### 2015-07

by Burtseva, L.; Werner, F. .

**Series:** 2015-07, Preprints

- MSC:
- 05B45 Tessellation and tiling problems
- 05B40 Packing and covering
- 52B05 Combinatorial properties (number of faces, shortest paths, etc.)

**Abstract:**

In recent years, the literature shows an increasing interest to tessellation methods based on Voronoi diagrams to model different structures as packing of spheres. Voronoi diagrams have found numerous practical and theoretical applications in a large number of fields in science and technology as well as in computer graphics. A useful property of Voronoi diagrams is that they represent cellular structures found in the nature and technology in a natural manner, easily to understand and to design. Although this approach is really not new, meanwhile its intensive use and, consecutively, a systematical study started around 2000 with advances in nanoscience and nanotechnology. In this chapter, two basic tessellation methods are considered in more detail: the Voronoi-Delaunay tessellation and the Voronoi diagram in Laguerre geometry, as well as some of their generalizations. The principal concepts of both tessellation methods are briefly explained for a better understanding of this approach. A review of the related literature is given, focusing mainly on new mathematical tools and several particularities of the applications considered.

**Keywords:**

tessellation, Voronoi diagram, Delaunay simplex, Laguerre geometry, sphere packing, structure modeling