再讲一遍52个四维平行六面体
Once more about the 52 four-dimensional parallelotopes
摘要：
有几个作品\cite{De}（和\cite{St}），\cite{En}，\cite{Co}和\cite{Va}枚举四维平行表。在这项工作中，我们给出了一个新的计数表明，任何四维平行体是一个zonotope或Minkowski和zonotope与正规24-细胞$\{3,4,3\}$。每一个带形平行体都是其生成向量形成一个幺模系统的线段的闵可夫斯基和。一共有17个四维幺模系统。因此，有17个四维的zonotopal平行。其他35个四维平行体是：规则的24-细胞$\{3,4,3\}$和34个总和的规则平行体与非零zonotopal平行体。对于这34个和的非平凡计数，我们利用一个定理描述了带线段的平行体的Minkowski和是平行体的充要条件.
英文摘要：
There are several works \cite{De} (and \cite{St}), \cite{En}, \cite{Co} and \cite{Va} enumerating four-dimensional parallelotopes. In this work we give a new enumeration showing that any four-dimensional parallelotope is either a zonotope or the Minkowski sum of a zonotope with the regular 24-cell $\{3,4,3\}$. Each zonotopal parallelotope is the Minkowski sum of segments whose generating vectors form a unimodular system. There are exactly 17 four-dimensional unimodular systems. Hence, there are 17 four-dimensional zonotopal parallelotopes. Other 35 four-dimensional parallelotopes are: the regular 24-cell $\{3,4,3\}$ and 34 sums of the regular parallelotope with non-zero zonotopal parallelotopes. For the nontrivial enumerating of the 34 sums we use a theorem discribing necessary and sufficient conditions when the Minkowski sum of a parallelotope with a segment is a parallelotope.
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