Escheresque parquet deformations of an aperiodic monotile

On March 20, it was announced that a team of mathematicians has found a 13-sided polygon that can tile the plane aperiodically (and that cannot tile the plane periodically), thus solving the so-called “einstein problem.” Craig Kaplan’s website has lots of details on the new shape — which so far has generally been dubbed the “hat,” even though to me it looks a lot more like a T-shirt.

One of the cool details about the “hat” is that it’s just one of a whole continuous family of shapes, each of which is still an aperiodic monotile. Here are seven representatives of that set: the first, fourth, and seventh are special cases that can tile either periodically or aperiodically, while the other four are representatives of the infinite family that tile only aperiodically. The fifth is the “hat”; the first, third, and seventh are known as the “comet,” “turtle,” and “chevron” respectively.

A family of monotiles

This got me thinking about M.C. Escher’s Metamorphosis series. Wouldn’t it be interesting to make a tessellation that morphs smoothly from one member of this family to another? (In fact, einstein coauthor Craig Kaplan, mentioned above, also happens to be big into parquet deformations!)

Detail of "Metamorphosis II"

So I wrote a little JavaScript thingie you can run in your browser to create your own Escher-like parquet deformations. Play with it here, to produce on-screen images like this one:

Escheresque parquet deformation

What would Escher do with this pattern?

Posted 2023-03-30