Hypercube embedding of Wythoffians

Michel Marie Deza, Mathieu Dutour Sikirić, Sergey Shpectorov


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 1/2 Hm. 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.


Coxeter group, embedding

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DOI: https://doi.org/10.26493/1855-3974.26.6c7

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications