"Einstein Tile" mystery solved: amateur mathematician discovers unique 13-sided "hat"

Originally published at: "Einstein Tile" mystery solved: amateur mathematician discovers unique 13-sided "hat" | Boing Boing


I need more Smithsonian in my life. Also, that video showing the morphing of the aperiodic tiling is pleasing :slight_smile:


I’m redoing a bathroom. Where can I buy some of this tile?


Worth pointing out that this shape is not as magical as the article makes out. You need the mirrorred version of the shape as well, to make the tiling work. That’s two shapes in my book!


Do you have an infinite bathroom? :smile:


Nah I’m assuming the contractor cut the tiles to fit.

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What am I missing here? This is just a form of tesselation (or semi regular tiling as @Crispy75 points out). I mean, a square fits the definition above. I even seem to remember a website where you could move lines around to create unique tesselations. Is it just that it specifically has 13 sides?

ETA: Yep, here’s one. I seem to remember one that would automatically adjust other lines to ensure it remained a tesselation. EETA: Oh wait, it does do that.

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Well, it’s also one of the reasons that this is interesting. If you only used one non-mirrored tile, it would have to be one of the ones on the standard list.


The einstein part is a joke: “one stone”, otherwise unrelated to the physicist. The other aperiodic tesselations need at least two tiles/stones. Periodic tesselations with one stone have been known for thousands of years.


Ok, but how is that different from an Escher drawing, for instance? He often used patterns that would reverse or rotate and still tesselate properly.

If it’s a question of it being a true polygon, it would be trivial to translate those arcs into lines and still maintain the tesselation.

Also: YIL (I first read this elsewhere yesterday), that Einstein means “one stone”.


These are all periodic. That is, you could grab a white and blue tile in those images, copy/paste them over and over without rotating and fill out your whole piece of paper no matter how big. (“tessellates the plane”)

Aperiodic means that this copy-paste scheme isn’t going to work. Lots of things don’t copy-paste, but the interesting part is that you can still fill out the whole plane anyway. While doing that, they are going to combine in an infinite number of different non-repeating ways. (Well, in small regions, there will only be a handful of ways that they can combine, but as you zoom out there will always be differences if the plane is going to end up covered.)


Ok, I think I’m getting it, thanks for indulging me. So the unique property is that the shape can be flipped or rotated in more than two orientations and still be made to fit? So in other words, multiple “tabs” can fit into multiple “notches” whether they are flipped or not.

Right? :grimacing:

From what I understand, it’s unique because it is the only tile that is impossible to make repeating patterns with.


Making non-repeating patterns

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No, the point is that the pattern of tiles a) fits together to perfectly cover the plane but b) the pattern is “aperiodic”; it doesn’t repeat itself over and over forever (the Escher tiles above do have such repeating “periodic” patterns)


Derek explains it pretty well here. The Infinite Pattern That Never Repeats - YouTube

The only one discovered so far. (Although the ratio of the two different edge lengths can be varied while still keeping the forced aperiodic tiling property, so it is a family of shapes with the property).


That’s part of the ‘local picture’, but what I mean is that you can draw lines like this:

and all the parallelograms look the same. That means the whole pattern can be shifted along the red or green lines by one parallelogram and it’s the same.

Edit: Now that I drew those lines and posted this, I see the faint remains of those that Escher used. They are slanted the other way and offset, but give the same translational structure.

“Aperiodic” means this doesn’t happen for the new tile set.

This may be of interest:


I’m not sure what result you’re referring to here. Maybe the periodic case?

The question is still open, of whether there exists a single (flips not allowed) contiguous tile with no holes, that can tile the plane but only nonperiodically.

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Another good write-up from the winner of the most wittily and meta-named publication prize …