Topology of a class of p2-crystallographicreplication tiles

Abstract

Abstract. We study the topological properties of a class of planar crystallographic replication tiles. Let M ∈ Z 2×2 be an expanding matrix with characteristic polynomial x 2 + Ax + B (A, B ∈ Z, B ≥ 2) and v ∈ Z 2 such that (v, M v) are linearly independent. Then the equationdefines a unique nonempty compact set T satisfying T o = T . Moreover, T tiles the plane by the crystallographic group p2 generated by the π-rotation and the translations by integer vectors. It was proved by Leung and Lau in the context of self-affine lattice tiles with collinear digit set that T ∪(−T ) is homeomorphic to a closed disk if and only if 2|A| < B + 3. However, this characterization does not hold anymore for T itself. In this paper, we completely characterize the tiles T of this class that are homeomorphic to a closed disk.

Elementary Fractal Geometry. 2. Carpets Involving Irrational Rotations

“…We shall now determine the combinatorial structure of our example IFS, the so-called neighbor graph. For the case of pieces of equal size, this object has been defined in various papers, including [37][38][39][40][41][42][43][44][45][46][47]. We introduce the concept briefly and refer to the literature for details.…”

“…This makes sense even without self-similarity. For self-similar crystallographic tilings, see [33,45,46]. Then there are aperiodic tilings as in Figure 2 which have more proper neighbors, but not all possible neighbors apply to each tile.…”

“…A large part of the literature on fractal tilings [25,26,41,42,46,[51][52][53] is concerned with self-affine tiles where neighbor maps are translations and symmetries play no part. However, there are also crystallographic fractal tiles where we have a crystallographic group which acts on the tiling [35,44,54]. Only very few fractal tiles have infinite symmetry groups [5,[28][29][30].…”

Elementary fractal geometry. Networks and carpets involving irrational rotations

“…We shall now determine the combinatorial structure of our example IFS, the so-called neighbor graph. For the case of pieces of equal size, this object has been defined in various papers, including [1,4,11,14,17,18,27,35,43,44,54]. We introduce the concept briefly and refer to the literature for details.…”

“…A large part of the literature on fractal tilings [5,17,18,29,30,33,44,53] is concerned with self-affine tiles where neighbor maps are translations and symmetries play no part. However, there are also crystallographic fractal tiles where we have a crystallographic group which acts on the tiling [24,34,35]. Only very few fractal tiles have infinite symmetry groups [42,15,22,10].…”