An aperiodic tiling of variable geometry made of two tiles, a triangle and a rhombus of any angle

TL;DR

This paper introduces a new class of infinitely many aperiodic tilings made of a triangle and a rhombus with any angle, enabling continuous transformation and linking periodic and aperiodic patterns.

Contribution

It presents a novel infinite family of variable geometry aperiodic tilings based on two tiles, expanding the understanding of tiling structures and their relation to quasicrystals.

Findings

Discovered an infinite set of aperiodic tilings with variable geometry.

Proposed a continuous transformation between periodic and aperiodic tilings.

Identified the underlying dodecagonal structure of the tilings.

Abstract

Aperiodic tiling is a well-know area of research. First developed by mathematicians for the mathematical challenge they represent and the beauty of their resulting patterns, they became a growing field of interest when their practical use started to emerge. This was mainly in the eighties when a link was established with quasi-periodic materials. Several aperiodic tilings made of two tiles were discovered, the first one being by Penrose in the seventies. Since then, scientists discovered other aperiodic tilings including the square-triangle one, a tiling that has been particularly useful for the study of dodecagonal quasicrystals and soft matters. Based on this previous work, we discovered an infinite number of aperiodic tilings made of two tiles, a triangle and a rhombus of any angle. As a result, a variable geometry, i.e. continuously transformable, aperiodic tiling is proposed, whose underlying structure is dodecagonal. We discuss this limit case where the rhombus is so thin that it becomes invisible. At the boundary of this infinite space of tilings are two periodic ones; this represents a uniform view of periodic and aperiodic tilings.