NUMERATION SYSTEMS FOR APERIODIC STRUCTURES

A tiling is the covering of a 𝑑-dimensional space using one or more geometric shapes called prototiles, with no overlaps and no gaps. Periodic tilings of ℝ3 by crystal shapes are well understood since the end of the 19th century when the point group classification was provided by Schoenflies. Although aperiodic tiling have been studied since the 1960s, the discovery of solids with crystallographically forbidden symmetries strongly motivates mathematicians to design aperiodic structures having the required properties, such as certain types of symmetry, repetition of arbitrarily large motifs and possibility to generate the tiling by a geometric substitution, i.e., an inflation rule that defines a self-similar tiling. The questions on structural features of tiling eventually lead to problems of numeration systems and combinatorics on words.