Digit tiling of Euclidean space. (English) Zbl 0972.52012

Baake, Michael (ed.) et al., Directions in mathematical quasicrystals. Providence, RI: AMS, American Mathematical Society. CRM Monogr. Ser. 13, 329-370 (2000).
Summary: This is an expository paper on digit tiling of Euclidean space, a special kind of self-affine tiling by translates of a single tile. In particular, the following topics are discussed: the construction of digit tiles and the construction of the boundary, the Hausdorff dimension of the boundary, the relation between digit tiles and positional number systems, the self-replicating properties of digit tiling, and lattice and crystallographic digit tiling.
In the last sections digit tiling is placed into the broader context of both periodic and nonperiodic self-affine tiling of Euclidean space by a finite set of proto-tiles. In particular, the following topics are discussed: general results on hierarchical tiling, results specific to self-affine and self-similar tiling, the construction of self-affine and self-similar tilings using graph iterated function systems, and some illustrative examples.
For the entire collection see [Zbl 0955.00025].


52C22 Tilings in \(n\) dimensions (aspects of discrete geometry)
52C23 Quasicrystals and aperiodic tilings in discrete geometry
28A80 Fractals
37B10 Symbolic dynamics