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56 57 This book started with the regular two and threedimensional polytopes, the regular polygons and Platonic solids. We end it by showing how analogy can be used to guess and prove the properties of higherdimensional regular polytopes. Basic Polytopes Using coordinates, it is easy to see that the tetrahedron, cube, and octahedron have the following relatives in ndimensional or nd space. The nd simplex has n1 vertices, two of which are the same distance apart. A 1, 2, 3, ..., nd simplex is bordered by 2, 3, 4, ..., n1 simplices of one less dimension. The vertices of an nd cube are the points with all coordinates 1 or 1. A 1, 2, 3, ..., nd cube has 2, 4, 8, ..., 2n vertices and is bordered by 2n cubes of one less dimension. The tesseract is the 4d cube. The vertices of the nd orthoplex are the 2n end points of an nd unit cross. A 1, 2, 3, ..., nd orthoplex has 2, 4, 6, ..., 2n vertices and is bordered by 2, 4, 8, ..., 2n simplices of one less dimension. Classification. Ludwig Schlafli 18141895 proved that apart from the simplex, cube, and orthoplex, there are three more regular polytopes in four dimensions and no more in any higher dimension. The regular 3d polytopes correspond to the five ways of fitting at least three identical regular 2d polytopes around a vertex with space left to fold up into the third dimension see opp. page 1. The regular 4d polytopes correspond to the six ways of fitting at least three identical regular 3d polytopes around an edge with space left to fold up into 4d space. aPPendIX v PolytoPes oF analogy Vital Statistics The regular 3d polytopes have identical vertices, 1d edges, and 2d faces. The regular 4d polytopes also have 3d cells. The numbers V vertices, E edges, F faces, C cells, fv faces per vertex, ce cells per edge, cv cells per vertex are The bold entries are easily extracted from the information on the opposite page and, in the case of the 3d polytopes, from some life models. Here is how you extract the numbers cv cells per vertex for the 4d polytopes by analogy. Cutting a regular 3d polytope close to a vertex gives a regular polygon whose sides are the cuts of the faces around the vertex. Thus, no. sides of polygon fv of 3d polytope Cutting a regular 4d polytope close to a vertex gives a regular 3d polytope whose faces are the cuts of the cells around the vertex. Hence, no. faces of 3d polytope cv of 4d polytope fv of 3d polytope ce of 4d polytope These relationships allow one to deduce the cuts of the different 4d polytopes on the right and thereby also their cv. So, for example, since the 4d cube has a tetrahedral cut, it has four cells per vertex. The numbers of cells of the three nonstandard regular 4d polytopes are built into their names, and no oneglance way to deduce these numbers has been discovered. However, once we know these numbers, the part of the 4d table that we have not yet touched can be easily filled in using some simple relationships such as 1 the 4d analogue of the Euler formula VFEC0 see page 44, 2 F C . faces per cell2, and last but not least, 3 E C . edges per cellcells per edge.
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