Fixed point theorem and aperiodic tilings

TL;DR

This paper introduces a simple, fixed point-based construction of an aperiodic tile set, highlighting its historical significance and suggesting it should have been discovered in the 1960s, predating many known constructions.

Contribution

The paper presents a novel, straightforward fixed point construction of an aperiodic tile set, emphasizing its historical importance and correcting the overlooked discovery timeline.

Findings

Construction based on fixed point argument is simpler and more natural.

The approach predates and could have influenced earlier aperiodic tiling discoveries.

Historical analysis suggests the method was known but overlooked for decades.

Abstract

We propose a new simple construction of an aperiodic tile set based on self-referential (fixed point) argument. People often say about some discovery that it appeared "ahead of time", meaning that it could be fully understood only in the context of ideas developed later. For the topic of this note, the construction of an aperiodic tile set based on the fixed-point (self-referential) approach, the situation is exactly the opposite. It should have been found in 1960s when the question about aperiodic tile sets was first asked: all the tools were quite standard and widely used at that time. However, the history had chosen a different path and many nice geometric ad hoc constructions were developed instead (by Berger, Robinson, Penrose, Ammann and many others. In this note we try to correct this error and present a construction that should have been discovered first but seemed to be unnoticed for more that forty years.