Self-Similar Tilings of Fractal Blow-Ups

TL;DR

This paper introduces a new method for creating infinite families of self-similar, non-periodic tilings from fractal blow-ups of certain iterated function systems, with properties like finiteness of prototiles and quasiperiodicity.

Contribution

It develops a novel construction of tilings associated with fractal attractors that are self-similar, repetitive, and often non-periodic, expanding the understanding of fractal tilings.

Findings

Finite prototile set for each tiling family

Tilings are repetitive and quasiperiodic

When the IFS is rigid, tilings have no non-trivial symmetry

Abstract

New tilings of certain subsets of $\mathbb{R}^{M}$ are studied, tilings associated with fractal blow-ups of certain similitude iterated function systems (IFS). For each such IFS with attractor satisfying the open set condition, our construction produces a usually infinite family of tilings that satisfy the following properties: (1) the prototile set is finite; (2) the tilings are repetitive (quasiperiodic); (3) each family contains self-similartilings, usually infinitely many; and (4) when the IFS is rigid in an appropriate sense, the tiling has no non-trivial symmetry; in particular the tiling is non-periodic.