Random Subdivisions of Space into Crystals

Abstract

This papers considers random subdivisions of a D-dimensional Euclidean space into disjoint regions. The regions will be called crystals because they represent individual crystal grains in a metal or mineral (D = 3) in one application. However, other interpretations are possible. The "crystals" may also be cells in living tissue, bubbles in a froth, and (when D is large) detection regions of a code. Section 3 describes random processes which have been used to subdivide space into crystals. Given such a process statistical properties of the crystals present interesting problems in geometric probability. For crystals in two different mineral models Meijering derived the mean surface area, mean number of faces, mean total edge length and many other related mean values. Continuing with the same two models, the present paper concerns itself with the distribution of crystal volumes. Section 4 finds the variance of the volume of a crystal and also finds some variances associated with plane or line sections through crystals. Curiously these two models have similar values for many statistical parameters but have different volume variances. Section 5 gives bounds on the distribution functions for crystal volumes. Section 2 and 3 review related probabilistic results from the literature on mineralogy and metallurgy.