Random division of spaces

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1. We study Poisson-Voronoi tessellations of 3-dimensional hyperbolic spaces, and find explicit formulas that give mean number of vertices, mean total length of edges, and mean surface area of their cells. These mean characteristics comprise, as a particular case, the corresponding formulas for the classical Euclidean case, and depend only on the ratio of curvature of hyperbolic space to intensity of Poisson process. Relying on this result, we develop a method of estimating curvatures of hyperbolic spaces from data on Poisson-Voronoi tessellations. ([1], [2])2. In the 3-dimensional Euclidean spaces, we investigate a problem of random sequential packing of rectangular rods. Assuming that these rods are placed parallel to any of three axes of Cartesian coordinates system. We find a method of reducing the problem to that of 6-dimensional Markov chain. A large simulation using this reduction reveals that configurations of rods are isotopic and their packing density equals 3/4. ([3])3. In the one-dimensional Euclidean spaces, we study a problem of random sequential packing of internals that are generated to a self-similar probability distribution P. Then the resulting probability distribution of packed intervals Q is proved to be self-similar but different from P. Moreover, when P is in particular a uniform distribution, we determine the Hausdorff dimension of the set that are not cover by packed intervals. ([4])4. We study the classical 13 spheres problem, and succeed in obtaining detailed information on the configuration of these spheres. We consider the graph of Delaunay tessellation that are determined by centers of spheres, and prove that only two graphs are possible, that is, the dodecahedron graph and the graph of rhombic dodecahedron. Furthermore we study a continuous deformation of among these graphs. ([5])