Disklikeness of planar self-affine tiles

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) ⊂ ℤ². Let f(x) = x² +px+q be the characteristic polynomial of A. We show that the tile T is disklike if and only if 2|p| ≤ |q+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. Copyright © 2007 American Mathematical Society.