The division of space and the Poisson distribution

Abstract

This letter reports on a fascinating new result concerning the Poisson distribution. The result arose during a study of cellular division within epithelia (fundamental 'sheet-like' tissues of the body) and has general relevance to the partitioning to two-dimensional space whenever the compartments of space are successively cleaved or fractured. A more complete account of the biological context from which the mathematical problem is abstracted is given in Cowan and Morris (1988) but, in this note, we focus on part of that work which (in a surprising way) involves the Poisson distribution. Consider a convex polygon with k sides. We propose to divide it with a straight line joining two of the k sides. Suppose that the choice of sides is stochastic with the simplest probability measure, namely, each of the kC2 choices being equally likely. Having chosen which two sides are to be joined, it is unimportant where on these sides the line's endpoints are positioned, except that neither should be placed at a vertex of the polygon. After the dividing line is placed we have two convex polygons, the 'daughters' of the original 'mother cell'. Next we independently apply the same stochastic rules to each daughter polygon, simultaneously dividing them both and so creating four convex polygons. Division proceeds in this way. Let X,, be the number of sides of a randomly chosen polygon from generation n (the initial k-sided polygon is in generation 0). As n tends to infinity, it turns out that X,, 3 becomes Poisson distributed with mean 1 for any k. To show this, let Y,(r) be the number of polygons with r sides in generation n. Given Y,(r), the chance that a randomly sampled member of generation n has r sides is Y,(r)/2. Unconditionally, this chance is EYn(r)/2. Let m, be a row vector whose elements are EY,(r), r = 3, 4, . This initial vector mo contains mainly zeros but with 1 in the appropriate position to indicate the starting polygon. It is easy to show that