How do Shapes Fill Space?

Shapes fill space all around us, from bathroom tilings to brick walls. The puzzling problem of how to fit shapes together so that they fill space to form what we call tilings has been considered throughout much of the history of humanity. The problem probably emerged first in the arts, where tiles were used to produce interesting patterns, like those of Islamic art. It has also been studied as part of mathematics since the ancient Greeks. For example, our understanding of symmetries and their mathematical description in terms of group theory originated from the investigation of tilings, and underlies the classification of crystal structures.The modern era for tilings began in the 1960s when Berger proved that the problem of whether a given set of shapes could tile the plane was undecidable, a result extended to the hyperbolic plane by Margenstern in 2007. This led directly to the discovery of new worlds of tiling theory and fascinating examples such as the Penrose tiling. The discovery of quasicrystals (crystals with 'forbidden' symmetry) gave additional impetus as the Penrose and related tilings provide models for non-periodic ordered structures. From a scattering of strange examples, our understanding is now evolving to coalesce into a coherent theory. In particular, the theory of substitution rules is giving a natural setting for the Penrose tiling.Tilings therefore offer a combination of deep mathematics with beautiful imagery, which makes them an ideal topic for public engagement activities. The visual appeal, the link to arts and architecture and the interactive character of related puzzle-type activities, as well as the link to current research on mysterious materials such as quasicrystals, fascinates audiences across all age groups. Because tilings are familiar objects, this topic avoids the barrier often caused by the mathematical language of symbols and equations, and enables us to communicate non-trivial mathematical concepts to a public audience.This project will create material for an exhibit at the Royal Society Summer Exhibition 2009, which is expected to attract in excess of 5000 visitors. After the exhibition in June/July 2009, the materials are adapted for continued use in UK-wide mathematics masterclasses (1 to 2.5 hour interactive sessions for 10-18 year olds) supported by the Royal Institution (Ri) and for use in Family Fun Days hosted at the Ri.