论文标题
常规$ 3 $ - 订单订单$ 2^np $
Regular $3$-polytopes of order $2^np$
论文作者
论文摘要
在[关于多面体，他们的群体和实现的问题，量化数学。匈牙利53（2006）231-255] Schulte和Weiss提出了以下问题：{\ em代表订单的常规多型$ 2^np $ for $ n $ a $ a a $ a正整数和$ p $ a Odd Prime}。在本文中，我们首先证明，如果$ 3 $ - 订单的订单$ 2^np $具有Schläflitype $ \ {k_1，k_2 \} $，则$ p \ mid k_1 $或$ p \ p \ mid k_2 $。对于Schläfli类型，这将导致两个类，即二元性，即（1）其中$ k_1 = 2^sp $ and $ k_2 = 2^t $和类型（2）其中$ k_1 = 2^sp^sp $ and $ k_2 = 2 = 2^tp $。然后，我们表明存在常规$ 3 $ - 订单的订单$ 2^np $，type（1）当$ s \ geq 2 $，$ t \ geq 2 $和$ n \ geq s+t+t+1 $来自常规$ 3 $ 3 $ -POLYTOPES的常规构建订单$ 2^N \ ell_1 \ ell_1 \ ell_2 $ typeschlä2$ 2 $ -polytopes $ \ {2^s \ ell_1,2^t \ ell_2 \} $，其中$ \ ell_1 $和$ \ ell_2 $都是奇数。此外，对于$ p = 3 $和$ n \ geq 7 $，我们表明存在常规的3个订单订单$ 3 \ cdot2^n $，type $ \ {6,2^s \} $，仅当$ 2 \ leq s \ leq s \ leq s \ leq n-2 $和$ s \ s \ s \ n-3 $。对于类型（2），我们证明存在常规$ 3 $ - 订单的订单$ 2^n \ cdot 3 $，带有schläfli型$ \ {6，6，6，6 \} $当$ n \ ge 5 $来自常规$ 3 $ 3 $ -POLYTOPES ofSCHLäfli-Polytopes type $ \ fludy $ \ fludy $ 3 $ \ \ \ {6，6，38 $^38 $^38 $或7.68亿美元^3 $，对于任何正整数$ m $。
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 $m$.
