In this first of a two-part series, we present the smallest known sets of aperiodic Wang tiles and review their properties. More specifically, we will present the following sets:
The set of 16 Wang tiles proposed by Ammann in the 1980s.
The sets of 14 and 13 Wang tiles proposed by Kari and Culik in the 1990s.
The smallest aperiodic Wang tile set discovered by Jeandel and Rao in 2015.
These three sets have different properties. In particular:
- Ammann's tiles are self-similar, but the others are not.
- The labels of the Kari and Culik tiles satisfy equations, but the others do not.
- The set of tilings with Kari and Culik tiles is strictly positive entropy, but the others are not.
- The Jeandel-Rao tessellations are explained by a polygonal partition, but not the others.
These very different properties make it difficult to create a unified theory to explain all aperiodic tile sets.