Regular $3$-polytopes of order $2^np$

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 $2^np$ for $n$ a positive integer and $p$ an odd prime}. In this paper, we first prove that if a $3$-polytope of order $2^np$ 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 $k_1=2^sp$ and $k_2=2^t$ and Type (2) where $k_1=2^sp$ and $k_2=2^tp$. We then show that there exists a regular $3$-polytope of order $2^np$ with Type (1) when $s\geq 2$, $t\geq 2$ and $n\geq s+t+1$ 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 $n \geq 7$, we show that there exists a regular 3-polytope of order $3\cdot2^n$ with type $\{6,2^s\}$ if and only if $2\leq s \leq n-2$ and $s \neq n-3$. 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 $m$.

Theorems & Definitions (31)

• Theorem 1.2
• Theorem 1.3
• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• Proposition 2.5
• Proposition 2.6
• Proposition 2.7
• Proposition 2.8