CPR graphs and regular polytopes

Regular polytopes

Regular Polytopes in Z

In [3], all embeddings of regular polyhedra in the three dimensional integer lattice were characterized. Here, we prove some results toward solving this problem for all higher dimensions. Similarly to [3], we consider a few special polytopes in dimension 4 that do not have analogues in higher dimensions. We then begin a classification of hypercubes, and consequently regular cross polytopes in t...

1 Classical Regular Polytopes

Our purpose in this introductory chapter is to set the scene for the rest of the book. We shall do this by briefly tracing the historical development of the subject. There are two main reasons for this. First, we wish to recall the historical traditions which lie behind current research. It is all too easy to lose track of the past, and it is as true inmathematics as in anything else that those...

Regular triangulations and resultant polytopes

We describe properties of the Resultant polytope of a given set of polynomial equations towards an outputsensitive algorithm for enumerating its vertices. In principle, one has to consider all regular fine mixed subdivisions of the Minkowski sum of the Newton polytopes of the given equations. By the Cayley trick, this is equivalent to computing all regular triangulations of another point set in...

Spectra of Regular Polytopes

Introduction As is well known, the combinatorial problem of counting paths of length n between two xed vertices in a graph reduces to raising the adjacency matrix A of the graph to the n-th power ((B], p. 11). For an undirected graph, A is symmetric and the problem above simpliies considerably if its spectrum (A) is known and contains few distinct elements. Spectra of graphs, meaning spectra of...