourth dimension. I heartily recommend his book The Fourth Dimension for further information on higher dimensions in science and spirituality. I also thank Dr. Thomas Banchoff, author of Beyond the Third Dimension, for his pioneering work in visualizing the fourth dimension. Among his many talents, Dr. Banchoff is also an expert on the nineteenth-century classic Flatland, which continues to be an excellent introduction to the interrelationship between worlds of different dimensions. The various works of Martin Gardner, listed in the Further Readings section, have also been influential in my formulating an eclectic view of the fourth dimension. I thank Kirk Jensen, my editor at Oxford University Press, for his continued support and encouragement, and Brian Mansfield, Lorraine Miro, Carl Speare, Arlin Anderson, Clay Fried, Gary Adamson, Ben Brown, Sean Henry, Michelle Sullivan, Greg Weiss, and Dan Platt for useful advice and comments. Brian Mansfield prepared many of the illustrations and April Pedersen drew the wonderful small cartoons used in the chapter openings on pages 8, 43, 69, 96, 129, 155, and 163. Some of the drawings of Earthly animals, such as the seashells and trilobites, come from the Dover Pictorial Archive; one excellent source is Ernst Haeckel's Art Forms in Nature. Many of the science-fiction books listed in Appendix B were suggested by Dr. Sten Odenwald. The twisted bottle in Figure 5.10 is courtesy of artist/writer Paul Ryan of the Earth Environmental Group and was drawn by Gary Allen. Figures 3.3, 3.8b, and 5.2 and are courtesy of the National Library of Medicine's Visible Human Project. Don Webb is the author of the poem, "Reflections on aTesseract Rose." The Chinese calligraphy in the Introduction was contributed by Dr. Siu-Leung Lee, who has been practicing the art of calligraphy for more than forty years. Capable of writing in many styles, Dr. Lee has created his own form evolving from those of the Han and Jin dynasties. Roughly translated, his calligraphy is: "We surf in higher dimensions." The calligraphy uses lettering that combines archaic structure and fluid movements to symbolize the dynamic nature of the universe. This book was not prepared, approved, or endorsed by any entity associated with the Federal Bureau of Investigation, nor was it prepared, approved, licensed, or endorsed by any entity involved in creating or producing the X-Files TV show. vii
An unspeakable horror seized me. There was a darkness; then a dizzy, sickening sensation of sight that was not like seeing; I saw a Line that was no Line; Space that was not Space; I was myself, and not myself. When I could find voice, I shrieked aloud in agony, "Either this is madness or it is Hell." "It is neither," calmly replied the voice of the Sphere, "it is Knowledge; it is Three Dimensions; Open your eye once again and try to look steadily." —Edwin Abbott Abbott, Flatland Even the mathematician would like to nibble the forbidden fruit, to glimpse what it would be like if he could slip for a moment into a fourth dimension. —Edward Kasner and James Newman, Mathematics and the Imagination May I pass along my congratulations for your great interdimensional break<strong>through</strong>. I am sure, in the miserable annals of the Earth, you will be duly inscribed. —Lord John Whorfin in The Adventures ofBuckaroo Banzai Across the 8th Dimension A man who devoted his life to it could perhaps succeed in picturing himself a fourth dimension. —Henri Poincare, "L'Espace et la geometrie"
Page 2 and 3: surfing through hyperspace
Page 4 and 5: surfing through hyperspace Understa
Page 7: This book is dedicated to Gillian A
Page 11 and 12: X contents Appendix E Four-Dimensio
Page 13 and 14: xii preface I first became excited
Page 15 and 16: XIV preface problems—from a four-
Page 17 and 18: XVI preface My soul is an entangled
Page 19 and 20: XVIII preface I've attempted to mak
Page 21 and 22: The trouble with integers is that w
Page 23 and 24: XXII introduction cal and national
Page 25 and 26: xxiv introduction snaps a few photo
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Page 30 and 31: degrees of freedom FBI Headquarters
Page 32 and 33: DEGREES OF FREEDOM 5 Figure 1.1 A f
Page 34 and 35: DEGREES OF FREEDOM 7 Figure 1.3 A f
Page 36 and 37: DEGREES OF FREEDOM -9 1800s. Howeve
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Page 42 and 43: DEGREES OF FREEDOM 15 with other my
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Page 46 and 47: DEGREES OF FREEDOM 19 Figure 1.5 An
Page 48 and 49: DEGREES OF FREEDOM 21 time-like cha
Page 50 and 51: the divinity of higher dimensions F
Page 52 and 53: THE DIVINITY OF HIGHER DIMENSIONS 2
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THE DIVINITY OF HIGHER DIMENSIONS 3
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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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Figure 6.2 The cross section of 4-D
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THE GODS OF HYPERSPACE 151 Figure 6
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THE GODS OF HYPERSPACE 153 You take
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THE GODS OF HYPERSPACE 155 "Let's d
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Figure 6.4 Cross-sections of higher
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THE GODS OF HYPERSPACE 159 The mana
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THE GODS OF HYPERSPACE 161 Kindling
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concluding remarks This completes o
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CONCLUDING REMARKS 165 If intellige
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CONCLUDING REMARKS 167 swered. I ho
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appendix a mind-bending four-dimens
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MIND-BENDING FOUR-DIMENSIONAL PUZZL
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Figure A.3 (right). The capture of
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appendix b higher dimensions in sci
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HIGHER DIMENSIONS IN SCIENCE FICTIO
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appendix c banchoff klein bottle It
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BANCHOFF KLEIN BOTTLE 187 Figure C.
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QUARTERNIONS 189 Figure D.I author.
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FOUR-DIMENSIONAL MAZES 191 graph. T
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SMORGASBORD FOR COMPUTER JUNKIES 19
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SMORGASBORD FOR COMPUTER JUNKIES 19
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EVOLUTION OF FOUR-DIMENSIONAL BEING
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appendix h challenging questions fo
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Figure H.2 An artist's renditions o
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CHALLENGING QUESTIONS FOR FURTHER T
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HYPERSPACE TITLES 213 tinctly diffe
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HYPERSPACE TITLES 215 meaningfully
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notes Only God truly exists; all ot
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notes 219 4. Unfortunately, there a
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notes 221 tiny discrete units. To q
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notes 223 less universes could be v
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notes 225 less mass, the universe e
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notes 227 Appendix C 1. For more in
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notes 229 Ben Brown suggests that 4
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further readings 231 Ehrenfest, P (
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about the author Clifford A. Pickov
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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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