Some exciting geometry news today.
Smith, Myers, Kaplan, and Goodman-Strauss just posted a new preprint on a long unsolved geometric problem. They discovered a single connected shape which copies of (including reflections) can be used to tile the plane aperiodically.
Read more about it here https://cs.uwaterloo.ca/~csk/hat/

and here’s a triangle mapping creating a chunk of the tiling in Grasshopper Aperiodic_Monotile.gh (9.5 KB)

How long ago was it since the last new tessellating polygon was discovered? I’m probably getting the terminology all wrong, but I remember it being in the news because was a rare discovery.

Indeed. Ever since Penrose found his pair, people have been looking for ways to do it with just one.
There was one ‘tile’ composed of disconnected pieces found (Socolar–Taylor tile - Wikipedia), but this is the first with just one proper continuous single tile.
The shape itself is remarkably simple, just 4 thirds of a regular hexagon, though the way it fits together over larger scales is quite complex.

This really only works on a non periodic developable surface. A defining feature of aperiodic tilings is that they can’t be translated onto a copy of themselves.

Here’s a definition with a much larger patch, showing how the mapping can be used to apply other geometry onto the tile. Aperiodic_Monotile2.gh (567.3 KB)

These patterns can help reveal some of the complexity of the arrangement.
I find it amazing how much structure can come from such a simple shape.

Yes, this includes flips (interestingly the ratio between unflipped and flipped tiles isn’t even, there are almost 7 times more of one than the other).
The question of existence of an aperiodic tiling with a single tile only allowing translation and rotation remains open.

The sequence here is that several people in the early seventies managed to create aperiodic tilings, with relatively large numbers of rather abstruse tiles. In 1975, Penrose came up with his kites and darts tiling, with only two simple tiles. My father and I visited him that year, and were given a photocopy of the pattern. I wrote a computer program to draw the pattern on a Calcomp pen-plotter. My father, being a crystallographer, reduced the pattern to 35mm slide size and did a light diffraction pattern. This got published in about 1981. Dan Schectman discovered actual quasi-crystals in 1984, for which he received the Nobel prize for Chemistry in 2011. My father was not a chemist!

There was another thread earlier on using Grasshopper for this: Penrose Tiling, which has some of the pictures.

Escher died in 1972. He had been in correspondence with Penrose, and had attended crystallographic meetings with my father, but never lived to see aperiodic tilings. Sad! He would have loved this stuff!

The question as to whether these can be made from carbon remains for the future! Certainly buckminsterfullerenes and carbon nanotubules exist, even in nature, but creating more complex minimal surfaces at the atomic level remains to be seen.

Thank you so much for your grasshopper model above. I am going to have to play around with this.

I see you have four different decorations. The fourth is of course the nicest, in showing a beautiful over-and-under weave, which is what I am into. The third one is more useful for investigating the path taken by an individual thread. One of the nice things about the Penrose weave pattern is that the threads continue all the way across the pattern, with parallel threads in five different directions. That is definitely not the case here! Taking your definition, joining curves and selecting just one, shows that the threads that are not vestigial ones near the edge are often large rambling blobs, forming a closed loop. Not great for crafting purposes!