Hexagonal Aperiodic Monotile
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.