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New Classes of Random Tessellations Arising from Iterative Division of Cells

Part of: Geometric probability and stochastic geometry Designs and configurations Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Richard Cowan
Richard Cowan*
Affiliation:
University of Sydney
*
∗ Postal address: School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia. Email address: rcowan@mail.usyd.edu.au
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Abstract

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We present new ideas about the type of random tessellation which evolves through successive division of its cells. These ideas are developed in an intuitive way, with many pictures and only a modicum of mathematical formalism–so that the wide application of the ideas is clearly apparent to all readers. A vast number of new tessellation models, with known probability distribution for the volume of the typical cell, follow from the concepts in this paper. There are other interesting models for which results are not presented (or presented only through simulation methods), but these models have illustrative value. A large agenda of further research is opened up by the ideas in this paper.

Keywords

Random geometrytessellationiterative divisionfractalshape distribution

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 05B45: Tessellation and tiling problems
Secondary: 60G55: Point processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 42 , Issue 1 , March 2010 , pp. 26 - 47
Copyright
Copyright © Applied Probability Trust 2010 

References

Abramowitz, M. and Stegun, I. A. (1970). Handbook of Mathematical Functions. Dover, New York.Google Scholar
Chen, F. K. C. and Cowan, R. (1999). Invariant distributions for shapes in sequences of randomly-divided rectangles. Adv. Appl. Prob. 31, 1–14.CrossRefGoogle Scholar
Cowan, R. (1997). Shapes of rectangular prisms after repeated random division. Adv. Appl. Prob. 29, 26–37.CrossRefGoogle 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. and Chen, F. K. C. (1999). Four interesting problems concerning Markovian shape sequences. Adv. Appl. Prob. 31, 954–968.CrossRefGoogle Scholar
Halmos, P. R. (1944). Random alms. Ann. Math. Statist. 15, 182–189.CrossRefGoogle Scholar
Kendall, D. G. (1977). The diffusion of shape. Adv. Appl. Prob. 9, 428–430.CrossRefGoogle Scholar
Kendall, D. G., Barden, D., Carne, T. K. and Le, H. (1999). Shape and Shape Theory. John Wiley, Chichester.CrossRefGoogle Scholar
Mannion, D. (1990). Convergence to collinearity of a sequence of random triangle shapes. Adv. Appl. Prob. 22, 831–844.CrossRefGoogle Scholar
Mannion, D. (1993). Products of 2×2 random matrices. Ann. Appl. Prob. 3, 1189–1218.CrossRefGoogle Scholar
Miles, R. E. (1983). On the repeated splitting of a planar domain. In Proc. Oberwolfach Conf. Stochastic Geometry, Geometric Statistics and Stereology, eds Ambartzumian, R. and Weil Teubner, W., Leipzig, pp. 110–123.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.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.CrossRefGoogle Scholar
Norberg, T. (1984). Convergence and existence of random set distributions. Ann. Prob. 12, 726–732.CrossRefGoogle Scholar
Small, C. G. (1996). The Statistical Theory of Shape. Springer, New York.CrossRefGoogle Scholar
Watson, G. S. (1986). The shapes of a random sequence of triangles. Adv. Appl. Prob. 18, 156–169.CrossRefGoogle Scholar