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

Richard Cowan; Christoph Thäle

doi:10.5566/ias.v33.p39-54

Authors

Richard Cowan School of Mathematics and Statistics University of Sydney
Christoph Thäle Faculty of Mathematics Ruhr-University Bochum

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.

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