Geometry of self-affine tiles II

Abstract

We continue the study in part I of geometric properties of self-similar and self-affine tiles. We give some experimental results from implementing the algorithm in part I for computing the dimension of the boundary of a self-similar tile, and we describe some conjectures that result. We prove that the dimension of the boundary may assume values arbitrarily close to the dimension of the tile. We give a formula for the area of the convex hull of a planar self-affine tile. We prove that the extreme points of the convex hull form a set of dimension zero, and we describe a natural gauge function for this set.