zbMATH — the first resource for mathematics

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)
Full Text:
References:
 [1] Arak, T., Clifford, P. and Surgailis, D. (1993). Point-based polygonal models for random graphs. Adv. Appl. Prob. 25 , 348-372. · Zbl 0772.60036 [2] Cowan, R. (1978). The use of ergodic theorems in random geometry. Suppl. Adv. Appl. Prob. 10, 47-57. · Zbl 0381.60012 [3] Cowan, R. (1980). Properties of ergodic random mosaic processes. Math. Nachr. 97, 89-102. · Zbl 0475.60010 [4] 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 J. Gani and E. Seneta, Applied Probability Trust, Sheffield, pp. 3-15. · Zbl 1051.60010 [5] Cowan, R. (2010). New classes of random tessellations arising from iterative division of cells. Adv. Appl. Prob. 42, 26-47. · Zbl 1205.60028 [6] Cowan, R. and Morris, V. B. (1988). Division rules for polygonal cells. J. Theoret. Biol. 131, 33-42. [7] Cowan, R. and Tsang, A. K. L. (1994). The falling-leaf mosaic and its equilibrium properties. Adv. Appl. Prob. 26, 54-62. · Zbl 0792.60012 [8] Grünbaum, B. and Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman, New York. · Zbl 0601.05001 [9] Kendall, W. S. and Mecke, J. (1987). The range of mean-value quantities of planar tessellations. J. Appl. Prob. 24, 411-421. · Zbl 0624.60018 [10] Leistritz, L. and Zähle, M. (1992). Topological mean value relationships for random cell complexes. Math. Nachr. 155, 57-72. · Zbl 0762.60010 [11] Mecke, J. (1980). Palm methods for stationary random mosaics. In Combinatorial Principles in Stochastic Geometry, ed. R. V. Ambartzumian, Armenian Academy of Science, Erevan, pp. 124-132. · Zbl 0471.60019 [12] Mecke, J. (1983). Zufällige Mosaike. In Stochastische Geometrie , Akademie, Berlin, pp. 232-298 [13] Mecke, J. (1984). Parametric representation of mean values for stationary random mosaics. Math. Operationsforch. Statist. Ser. Statist. 15, 437-442. · Zbl 0547.60019 [14] 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. · Zbl 1155.60005 [15] Miles, R. E. (1970). On the homogeneous planar Poisson point process. Math. Biosci . 6, 85-127. · Zbl 0196.19403 [16] Miles, R. E. (1988). Matschinski’s identity and dual random tessellations. J. Microscopy 151, 187-190. [17] Miles, R. E. and Mackisack, M. S. (2002). A large class of random tessellations with the classic Poisson polygon distribution. Forma 17, 1-17. [18] Møller, J. (1989). Random tessellations in $$\mathbbR^d$$. Adv. Appl. Prob. 21, 37-73. · Zbl 0684.60007 [19] Muche, L. (1996). The Poisson-Voronoi tessellation. II. Edge length distribution functions. Math. Nachr. 178, 271-283. · Zbl 0846.60011 [20] Muche, L. (1998). The Poisson-Voronoi tessellation. III. Miles’ formula. Math. Nachr. 191, 247-267. · Zbl 0906.60009 [21] Muche, L. (2005). The Poisson-Voronoi tessellation: relationships for edges. Adv. Appl. Prob. 37, 279-296. · Zbl 1079.60020 [22] 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. · Zbl 1098.60012 [23] Nagel, W. and Weiss, V. (2008). Mean values for homogeneous STIT tessellation in 3D. Image Analysis Stereol. 27, 29-37. · Zbl 1168.60006 [24] Okabe, A., Boots, B., Sugihara, K. and Chiu, S. N. (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams , 2nd edn. John Wiley, Chichester. · Zbl 0946.68144 [25] Radecke, W. (1980). Some mean value relations on stationary random mosaics in the space. Math. Nachr. 97, 203-210. · Zbl 0475.60011 [26] Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry . Springer, Berlin. · Zbl 1175.60003 [27] Stoyan, D. (1986). On generalized planar random tessellations. Math. Nachr. 128, 215-219. · Zbl 0629.60018 [28] Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications , 2nd edn. John Wiley, Chichester. · Zbl 0838.60002 [29] Thäle, C. and Weiss, V. (2010). New mean values for homogeneous spatial tessellations that are stable under iteration. Image Analysis Stereol. 29, 143-157. · Zbl 1215.60008 [30] Weiss, V. and Zähle, M. (1988). Geometric measures for random curved mosiacs of $$\mathbbR^d$$. Math. Nachr. 138, 313-326. · Zbl 0663.60008 [31] Zähle, M. (1988). Random cell complexes and generalised sets. Ann. Prob. 16, 1742-1766. · Zbl 0656.60024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.