Crystallographic groups and topology in Escher. (Spanish:Grupos cristalográficos y topología en Escher).

Abstract

In the late 19th century Fedorov, Schoenflies, and Barlow classified the seventeen wallpaper groups (two-dimensional crystallographic groups, five of them direct movements and twelve of them inverse movements) and the 320 three-dimensional crystallographic groups. In order to get the lists of groups, they all used the same geometric strategy: to combine all possible movements and study them case by case. Later on, Zassenhaus developed a purely algebraic algorithm which allowed him to use the computer in order to get the groups, but all geometric information was lost in that process. More recently, Bonahon and Siebenmann introduced a topological method, based on ideas introduced by Thurston around 1970, to get a complete list of the three-dimensional groups. In this article, the author explains how to use this last method in order to topologically obtain the five direct two-dimensional crystallographic groups and indicates how to obtain the twelve inverse ones. All groups are illustrated with designs of Dutch artist M. C. Escher.