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En Garde, whole no. 16, January 1946
Page 13
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page 13. visualize it, unless you are Quintus Teal. But to show that we can tell a good deal about what would happen, let's consider another aspect of this cube-generation. When the point moved, it generated, first, a line, and second, 2 points, the ends of the line, one representing its initial position and one representing its final position. Any point in any of the n-cubes, when the cube generates an (n+1)-cube, will similarly give rise to 1 line and 2 points. For example, the 4 points at the vertices of the square generate, in the cube, 8 vertices and 4 edges. (If you can't visualize this immediately, it'll help if you'll make the slight effort necessary to do so.) Similarly, when the lines is moved out of its own path it produces 1 area, representing the space through which it traveled, and 2 lines, its initial and final positions; and the squares's interior produces 1 volume (the interior of the cube) and 2 squares (2 of the cube's 6 faces). Extrapolating, quite legitimately, let's consider the hypothetical generation of the 4- by the 3-cube. Each of the cube's 8 vertices gives 2 vertices (making a total of 16 and 1 edge. Each of the cube's 12 edges gives 2 edges (making, with those obtained already, 32) and 1 area, or face. Each of the cube's 6 faces gives 2 faces (or an overall total of 24 and one volume.( The volumes are not the interior of the 4-cube, of course). The cube's interior volume gives 2 volumes (an overall total of 8) and 1 4-volume, the interior of the new figure. So ins pite of not being able to visualize this figure, we have a pretty complete idea of its structure. Just to show the generality with which we can study n-cubes, I'll set down the complete box-score for all them through the 5-cube. Vertices Edges Faces Volumes 4-Volumes 5-Volumes 0-cube (point) 1 0 0 0 0 0 1-cube (line segment) 2 1 0 0 0 0 2-cube (square) 4 4 1 0 0 0 3-cube (cube) 8 12 6 1 0 0 4-cube (tesseract) 16 32 24 8 1 0 5-cube 32 80 8 40 10 1 We can further state with regard to all of these, and the higher n-cubes, that all their component edges, or faces, or volumes, etc, are congruent; that their component (n-1)-volumes (as the edges of a 2-cube, or the faces of a 3-cube, or the volumes of the 4-cube, etc.) occur in n parallel pairs, parallel having the same meaning in hyper-space as it does in 2- or 3-space; that the number of vertices of an n-cube is 2[[n exponent]], and the number of n-volumes is 1; and many other facts.
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page 13. visualize it, unless you are Quintus Teal. But to show that we can tell a good deal about what would happen, let's consider another aspect of this cube-generation. When the point moved, it generated, first, a line, and second, 2 points, the ends of the line, one representing its initial position and one representing its final position. Any point in any of the n-cubes, when the cube generates an (n+1)-cube, will similarly give rise to 1 line and 2 points. For example, the 4 points at the vertices of the square generate, in the cube, 8 vertices and 4 edges. (If you can't visualize this immediately, it'll help if you'll make the slight effort necessary to do so.) Similarly, when the lines is moved out of its own path it produces 1 area, representing the space through which it traveled, and 2 lines, its initial and final positions; and the squares's interior produces 1 volume (the interior of the cube) and 2 squares (2 of the cube's 6 faces). Extrapolating, quite legitimately, let's consider the hypothetical generation of the 4- by the 3-cube. Each of the cube's 8 vertices gives 2 vertices (making a total of 16 and 1 edge. Each of the cube's 12 edges gives 2 edges (making, with those obtained already, 32) and 1 area, or face. Each of the cube's 6 faces gives 2 faces (or an overall total of 24 and one volume.( The volumes are not the interior of the 4-cube, of course). The cube's interior volume gives 2 volumes (an overall total of 8) and 1 4-volume, the interior of the new figure. So ins pite of not being able to visualize this figure, we have a pretty complete idea of its structure. Just to show the generality with which we can study n-cubes, I'll set down the complete box-score for all them through the 5-cube. Vertices Edges Faces Volumes 4-Volumes 5-Volumes 0-cube (point) 1 0 0 0 0 0 1-cube (line segment) 2 1 0 0 0 0 2-cube (square) 4 4 1 0 0 0 3-cube (cube) 8 12 6 1 0 0 4-cube (tesseract) 16 32 24 8 1 0 5-cube 32 80 8 40 10 1 We can further state with regard to all of these, and the higher n-cubes, that all their component edges, or faces, or volumes, etc, are congruent; that their component (n-1)-volumes (as the edges of a 2-cube, or the faces of a 3-cube, or the volumes of the 4-cube, etc.) occur in n parallel pairs, parallel having the same meaning in hyper-space as it does in 2- or 3-space; that the number of vertices of an n-cube is 2[[n exponent]], and the number of n-volumes is 1; and many other facts.
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