Rutgers Discrete Mathematics Seminar

Title: The Law of Aboav--Weaire and its Analogue in Three Dimensions

Speaker: Richard Ehrenborg, Princeton and University of Kentucky

Date: Monday, April 13, 2015 11:00 am

Location: CoRE Bldg, Room 431, Rutgers University, Busch Campus, Piscataway, NJ


When investigating the structure of metals it is known that the atoms lie in a lattice structure. However, the lattice property only holds locally, that is, in a three dimensional cell called a grain. Bordering the grain is a boundary where the atoms lie chaotically, and beyond that is a new grain where the lattice has a different orientation. The structure of these grains amounts to a three dimensional simple subdivision of space.

Looking at the two dimensional analogue, 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 states that 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 and discuss its extension to three dimensions.

This is joint work with Menachem Lazar and Jeremy Mason. Moreover, selected work of von Neumann, MacPherson and Srolovitz will be presented.