Regular $3$-polytopes of order $2^{n}p$

Abstract

In [Problems on polytopes, their groups, and realizations, Periodica Math. Hungarica 53 (2006) 231-255] Schulte and Weiss proposed the following problem: {\em Characterize regular polytopes of orders $2^{n}p$ for $n$ a positive integer and $p$ an odd prime}. In this paper, we first prove that if a $3$-polytope of order $2^{n}p$ has Schläfli type $\{k_1,k_2\}$, then $p\mid k_1$ or $p\mid k_2$. This leads to two classes, up to duality, for the Schläfli type, namely Type (1) where $k_1=2^{s}p$ and $k_2=2^t$ and Type (2) where $k_1=2^{s}p$ and $k_2=2^{t}p$. We then show that there exists a regular $3$-polytope of order $2^{n}p$ with Type (1) when $s\ge 2$, $t\ge 2$ and $n\ge s+t+1$ coming from a general construction of regular $3$-polytopes of order $2^n{\ell }_1{\ell }_2$ with Schläfli type $\{2^s{\ell }_1,2^t{\ell }_2\}$ where both ${\ell }_1$ and ${\ell }_2$ are odd. Furthermore, for $p=3$ and $n\ge 7$, we show that there exists a regular 3-polytope of order $3\cdot 2^n$ with type $\{6,2^s\}$ if and only if $2\le s\le n-2$ and $s\ne n-3$. For Type (2), we prove that there exists a regular $3$-polytope of order $2^n\cdot 3$ with Schläfli type $\{6,6\}$ when $n\ge 5$ coming from a general construction of regular $3$-polytopes of Schläfli type $\{6,6\}$ with orders $192m^3$, $384m^3$ or $768m^3$, for any positive integer $m$.

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