Monday Morning Math: Einstein found

Good morning!  There’s been some exciting math in the news lately about an einstein.  Not a  person, but a shape. A tile.

First, some background.  A tessellation is a way of covering a floor (or a whole plane) with tiles so there are no gaps or overlaps.  By putting a few or a lot of restrictions on the tiles, like that the tiles have to match edge-to-edge and that you can only use one shape, you can get some interesting results.

For example, if you only want to use one shape but you want that shape to be regular (all the sides the same and all the angles the same), you might end up with something like this:

(CC by Jeff Weeks)

If you don’t require that the shape be regular, there are more options.  Here’s a tiling made up of rhombuses (where the sides are all the same, but not the angles).  This is actually a Roman mosaic so it’s a tile made up of tiles, but for the purposes of this we’re just looking at the diamond shapes, which look like they make up cubes.

(“An ancient roman geometric mosaic from the “Palazzo Massimo alle Terme” National Roman Museum of Rome, Italy”  CC by Mbellaccini)

Then there are aperiodic tessellations, which avoid repeating patterns.  Until recently people didn’t know if it was possible to have a single tile that would form an aperiodic tiling. Such a potential shape was called by the German name ein Stein, which mean “one stone” (like one tile). 

It turns out that there IS a tile that does this! This is pretty recent, so it hasn’t been peer reviewed yet, but last year David Smith found a tile that seemed to work, and with a little help from his friends wrote a paper proving it (twice, in two different ways).  

(The yellow tiles are reflections of the blue tiles.  CC by Ginger)

Cool! Even cooler, it’s part of a whole family he discovered, indicating that there are many ways to be an einstein.

Thanks to Arianna who mentioned this last week, and Joe who sent me a New York Times article about it.
Sources:

• Wikipedia
• “Elusive ‘Einstein’ Solves a Longstanding Math Problem” by Siobhan Roberts, New York Times, March 28, 2023.