• Richard Cowan School of Mathematics and Statistics University of Sydney
  • Viola Weiss Ernst-Abbe-Hochschule, Jena



combinatorial topology, edge types, planar tessellations, STIT tessellation, stochastic geometry, superposition/nesting


Planar tessellation structures occur in material science, geology (in rock formations), physics (of foams, for example), biology (especially in epithelial studies) and in other sciences. Their mathematical and statistical study has many aspects to consider. In this paper, line-segments which are either a tessellation edge or a finite union of edges are studied. Our focus is on a sub-class of such line-segments – those we call M-segments – that are not contained in a longer union of edges. These encompass the so-called I-segments that have arisen in many recent tessellation models. We study the expected numbers of edges and cell-sides contained in these M-segments, and the prevalence of these entities. Many examples and figures, including some based on tessellation nesting and superposition, illustrate the theory. M-segments are much more prevalent when a tessellation is not side-to-side, so our paper has theoretical connections with the recent IAS paper by Cowan and Thäle (2014); that paper characterised non side-to-side tessellations.


Blizard WD (1989) Multiset Theory. Notre Dame J Formal Logic 30:36--66.

Chiu SN, Stoyan D, Kendall WS, Mecke J (2013) Stochastic Geometry and

its Applications. 3rd Edition. Wiley, Chichester.

Cowan R (1978) The use of ergodic theorems in random geometry. Suppl

Adv Appl Prob 10:47--57.

Cowan R (1979) Homogeneous line-segment processes. Math Proc Camb

Phil Soc 86:481--9.

Cowan, R (1980) Properties of ergodic random mosaic processes. Math

Nachr 97:89--102.

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. Adv Appl Prob 45:295--311.

Cowan R, Th"{a}le C (2014) The character of planar

tessellations which are not side to side. Image Anal Steriol 33:39--54.

Cowan R, Weiss V (2015). Constraints on the fundamental topological parameters of spatial

tessellations. Math Nachr 288:540--65.

Gilbert EN (1967) Surface films of needle-shaped crystals. In: Noble B, Ed. Applications of Undergraduate Mathematics in Engineering, 329--46. New York: Macmillan.

Kallenberg O (1977) A counterexample to R. Davidson's conjecture on line processes. Math Camb Proc Phil Soc 82:301--7.

Kline M (1980) Mathematics: The Loss of Certainty. Oxford University Press, Oxford.

Mackisack M, Miles RE (1996) Homogeneous rectangular tessellations.

Adv. in Appl. Probab.,28:993--1013, 1996.

Mecke J (1979) An explicit description of Kallenberg's lattice type point process. Math Nachr 89:185--95.

Mecke J, Nagel W, Weiss V (2007) Length distributions of edges in

planar stationary and isotropic STIT tessellations. J Contemp Math Anal 42:28--43.

Mecke J, Nagel W, Weiss V (2011) Some distributions for

I-segments of planar random homogeneous STIT tessellations. Math

Nachr 284:1483--95.

Miles RE, Mackisack M (2002) A large class of random tessellations with the classic Poisson polygon distributions. Forma 17:1--17.

Muche L (2005). The Poisson-Voronoi tessellation. Adv Appl Prob 37:279--96.

Neuh"{a}user D, Hirsch C, Gloaguen C, Schmidt V (2014) Ratio limits and simulation algorithms for the Palm version of stationary iterated tessellations. Journal of Statistical Computation and Simulation 84:1486--504.

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

Weiss V, Cowan R (2011). Topological relationships in spatial

tessellations. Adv Appl Prob 43:963--84.




How to Cite

Cowan, R., & Weiss, V. (2018). LINE SEGMENTS WHICH ARE UNIONS OF TESSELLATION EDGES. Image Analysis and Stereology, 37(1), 83–98.



Original Research Paper