An aperiodic set of 11 Wang tiles

TL;DR

This paper introduces a minimal aperiodic Wang tileset with 11 tiles and 4 colors, providing a definitive solution to Wang's 1961 problem about the smallest such set.

Contribution

It presents the first minimal aperiodic tileset with 11 tiles on 4 colors, proving no smaller set exists.

Findings

The tileset is proven to be aperiodic.

No aperiodic set with fewer than 11 tiles or 4 colors exists.

This work resolves a long-standing open problem.

Abstract

We present a new aperiodic tileset containing 11 Wang tiles on 4 colors, and we show that this tileset is minimal, in the sense that no Wang set with either fewer than 11 tiles or fewer than 4 colors is aperiodic. This gives a definitive answer to the problem raised by Wang in 1961.