Disklikeness of planar self-affine tiles

By K. Leung, Ka-Sing Lau. 2007

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We consider the disklikeness of the planar self-affine tile T generated by an integral expanding matrix A and a consecutive collinear digit set V = {0,v,2v,- " ,(|q| - 1) v ) ⊂ Z 2 . Let f(x) = x 2 +px+q be the characteristic polynomial of A. We show that the tile T is disklike if and only if 2|p| ≤ |g+2|. Moreover, T is a hexagonal tile for all the cases except when p = 0, in which case T is a square tile. The proof depends on certain special devices to count the numbers of nodal points and neighbors of T and a criterion of Bandt and Wang (2001) on disklikeness.

Subfields: Electronic, Optical and Magnetic Materials, Materials Chemistry, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cb966fe5455ba1e73fdec5c8d0c4486066b697b0