Constraints on the fundamental topological parameters of spatial tessellations |
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Authors: | Richard Cowan Viola Weiss |
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Institution: | 1. +61 2. 2 3. 9043 4. 6710;5. University of Sydney, School of Mathematics and Statistics, Australia;6. Ernst‐Abbe‐Hochschule Jena, Fachbereich Grundlagenwissenschaften, Jena, Germany |
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Abstract: | Tessellations of 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 15 , 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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Keywords: | Random geometry tessellations tilings packing of polyhedra space‐filling combinatorial topology cell complex parameter constraints Primary: 05B45 52C17 60D05 Secondary: 52B10 60G55 51M20 |
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