Advances in Applied Probability

Topological relationships in spatial tessellations

Viola Weiss and Richard Cowan

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Tessellations of R3 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 neighbours. Adjacency concepts between neighbouring cells (or between neighbouring cell elements) are not easily formulated when facets do not coincide. In this paper we make the first systematic study of these topological relationships when a tessellation of R3 is not facet-to-facet. The results derived can also be applied to the simpler facet-to-facet case. Our study deals with both random tessellations and deterministic `tilings'. Some new theory for planar tessellations is also given.

Article information

Adv. in Appl. Probab., Volume 43, Number 4 (2011), 963-984.

First available in Project Euclid: 16 December 2011

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Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 05B45: Tessellation and tiling problems [See also 52C20, 52C22] 52C17: Packing and covering in $n$ dimensions [See also 05B40, 11H31]
Secondary: 60G55: Point processes 51M20: Polyhedra and polytopes; regular figures, division of spaces [See also 51F15] 52B10: Three-dimensional polytopes

Random geometry tessellation tilings packing of polyhedra space filling topology


Weiss, Viola; Cowan, Richard. Topological relationships in spatial tessellations. Adv. in Appl. Probab. 43 (2011), no. 4, 963--984. doi:10.1239/aap/1324045694.

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