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Topological relationships in spatial tessellations

Part of:Designs and configurationsPolytopes and polyhedraStochastic processesDiscrete geometryReal and complex geometryGeometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Viola Weiss and
Richard Cowan
Viola Weiss*
Affiliation:
Fachhochschule Jena
Richard Cowan*
Affiliation:
University of Sydney
*
∗ Postal address: Fachhochschule Jena, D-07703 Jena, Germany. Email address: viola.weiss@fh-jena.de
∗∗ Postal address: School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia. Email address: rcowan@usyd.edu.au
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Abstract

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Tessellations of 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 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 R 3 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.

Keywords

Random geometrytessellationtilingspacking of polyhedraspace fillingtopology

MSC classification

Primary:60D05: Geometric probability and stochastic geometry05B45: Tessellation and tiling problems52C17: Packing and covering in $n$ dimensions
Secondary:60G55: Point processes51M20: Polyhedra and polytopes; regular figures, division of spaces52B10: Three-dimensional polytopes

Information

Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 43 , Issue 4 , December 2011, pp. 963 - 984
Copyright
Copyright © Applied Probability Trust 2011 

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