We describe the global dynamics of some pointwise periodic piecewise linear maps in plane that exhibit interesting dynamic features. For each these we find a first integral. integrals set values are discrete, thus quantized. Furthermore, level sets bounded whose interior is formed by finite number open tiles certain regular or uniform tessellations. The action on invariant described geometrical...

We analyse in a common framework the properties of the Voronoi tessellations resulting from regular 2D and 3D crystals and those of tessellations generated by Poisson distributions of points, thus joining on symmetry breaking processes and the approach to uniform random distributions of seeds. We perturb crystalline structures in 2D and 3D with a spatial Gaussian noise whose adimensional streng...

We consider stationary Poisson-Voronoi tessellations (PVT) in the Euclidean plane and study properties of Voronoi tessellations induced by linear Poisson processes on the edges of the PVT. We are especially interested in simulation algorithms for the typical cell. Two different simulation algorithms are introduced. The first algorithm directly simulates the typical cell whereas the second algor...

This paper deals with the design of multi-(micro)computer interconnection networks modelled by graphs. In particular we concentrate upon the optimization of a new family of such networks that turn out to be a generalization of the well-known Arden and Lee's chordal ring networks. This optimization problem leads to the search for certain 3-regular graphs with minimumdiameter for a given order an...

We consider regular tessellations of the plane as infinite graphs in which q edges and q faces meet at each vertex, and in which p edges and p vertices surround each face. For 1/p + 1/q = 1/2, these are tilings of the Euclidean plane; for 1/p + 1/q < 1/2, they are tilings of the hyperbolic plane. We choose a vertex as the origin, and classify vertices into generations according to their distanc...

Shape Delaunay tessellations are a generalization of the classical Delaunay triangulation of a finite set of points in the plane, where the empty circle condition is replaced by emptiness of an arbitrary convex compact shape. We present some new and basic properties of shape Delaunay tessellations, concerning flipping, subgraph structures, and recognition.

By a regular tessellation, we mean any hyperbolic 3-manifold tessellated by ideal Platonic solids such that the symmetry group acts transitively on oriented flags. A regular tessellation has an invariant we call the cusp modulus. For small cusp modulus, we classify all regular tessellations. For large cusp modulus, we prove that a regular tessellations has to be infinite volume if its fundament...