# Voronoi manuscripts of Ken Brakke

These are five unpublished manuscripts I wrote in the mid 1980's when I was temporarily infatuated with generating random Voronoi tessellations on my new personal computer and the wonders of TeX. My attention was soon absorbed by my Surface Evolver project, but word of my Voronoi manuscripts got circulated somehow, and there have been requests for copies sporadically. Therefore I have scanned the manuscripts (the original TeX source being long lost), used OCR to convert them to LaTeX, and then to PDF. For the OCR, I used InftyReader, which does a very nice job of scanning math formulas and converting them to LaTeX. Some of the figures had to be regenerated, and I fixed a few misprints, but these are pretty close to the originals. Beware that the scanning process may have misread digits in the various numerical tables; I found many such and fixed them, but there may be others.
Statistics of Random Plane Voronoi Tessellations
Abstract: If S is a discrete set of points in a space, and each point of the space is associated with the nearest point of S, then the resulting partition is called a Voronoi tessellation. This paper derives a general scheme for setting up integrals for statistics for tessellations generated from a Poisson point process. For the case of the plane, the integrals are evaluated to find the variances of cell area, edge length, perimeter, and number of sides. The distributions of several parameters, including edge length, are also found.
200,000,000 Random Voronoi Polygons
Abstract: The results of computer simulation of random Voronoi tessellations of the plane are presented. Statistics tallied include frequencies, area, and perimeter of n-sided cells, and the frequencies of n- and m-sided cells abutting.
Statistics of Three Dimensional Random Voronoi Tessellations
Abstract: This paper derives some integral formulas for first and second order statistics of 3D Poisson Voronoi tessellations.
Statistics of Non-Poisson Point Processes in Several Dimensions
Abstract: The Poisson point process is the most commonly studied random point process, but there are others. It is not always justifiable to assume that a random point process will have the same statistics as a Poisson point process. This paper derives the relationship between Poisson processes and some other random processes for some types of geometric statisics.
Random Voronoi Tessellations in Arbitrary Dimension
Abstract: Voronoi tessellations generated by Poisson point processes in n-dimensional Euclidean space are studied. Formulas for the expected measure of the k-dimensional skeleton of the tessellation are developed, along with formulas for q-dimensional cross sections. As n goes to infinity with q fixed, there is a limiting tessellation process, which is intuitively a finite dimensional cross section of an infinite dimensional tessellation.
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