Hinton - The Fourth Dimension.pdf

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130 THE FOURTH DIMENSION cube, is written its name. It will be noticed that the figures are symmetrical right and left; and right and left the first two numbers are simply interchanged. Now this being our selection of points, what figure do they make when all are put together in their proper relative positions? To determine this we must find the distance between corresponding corners of the separate hexagons. 2103 3102 3201 i 3021 3120 1023 h 1203 1302 2301 j 2013 0k 1k 2k 0123 1320 0321 3012 3210 3k 0213 2310 0312 1032 0132 2031 2130 Fig. 73. 0231 1230 To do this let us keep the axes i, j, in our space, and draw h instead of k, letting k run out in the fourth dimension, fig. 73. Here we have four cubes again, in the first of which all the points are 0k points; that is, points at a distance zero in the k direction from the space of the three dimensions ijh. We have all the points selected before, and some of the distances, which in the last diagram led from figure to figure are shown here in the same figure, and so capable
A FOUR-DIMENSIONAL FIGURE 131 of measurement. Take for instance the points 3120 to 3021, which in the first diagram (fig. 72) lie in the first and second figures. Their actual relation is shown in fig. 73 in the cube marked 2k, where the points in question are marked with a * in fig. 73. We see that the distance in question is the diagonal of a unit square. In like manner we find that the distance between corresponding points of any two hexagonal figures is the diagonal of a unit square. The total figure is now easily constructed. An idea of it may be gained by drawing all the four cubes in the catalogue figure in one (fig. 74). These cubes are exact repetitions of one another, so one drawing will serve as a representation of the whole series, if we take care to remember where we are, whether in a 0h, a 1h, a 2h, or a 3h figure, when R T V P Q R S we pick out the points required. Fig 74 is a representation of all the catalogue cubes put in one. For the sake of clearness the front faces and the back faces of this cube are represented separately. The figure determined by the selected points is shown below. In putting the sections together some of the outlines in them disappear. The line TW for instance is not wanted. We notice that PQTW and TWRS are each the half of a hexagon. Now QV and VR lie in one straight line. S W Fig. 74.
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THE FOURTH DIMENSION
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THE FOURTH DIMENSION BY C. HOWARD H
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PREFACE I HAVE endeavoured to prese
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CONTENTS CHAP. PAGE. I. FOUR-DIMENS
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THE FOURTH DIMENSION ————
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FOUR-DIMENSIONAL SPACE 3 existence
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FOUR-DIMENSIONAL SPACE 5 Bringing i
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THE ANALOGY OF A PLANE WORLD 7 him.
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THE ANALOGY OF A PLANE WORLD 9 to a
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THE ANALOGY OF A PLANE WORLD 11 Aga
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THE ANALOGY OF A PLANE WORLD 13 sec
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CHAPTER III THE SIGNIFICANCE OF A F
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SIGNIFICANCE OF A FOUR-DIMENSIONAL
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CHAPTER IV THE FIRST CHAPTER IN THE
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THE FIRST CHAPTER IN THE HISTORY OF
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CHAPTER V THE SECOND CHAPTER IN THE
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SECOND CHAPTER IN THE HISTORY OF FO
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CHAPTER VI THE HIGHER WORLD IT is i
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THE HIGHER WORLD 63 dimensions. His
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THE HIGHER WORLD 65 If now he turns
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THE HIGHER WORLD 67 to him that, by
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THE HIGHER WORLD 69 simply the plan
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THE HIGHER WORLD 71 turn these rods
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THE HIGHER WORLD 73 revolution, the
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THE HIGHER WORLD 75 the mode of act
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THE EVIDENCES FOR A FOURTH DIMENSIO
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Page 95 and 96: CHAPTER VIII THE USE OF FOUR DIMENS
Page 97 and 98: 5 THE USE OF FOUR DIMENSIONS IN THO
Page 99 and 100: THE USE OF FOUR DIMENSIONS IN THOUG
Page 117 and 118: CHAPTER IX APPLICATION TO KANT’S
Page 119 and 120: APPLICATION TO KANT’S THEORY OF E
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Page 133 and 134: A FOUR-DIMENSIONAL FIGURE 123 The r
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Page 167 and 168: CHAPTER XII THE SIMPLEST FOUR-DIMEN
Page 169 and 170: THE SIMPLEST FOUR-DIMENSIONAL SOLID
Page 185 and 186: Purple Red THE SIMPLEST FOUR-DIMENS
Page 187 and 188: Light blue Gren Light purple THE SI
Page 189 and 190: REMARKS ON THE FIGURES 179 ochre cu
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REMARKS ON THE FIGURES 181 for it i
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REMARKS ON THE FIGURES 183 that is
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REMARKS ON THE FIGURES 185 yellow l
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REMARKS ON THE FIGURES 187 As the r
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REMARKS ON THE FIGURES 189 cubes, w
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4 4 REMARKS ON THE FIGURES 191 The
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REMARKS ON THE FIGURES 193 face or
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REMARKS ON THE FIGURES 195 face in
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REMARKS ON THE FIGURES 197 space, a
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REMARKS ON THE FIGURES 199 Hence wh
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REMARKS ON THE FIGURES 201 sections
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CHAPTER XIV * A RECAPITULATION AND
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RECAPITULATION AND EXTENSION 205 ou
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RECAPITULATION AND EXTENSION 207 ha
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RECAPITULATION AND EXTENSION 209 Bu
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RECAPITULATION AND EXTENSION 211 We
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RECAPITULATION AND EXTENSION 213 Th
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RECAPITULATION AND EXTENSION 215 th
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RECAPITULATION AND EXTENSION 217 In
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RECAPITULATION AND EXTENSION 221 ta
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RECAPITULATION AND EXTENSION 223 If
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RECAPITULATION AND EXTENSION 227 di
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RECAPITULATION AND EXTENSION 229 If
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APPENDIX I THE MODELS IN Chapter XI
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APPENDIX I: THE MODELS 233 in an op
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APPENDIX I: THE MODELS 235 would be
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APPENDIX I: THE MODELS 237 have to
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APPENDIX I: THE MODELS 239 can be u
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APPENDIX I: THE MODELS 241 The four
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APPENDIX I: THE MODELS 243 orange f
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APPENDIX I: THE MODELS 245 we make
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APPENDIX I: THE MODELS 247 sponding
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APPENDIX II: A LANGUAGE OF SPACE 24
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Sil Sit Sin Sil Sit Sin Silan Sitan
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TRANSCRIBER’S NOTE. Do what thou
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