Equifacetted polytopes whose facets are products of simplices: The simplicial prism case

Abstract

We study \emph{prismatical} polytopes, those all of whose facets are combinatorially isomorphic to the simplicial prism ${\Delta }^{d-2}\times {\Delta }^1$. In dimension three these are exactly the cubical polytopes, so the class sits naturally between simplicial and cubical polytopes. In dimension four, ${\Delta }^2\times {\Delta }^2$ is the simplest prismatical polytope and we can construct infinitely many by using connected sums. By fibreing a triangle over a triangulated polygon we construct infinitely many connected-sum-prime prismatical $4$-polytopes. Finally, we show that there are no prismatical $d$-polytopes for $d\ge 5$, answering a question of Perles and Shephard.

arXiv comment26 pages, 10 figures, comments welcome