Applied and Computational Mathematics
|Topic:||The Combinatorics of Periodic Tilings|
|Date:||Thursday, February 11, 1999|
|Place:||Gould-Simpson, Room 701|
Interest in periodic tilings and patterns goes back a long time and one of the oldest known mathematical results is the enumeration of the five Platonic solids, which is over 2000 years old.
Many results have been obtained since then. But only recently, over the past 15 years, has an adequate combinatorial approach to the study of periodic tilings been developed. The fundamental observation is that periodic tilings can be described by finite, connected, edge-colored graphs called Delaney symbols. Indeed, two tilings are equivalent, if and only if they have the same symbol (and this is true for any simply-connected space of any dimension).
Using this approach, the problem of systematically enumerating periodic tilings of the plane, sphere or hyperbolic plane has been solved by the development (and implementation) of appropriate algorithms. Recently, a number of results concerning three-dimensional tilings have been obtained.
This talk gives an introduction to combinatorial tiling theory and includes a demonstration of software based on this approach.