Project III 2018-19

FRIEZE PATTERNS, OLD AND NEW

Classically, a frieze pattern is a line of pictures or motifs, drawn from a finite set, arranged in a line going off to infinity in both directions. Importantly, they are expected to have some form of symmetry, at the very least a periodic repetition (translational symmetry) as you move along the line, but there may be additional symmetry as well. The possible symmetries have long been classified – all such patterns fall into one of 8 possible symmetry structures.

The project has three stages. The first is to look at this classical structure and review the classification theorem indicated. The second stage is to look at a more recent set of geometric objects – patterns arranged along a line which are highly structured but have no (translational) symmetry. These aperiodically ordered patterns were first identified in the 1960`s (though with a history that goes back long before), and have become more significant in the last few decades as they are key parts of models for so-called quasicrystals, materials only recently discovered which contradict the traditional framework of crystallography. The third stage is to bring these two parts together – to begin to explore aperiodic frieze patterns.

PREREQUISITES

The second year work on Group Theory (Algebra II, MATH2581) would be useful. The second year work on Geometric Topology could also prove useful, but probably less necessary: however, the project might nevertheless have more of the flavor of that subject as it is quite topological. The modules Topology III MATH3281 and Geometry III MATH3201 may be generally interesting to take alongside this project.

RESOURCES

Good places to start with are the texts

Branko Grunbaum and G. C. Shephard, Tilings and Patterns: An Introduction, W.H.Freeman & Co (1989)

John Conway, Heidi Burgiel and Chaim Goodman-Strauss, The Symmetries of Things, CRC Press 2008

L. Sadun, Topology of Tiling Spaces, American Maths Society, University Lecture Series 46 (2008)

EMAIL

John Hunton (john.hunton@durham.ac.uk)