Hypercube Embeddings of Wythoffians

The Wythoff construction takes a d-dimensional polytope P , a subset S of {0, . . . , d} and returns another d-dimensional polytope P (S). If P is a regular polytope, then P (S) is vertex-transitive. This construction builds a large part of the Archimedean polytopes and tilings in dimension 3 and 4. We want to determine, which of those Wythoffians P (S) with regular P have their skeleton or dual skeleton isometrically embeddable into the hypercubes Hm and half-cubes 12Hm. We find six infinite series, which, we conjecture, cover all cases for dimension d > 5 and some sporadic cases in dimension 3 and 4 (see Tables 1 and 2). Three out of those six infinite series are explained by a general result about the embedding of Wythoff construction for Coxeter groups. In the last section, we consider the Euclidean case; also, zonotopality of embeddable P (S) are addressed throughout the text.