Abstract
We study self-affine tiles which tile the n-dimensional real vector space with respect to a crystallographic group. First we define classes of graphs that allow to determine the neighbors of a given tile algorithmically. In the case of plane tiles these graphs are used to derive a criterion for such tiles to be homeomorphic to a disk. As particular application, we will solve a problem of Gelbrich, who conjectured that certain examples of tiles which tile \({\mathbb{R}}^2\) with respect to the ornament group p2 are homeomorphic to a disk.
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Loridant, B., Luo, J. & Thuswaldner, J.M. Topology of crystallographic tiles. Geom Dedicata 128, 113–144 (2007). https://doi.org/10.1007/s10711-007-9186-0
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DOI: https://doi.org/10.1007/s10711-007-9186-0