空间镶嵌基本拓扑参数的约束条件
Constraints on the fundamental topological parameters of spatial tessellations
摘要：
使用凸多面体单元来填充空间的$R^3$的镶嵌可能极其复杂，特别是如果它们不是面到面的，即，如果单元的面不一定与该单元的邻居的面重合。在最近的一篇论文中（Weiss和Cowan，Adv.Appl.Prob. 2011），我们已经开发了一个理论，它涵盖了这些复杂的情况下，至少在他们的组合拓扑结构。这个理论需要七个参数，其中三个足以满足面到面的情况;剩下的四个参数是在一般情况下出现的笨拙的邻接概念所需要的。本文建立了适用于这七个参数的约束条件，并在其七维空间内定义了一个允许区域，我们发现这个区域是没有边界的。我们在相对简单的面到面情况下的约束也是新的。
英文摘要：
Tessellations of $R^3$ that use convex polyhedral cells to fill the space can be extremely complicated, especially if they are not facet-to-facet, that is, if the facets of a cell do not necessarily coincide with the facets of that cell's neighbours. In a recent paper (Weiss and Cowan, Adv.Appl.Prob. 2011), we have developed a theory which covers these complicated cases, at least with respect to their combinatorial topology. The theory required seven parameters, three of which suffice for facet-to-facet cases; the remaining four parameters are needed for the awkward adjacency concepts that arise in the general case. This current paper establishes constraints that apply to these seven parameters and so defines a permissible region within their seven-dimensional space, a region which we discover is not bounded. Our constraints in the relatively simple facet-to-facet case are also new.
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