Monotile

Mathematicians have finally discovered an aperiodic monotile, that is, a polygon that can only tile the plane in aperiodic tessellations.

In the following section we will describe families of tilings by a curious three-dimensional aperiodic monotile whose matching rule — for some values of its parameter — is weak but not strong.

And I am sorry I never delivered on the four-dimensional Poultry design. It turns out John Horton Conway found four-dimensional and five-dimensional aperiodic monotiles sixty years ago, but it's not too well documented.

This naturally led to serious rumination about the possible existence of a single tile, or monotile, that forces non-periodic global structure.

An aperiodic monotile never repeats a formation, no matter how long the pattern.