In the classical setting, a convex polytope is said to be semiregular if its facets are regular and its symmetry group is transitive on vertices. This paper continues our study of alternating semiregular abstract polytopes, which have abstract regular facets, still with combinatorial automorphism group transitive on vertices and with two kinds of regular facets occurring in an alternating fashion.

Our main concern here is the universal polytope ${\mathcal{U}}_{{\mathcal{P}},{\mathcal{Q}}}$, an alternating semiregular $(n+1)$-polytope defined for any pair of regular $n$-polytopes ${\mathcal{P}},{\mathcal{Q}}$ with isomorphic facets. After a careful look at the local structure of these objects, we develop the combinatorial machinery needed to explain how ${\mathcal{U}}_{{\mathcal{P}},{\mathcal{Q}}}$ can be constructed by “freely assembling” unlimited copies of ${\mathcal{P}}$, ${\mathcal{Q}}$ along their facets in alternating fashion. We then examine the connection group of ${\mathcal{U}}_{{\mathcal{P}},{\mathcal{Q}}}$, and from that prove that ${\mathcal{U}}_{{\mathcal{P}},{\mathcal{Q}}}$ covers any $(n+1)$-polytope ${\mathcal{B}}$ whose facets alternate in any way between various quotients of ${\mathcal{P}}$ or ${\mathcal{Q}}$.

在经典环境中，如果凸多面体的小面是规则的并且其对称组在顶点上是可传递的，则称其为半规则的。本文继续我们对交替的半规则抽象多面体的研究，它们具有抽象的规则面，组合自同构群仍在顶点上传递，并且两种规则面以交替的方式出现。

我们这里主要关注的是普遍的多面体$ {\ mathcal {蓝}} _ {{\ mathcal {P}}，{\ mathcal {Q}}} $，交替半规则$（N + 1）$ -polytope的定义具有同构面的任意一对常规$ n $-多面体$ {\ mathcal {P}}，{\ mathcal {Q}} $。在这些物体的局部结构仔细一看后，我们开发讲解如何需要的组合机械$ {\ mathcal {蓝}} _ {{\ mathcal {P}}，{\ mathcal {Q}}} $可以通过将$ {\ mathcal {P}} $，$ {\ mathcal {Q}} $的副本沿其各个面以交替的方式“自由组合”无限的数量来构造 。然后，我们检查$ {\ mathcal {U}} _ {{\ mathcal {P}}，{\ mathcal {Q}}} $$的连接组，并由此证明$ {\ mathcal {U}} _ {{\ mathcal {P}}，{\ mathcal {Q}}} $$可以覆盖任何$（n + 1）$-多面体$ {\ mathcal {B }} $的构面在$ {\ mathcal {P}} $或 $ {\ mathcal {Q}} $的各个商之间以任何方式交替。