Forcing nonperiodicity with a single tile

TL;DR

This paper discusses the concept of an 'einstein' tile, a single shape that can tile space nonperiodically, and reviews a recently introduced prototile that fits this definition, advancing understanding in aperiodic tilings.

Contribution

The paper clarifies features of a recently introduced single-tile prototile that is an aperiodic tile, supporting the existence of an 'einstein' shape.

Findings

Identification of a single prototile that enforces nonperiodic tiling

Clarification of features that qualify the prototile as an 'einstein'

Advancement in the theory of aperiodic tilings

Abstract

An aperiodic prototile is a shape for which infinitely many copies can be arranged to fill Euclidean space completely with no overlaps, but not in a periodic pattern. Tiling theorists refer to such a prototile as an "einstein" (a German pun on "one stone"). The possible existence of an einstein has been pondered ever since Berger's discovery of large set of prototiles that in combination can tile the plane only in a nonperiodic way. In this article we review and clarify some features of a prototile we recently introduced that is an einstein according to a reasonable definition. [This abstract does not appear in the published article.]