The law of Aboav--Weaire and extensions

In two-dimensional grain structure, one observes that grains with a small number of sides tend to be surrounded by grains with a large number of sides, and vice-versa. The Law of Aboav--Weaire gives this observation a mathematical formulation, that is, the average number of sides of the neighbors of an $n$-sided grain should be roughly $5 + 6/n$. By introducing the correct error term we prove this law of Material Science. We extend this law to three-dimensional grain structures. We also discuss the average number of sides of grains neighboring to two adjacent grains and three adjacent grains. This is joint work with Menachem Lazar and Jeremy Mason.