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Topological relationships in spatial tessellations. (English) Zbl 1238.60017
Tessellations of the 3-dimensional space \({\mathbb R}^3\) that use convex polyhedral cells to fill the space can be extremely complicated. This is especially so for tessellations which are not “facet-to-facet”, that is, for those where the facets of a cell do not necessarily coincide with the facets of that cell’s neighbors. Adjacency concepts between neighboring cells (or between neighboring cell elements) are not easily formulated when facets do not coincide. The authors develop new mean-value formulae for various topological parameters. The formulae generalize one of the identities for stationary planar tessellations presented by the second author (see [R. Cowan, “The use of ergodic theorems in random geometry”, Adv. Appl. Probab. 10, 47–57 (1978; Zbl 0381.60012); “Properties of ergodic random mosaic processes”, Math. Nachr. 97, 89–102 (1980; Zbl 0475.60010)]). The results derived can also be applied to the simpler facet-to-fact case. The study deals with both random tessellations and deterministic “tiling”-s. Some new theory for planar tessellations is also given.

MSC:
60D05 Geometric probability and stochastic geometry
05B45 Combinatorial aspects of tessellation and tiling problems
52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
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