An aperiodic tiling made of one tile, a triangle

TL;DR

This paper introduces a novel aperiodic tiling using a single isosceles right triangle with substitution rules, demonstrating its aperiodicity and underlying dodecagonal structure, advancing understanding of minimal tile sets.

Contribution

It presents the first known aperiodic tiling made of one tile with substitution rules, addressing the open Stein problem.

Findings

The tiling is proven to be aperiodic.

The tiling exhibits a dodecagonal structure.

The tile set consists of a single isosceles right triangle.

Abstract

How many different tiles are needed at the minimum to create aperiodicity? Several tilings made of two tiles were discovered, the first one being by Penrose in the seventies. Since then, scientists discovered other aperiodic tilings made of two tiles, including the square-triangle one, a tiling that has been particularly useful for the study of dodecagonal quasicrystals and soft matters. An open problem still exists: Can one tile be sufficient to create aperiodicity? This is known as the ein stein problem. We present in this paper an aperiodic tiling made of one single tile: an isosceles right triangle. The tile itself is not aperiodic and therefore not a solution to the ein stein problem but we present a set of substitution rules on the same tile that forces the tiling to be aperiodic. This paper presents its construction rules that proves its aperiodicity. We also show that this tiling offers an underlying dodecagonal structure close to the one of square-triangle tiling.