The Renewal Equation

Today I watched three videos about tilings, specifically non-periodic tilings. These are arrangements of tiles that cover an infinite plane in a pattern that never repeats.

Two of the videos were quite accessible. The first was from the popular science channel Veritasium¹ and covered the state of this subject three years ago:

The second was from a smaller channel called Up and Atom and explained recent developments that have caused excitement among math enthusiasts.

The third video was more technical and probably of less interest to nonspecialist readers. It included a little more detail about why some collections of shapes form non-periodic tilings.

I’m still very tired from yesterday and don’t have much bright to say about all this. I feel like there is something to understand, though, about the relationship between rotational symmetry and periodicity of tilings. That is, between the ability of a tile pattern to match its original configuration when rotated versus when translated.

1. Note that Veritasium has sometimes shaped its educational content to reflect well on its sponsors. ↩︎