Universal topological properties of shells in soap froth

An analysis of the shell distribution function of two-dimensional soap froth reveals a universal topological relation on the average number $M(j, n)$ of sides per cell to the number of cells $K(j, n)$ in the $j\mathrm{th}$ shell of a given center cell with $n$ sides. A plot of $M(j, n)K(j, n)$ vs $K(j, n)$ shows a slope of 5 for $j=1\mathrm{}$ and a slope of 6 for $j\ge 2$, for all samples. The results are universal for soap froths in the scaling state with different preparations, different times, and different temperatures. A theoretical justification is given based on general topological arguments, which are independently supported by experiments.