Questions tagged [abstract-polytopes]

Are there any finite regular abstract 5-polytopes whose facets are 11-cells and whose vertex figures are 57-cells?

Let $(P,\le)$ be an abstract $n$-polytope, with $n\ge 2$. Let $H,H',K$ be $m$-faces, with $0\le m \le n-2$. Is it true that there is a sequence $\{H_0=H,H_1,\ldots,H_{r-1},H_r=H' \} \subseteq P$ so ...

Let $C$ be a $d$-dimensional (abstract) polytopal complex. Most of what I say below could be asked in this general setting, but for a start, let's further restrict to simple polytopal spheres, that is,...

Take a set of reflections $\{r_1,\ldots,r_k\}$ of $\mathbb R^n$. Sometimes, the group presentation will turn out to be a C-group – this is where the regular planar polytopes in Euclidean space, ...

What are the conditions that allow an abstract polytope to have a non-skew (but not necessarily convex) realization in Euclidean space?