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| Format: | Preprint |
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2020
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| Accès en ligne: | https://arxiv.org/abs/2001.02945 |
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| _version_ | 1866916320947208192 |
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| author | Hou, Dong-Dong Feng, Yan-Quan Leemans, Dimitri |
| author_facet | Hou, Dong-Dong Feng, Yan-Quan Leemans, Dimitri |
| contents | 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 $m$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2001_02945 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Regular $3$-polytopes of order $2^np$ Hou, Dong-Dong Feng, Yan-Quan Leemans, Dimitri Combinatorics 20B25, 20D10, 52B10, 52B15 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 $m$. |
| title | Regular $3$-polytopes of order $2^np$ |
| topic | Combinatorics 20B25, 20D10, 52B10, 52B15 |
| url | https://arxiv.org/abs/2001.02945 |