Hostname: page-component-848d4c4894-wzw2p Total loading time: 0 Render date: 2024-05-01T06:09:26.241Z Has data issue: false hasContentIssue false
  • English
  • Français

Topological relationships in spatial tessellations

Part of: Designs and configurations Real and complex geometry Polytopes and polyhedra Stochastic processes Geometric probability and stochastic geometry Discrete 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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

Random geometrytessellationtilingspacking of polyhedraspace fillingtopology

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 05B45: Tessellation and tiling problems 52C17: Packing and covering in $n$ dimensions
Secondary: 60G55: Point processes 51M20: Polyhedra and polytopes; regular figures, division of spaces 52B10: Three-dimensional polytopes
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 

References

Arak, T., Clifford, P. and Surgailis, D. (1993). Point-based polygonal models for random graphs. Adv. Appl. Prob. 25, 348–372.CrossRefGoogle Scholar
Cowan, R. (1978). The use of ergodic theorems in random geometry. Suppl. Adv. Appl. Prob. 10, 47–57.Google Scholar
Cowan, R. (1980). Properties of ergodic random mosaic processes. Math. Nachr. 97, 89–102.Google Scholar
Cowan, R. (2004). A mosaic of triangular cells formed with sequential splitting rules. In Stochastic Methods and Their Applications (J. Appl. Prob. Spec. Vol. 41A), eds Gani, J. and Seneta, E., Applied Probability Trust, Sheffield, pp. 3–15.Google Scholar
Cowan, R. (2010). New classes of random tessellations arising from iterative division of cells. Adv. Appl. Prob. 42, 26–47.CrossRefGoogle Scholar
Cowan, R. and Morris, V. B. (1988). Division rules for polygonal cells. J. Theoret. Biol. 131, 33–42.Google Scholar
Cowan, R. and Tsang, A. K. L. (1994). The falling-leaf mosaic and its equilibrium properties. Adv. Appl. Prob. 26, 54–62.Google Scholar
Grünbaum, B. and Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman, New York.Google Scholar
Kendall, W. S. and Mecke, J. (1987). The range of mean-value quantities of planar tessellations. J. Appl. Prob. 24, 411–421.CrossRefGoogle Scholar
Leistritz, L. and Zähle, M. (1992). Topological mean value relationships for random cell complexes. Math. Nachr. 155, 57–72.CrossRefGoogle Scholar
Mecke, J. (1980). Palm methods for stationary random mosaics. In Combinatorial Principles in Stochastic Geometry, ed. Ambartzumian, R. V., Armenian Academy of Science, Erevan, pp. 124–132.Google Scholar
Mecke, J. (1983). Zufällige Mosaike. In Stochastische Geometrie, Akademie, Berlin, pp. 232–298 Google Scholar
Mecke, J. (1984). Parametric representation of mean values for stationary random mosaics. Math. Operationsforch. Statist. Ser. Statist. 15, 437–442.Google Scholar
Mecke, J., Nagel, W. and Weiss, V. (2007). Length distributions of edges in planar stationary and isotropic STIT tessellations. Izv. Akad. Nauk Armenii Mat. 42, 39–60. English translation: J. Contemp. Math. Anal. 42, 28-43.Google Scholar
Miles, R. E. (1970). On the homogeneous planar Poisson point process. Math. Biosci. 6, 85–127.Google Scholar
Miles, R. E. (1988). Matschinski's identity and dual random tessellations. J. Microscopy 151, 187–190.Google Scholar
Miles, R. E. and Mackisack, M. S. (2002). A large class of random tessellations with the classic Poisson polygon distribution. Forma 17, 1–17.Google Scholar
Møller, J. (1989). Random tessellations in R d . Adv. Appl. Prob. 21, 37–73.Google Scholar
Muche, L. (1996). The Poisson–Voronoi tessellation. II. Edge length distribution functions. Math. Nachr. 178, 271–283.CrossRefGoogle Scholar
Muche, L. (1998). The Poisson–Voronoi tessellation. III. Miles' formula. Math. Nachr. 191, 247–267.Google Scholar
Muche, L. (2005). The Poisson–Voronoi tessellation: relationships for edges. Adv. Appl. Prob. 37, 279–296.Google Scholar
Nagel, W. and Weiss, V. (2005). Crack STIT tessellations: characterization of stationary random tessellations stable with respect to iteration. Adv. Appl. Prob. 37, 859–883.Google Scholar
Nagel, W. and Weiss, V. (2008). Mean values for homogeneous STIT tessellation in 3D. Image Analysis Stereol. 27, 29–37.CrossRefGoogle Scholar
Okabe, A., Boots, B., Sugihara, K. and Chiu, S. N. (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, 2nd edn. John Wiley, Chichester.Google Scholar
Radecke, W. (1980). Some mean value relations on stationary random mosaics in the space. Math. Nachr. 97, 203–210.Google Scholar
Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.CrossRefGoogle Scholar
Stoyan, D. (1986). On generalized planar random tessellations. Math. Nachr. 128, 215–219.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.Google Scholar
Thäle, C. and Weiss, V. (2010). New mean values for homogeneous spatial tessellations that are stable under iteration. Image Analysis Stereol. 29, 143–157.Google Scholar
Weiss, V. and Zähle, M. (1988). Geometric measures for random curved mosiacs of R d . Math. Nachr. 138, 313–326.CrossRefGoogle Scholar
Zähle, M. (1988). Random cell complexes and generalised sets. Ann. Prob. 16, 1742–1766.Google Scholar