# THE CHARACTER OF PLANAR TESSELLATIONS WHICH ARE NOT SIDE-TO-SIDE

## DOI:

https://doi.org/10.5566/ias.v33.p39-54## Keywords:

combinatorial topology, Delaunay tessellation, random tessellations, STIT tessellation, stochastic geometry, tilings## Abstract

This paper studies stationary tessellations and tilings of the plane in which all cells are convex polygons. The focus is on the class of tessellations which are not side-to-side. The character of these tessellations is explored, with special attention paid to the relationship between edges of the tessellation and sides of the polygonal cells and to the combinatorial topology between the ‘adjacent’ geometric elements of the tessellation. Three new parameters, e0,e1 and e2 summing to unity, are introduced. These capture the essence of non side-to-side tessellations and play a role in understanding the adjacency of sides and cells. Examples illustrate the theory.## References

Cowan R (1978) The use of ergodic theorems in random geometry. Suppl Adv Appl Prob 1:47–57.

Cowan, Richard. Homogeneous line-segment processes. Mathematical Proceedings of the Cambridge Philosophical Society. Vol. 86. No. 03. Cambridge University Press, 1979.

Cowan, R Properties of ergodic random mosaic processes. Math. Nachr. 97:89–102.

Cowan R and Morris VB (1988) Division Rules for Polygonal Cells. J. Theor. Biol. 131:33–42.

Cowan R (2004) A mosaic of triangular cells formed with sequential splitting rules. J. Appl Prob Special Volume, 41A:3–15.

Cowan R (2010) New classes of random tessellations arising from iterative division of cells. Adv Appl Prob, 42:26–47.

Cowan R (2013) Line-segments in the isotropic planar STIT tessellation. To appear Adv Appl Prob.

Cowan R and Tsang AKL (1994) The falling-leaf mosaic and its equilibrium properties. Adv Appl Prob, 26:54–62.

Grünbaum B and Shephard GC (1987) Tilings and Patterns.WH Freeman and Co, New York.

Leistritz L and Zähle M (1992) Topological Mean Value Relationships for Random Cell Complexes. Math Nachr 155:57–72.

Maier R and Schmidt V (2003) Stationary iterated tessellations. Adv Appl Prob 35:337–353.

Mecke J (1980) Palm methods for stationary random mosaics. In Combinatorial Principles in Stochastic Geometry, Ed. RA Ambartzumian. Armenian Academy of Science, Erevan.

Mecke J (1984) Parametric representation of mean values for stationary random mosaics. Math Operations Statist Ser Statist 15:437–442.

Mecke J, Nagel W and Weiss V (2007) Length distributions of edges in planar stationary and isotropic STIT tessellations. Izvestija Akademii Nauk Armenii, Matematika 42:39–60.

Mecke J, Nagel W and Weiss V (2011) Some distributions for I-segments of planar random homogeneous STIT tessellations. Math Nachr 284:1483–1495.

Miles RE (1970) On the homogeneous planar Poisson point process. Math Biosciences 6:85–127.

Miles RE (1973) The various aggregates of random polygons determined by random lines in a plane. Adv Math 10:256–290.

Moller J (1989) Random tessellations in Rd. Adv Appl Prob 21:37–73.

Nagel W and Weiss V (2003) Limits of sequences of stationary planar tessellations. Adv Appl Prob 35:123–138.

Nagel W andWeiss V (2005) Crack STIT tessellations: characterization of stationary random tessellations stable with respect to iteration. Adv Appl Prob 37:859–883.

Santaló LA (1984) Mixed Random Mosaics. Math Nachr 117:129–133.

Schneider R and Weil W (2008) Stochastic and Integral Geometry. Springer, Berlin Heidelberg.

Schreiber T and Thäle C (2010) Second-order properties and central limit theory for the vertex process of iteration infinitely divisible and iteration stable random tessellations in the plane. Adv Appl Prob 42:913–935.

Schreiber T and Thäle C (2013) Limit theorems for iteration stable tessellations. To appear Ann Prob.

Stoyan D, Kendall WS and Mecke J (1995) Stochastic Geometry and its Applications. 2nd Edition.Wiley, Chichester.

Weiss V and Cowan R (2011). Topological relationships in spatial tessellations. Adv Appl Prob 43:963–984.

Weiss V and Zähle M (1988) Geometric Measures for Random Curved Mosiacs of Rd. Math Nachr 138:313–326.

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*Image Analysis and Stereology*,

*33*(1), 39–54. https://doi.org/10.5566/ias.v33.p39-54