Constraints on the fundamental topological parameters of spatial tessellations

TL;DR

This paper establishes constraints on seven fundamental parameters describing 3D space tessellations with convex polyhedral cells, revealing an unbounded permissible parameter region and providing new constraints even for simpler facet-to-facet cases.

Contribution

It derives the first constraints on seven key parameters of spatial tessellations, including the unbounded nature of their permissible region and new constraints for facet-to-facet tessellations.

Findings

Permissible parameter region is unbounded.

New constraints established for facet-to-facet tessellations.

Seven parameters are necessary to describe complex tessellations.

Abstract

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.