## Hexagonal Aperiodic Monotile

### Info

In 2009 Joan Taylor (Burnie, Tasmania) found a decoration of the hexagon, which - together with few local matching rules - allows only aperiodic tilings of the plane. This is certainly the best example of an aperiodic monotile we have today. This decorated hexagonal tile, together with the local matching rule, is shown in Fig 1 of a joint paper by J. Socolar and J. Taylor: http://arxiv.org/pdf/1003.4279v2. It contains also a non-decorated version of the prototile, but then the prototile is not longer a connected set. This paper contains also a substitution rule yielding the same tilings as the local matching rule. A slightly different version of this substitution rule is shown here. This is not longer a non-self-similar substitution. It uses 7 hexagonal prototiles (Taylor uses 14 half-hex tiles, making the substitution self-similar). The 7 prototiles are distinguished by colour here. If we generate a (large patch of) a tiling using the substitution, and then delete all colours, we get the same tilings as the ones generated by the local matching rule (in the stricter version of the manuscript cited below). The above image on the left shows the substitution rule for the 7 prototiles, and below the original decoration of Socolar and Taylor (right) and our humble version of it. A manuscript by Joan Taylor with hand-drawn images can be found at https://sfb701.math.uni-bielefeld.de/preprints/sfb10015.pdf

### Substitution Rule

### Patch

### References