# Grain Packing

### Derivation of packing fraction for jammed distribution of ellipses

This project took the first steps in an effort to theoretically derive the packing fraction for random packings of identical ellipses. We separate the problem into two parts, one depending on the packing procedure and one that does not, and we outline a method for calculating the relevant probability density function for the latter. The method is based on analysis of the graph formed by connecting contact points between ellipses, as shown in the figure opposite. This graph consists of polygons that tessellate the plane fully. Some polygons, labelled "g" in the figure cover a grain, others, labelled "c" cover an inter-grain space. By deriving the distribution of areas of these polygons under certain assumptions, we will be able to compute the probability density of the packing fraction.

In this work, we derived: (i) the probability density function for the length of a chord connecting two randomly distributed neighbouring contact points on the ellipse boundary and (ii) the probability density function for the area of a triangle circumscribed by an ellipse of given aspect ration. We then outline the procedure for derivation of the area distributions of polygons of more edges similarly and relate these to the calculation of the packing fraction.