Fundamental Properties and Constructions
Despite the simplicity of the notion of a polytope, our understanding of what properties a polytope may, or may not, have is severely hampered by the difficulty of producing polytopes having certain desired features. One example of such a difficulty will help to focus the problem. Let us consider 4-polytopes with 8 vertices; with some patience, combining the different types of polytopes considered in Chapter 4, the reader will probably find 4-polytopes with 8 vertices and 16 edges, or with any number of edges between 18 and 28. Now he may ask himself whether it is possible to complete the list by a polytope with 17 edges. But how is one to start looking for such a polytope?
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