Fixed Point and Aperiodic Tilings

TL;DR

This paper introduces a new flexible method for constructing aperiodic tile sets using fixed-point techniques, resulting in robust tilings resistant to sparse errors, with implications across logic, physics, and computation.

Contribution

It presents a novel fixed-point based construction of aperiodic tile sets that are robust against sparse errors, expanding the understanding of aperiodic tilings.

Findings

Constructed a new aperiodic tile set using fixed-point methods.

The tile set remains aperiodic even with sparse tiling errors.

The construction bridges logic, physics, and computation domains.

Abstract

An aperiodic tile set was first constructed by R.Berger while proving the undecidability of the domino problem. It turned out that aperiodic tile sets appear in many topics ranging from logic (the Entscheidungsproblem) to physics (quasicrystals) We present a new construction of an aperiodic tile set that is based on Kleene's fixed-point construction instead of geometric arguments. This construction is similar to J. von Neumann self-reproducing automata; similar ideas were also used by P. Gacs in the context of error-correcting computations. The flexibility of this construction allows us to construct a "robust" aperiodic tile set that does not have periodic (or close to periodic) tilings even if we allow some (sparse enough) tiling errors. This property was not known for any of the existing aperiodic tile sets.