The article deals with regular, semi-regular and irregular plane partitions, among the latter are partitions into equal polygons arbitrary simple knots. It continues author's supplementary lectures to university textbook «Crystallography» by G. M. Popov I. Shafranovsky, which was recommended students of geology St. Petersburg Mining University. new topic is interesting because first problem sol...

Centroidal Voronoi tessellations (CVTs) are Voronoi tessellations of a region such that the generating points of the tessellations are also the centroids of the corresponding Voronoi regions. Such tessellations are of use in very diverse applications, including data compression, clustering analysis, cell biology, territorial behavior of animals, and optimal allocation of resources. In this pape...

We make use of the recent proof that the critical probability for percolation on random Voronoi tessellations is 1/2 to prove the corresponding result for random Johnson–Mehl tessellations, as well as for twodimensional slices of higher-dimensional Voronoi tessellations. Surprisingly, the proof is a little simpler for these more complicated models.

In the authors’ recent paper (de Lacy Costello et al., 2010) the authors described the formation of complex tessellations of the plane arising from the various reactions of metal salts with potassium ferricyanide and ferrocyanide loaded gels. In addition to producing colourful tessellations these reactions are naturally computing generalised Voronoi diagrams of the plane. The reactions reported...

In our recent paper [de Lacy Costello et al. 2010] we described the formation of complex tessellations of the plane arising from the various reactions of metal salts with potassium ferricyanide and ferrocyanide loaded gels. In addition to producing colourful tessellations these reactions are naturally computing generalised Voronoi diagrams of the plane. The reactions reported previously were ca...

The hyperbolic space affords an infinite number of regular tessellations, as opposed to the Euclidean space. Thus, significantly extends design lattices, potentially providing access unexplored wave phenomena. Here we investigate dynamic behavior tessellations governed by interactions whose strengths depend upon distances between neighboring nodes. We find eigen-modes that are primarily localiz...

Although the map algebra framework is very popular within the GIS community for modelling fields, the fact that it is solely based on raster structures has been severely criticised. Instead of representing fields with a regular tessellation, we propose in this paper using the Voronoi diagram (VD), and argue that it has many advantages over other tessellations. We also present a variant of map a...

We introduce a class of random fields that can be understood as discrete versions of multicolour polygonal fields built on regular linear tessellations. We focus first on a subclass of consistent polygonal fields, for which we show Markovianity and solvability by means of a dynamic representation. This representation is used to design new sampling techniques for Gibbsian modifications of such f...

A centroidal Voronoi tessellation is a Voronoi tessellation whose generating points are the centroids (centers of mass) of the corresponding Voronoi regions. We give some applications of such tessellations to problems in image compression, quadrature, finite difference methods, distribution of resources, cellular biology, statistics, and the territorial behavior of animals. We discuss methods f...