Self-similarity viewed as a local property via tile sets

Abstract

A self-similar image is defined by its global invariance through a finite number of contractive transformations. We show that, in the plane, a self-similar figure can also be constructed according to local rules: given a finite number of similarities (also called an Iterated Function System, IFS), we construct a tile set with the following properties:

• any nth iterate of the IFS can be represented by a finite tiling of the plane,

• any finite non-trivial tiling of the plane is formed only by iterates of the IFS.

Furthermore, we construct another tile set such that all “correctly initialized” tilings of the whole plane represent the IFS attractor.