Mathematicians create a non-repeating pattern from a new 13-sided polygon dubbed 'the hat'

Cal Jeffrey

Posts: 3,846   +1,236
Staff member
What just happened? A group of mathematicians created a "new" polygon previously known to exist only in theory. It's a 13-sided shape that they dubbed "the hat," even though it only vaguely resembles one. What is unique about this geometrical figure is that it can tile a plane without creating a repeating pattern.

The hat can tile a surface without creating transitional symmetry. In other words, the resulting pattern does not repeat. It is similar to the Penrose configuration in this regard. At first glance, you might see what you think is a repeating pattern, but consider it more closely.

Imagine a floor covered in square or triangular tiles. You can lift any section and fit it on another area so long as you don't rotate it. So there is a transitional symmetry that repeats infinitely. The hat is a different bird.

Just like the Penrose, you can identify matching patterns on a small scale. However, imagine lifting that supposedly repeating series of tiles and those around it and moving them to overlay the other matching design—the smaller pattern lines up as expected, but moving further from the identical sections shows the rest of the layout does not match.

The primary distinction between the Penrose pattern and the hat is that it only requires one prototile instead of two. This monotile is called an "einstein"—not named for the famous physicist, but for the German word meaning "one stone." Ironically, the hat is actually a polykite, meaning that it was created from multiple kite shapes—specifically, eight kites connected at their edges.

The existence of an einstein has for decades been purely theoretical. The math proved it existed, but nobody had found one until now.

"You're literally looking for like a one-in-a-million thing. You filter out the 999,999 of the boring ones, then you've got something that's weird, and then that's worth further exploration," the study's co-author Chaim Goodman-Strauss, a mathematician at the National Museum of Mathematics, told New Scientist. "And then by hand, you start examining them and try to understand them and start to pull out the structure. That's where a computer would be worthless as a human had to be involved in constructing a proof that a human could understand."

If you're interested in all the geeky math details, the researchers pre-published their paper on Cornell University's arXiv repository. They also have a dedicated webpage with more understandable layman's information and sample images regarding the elusive shape.

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Plutoisaplanet

Posts: 959   +1,471
And then by hand, you start examining them and try to understand them and start to pull out the structure. That's where a computer would be worthless as a human had to be involved in constructing a proof that a human could understand.
I’m going to guess the rationale behind this is that computers cannot ingest psychedelics.
 

kiwigraeme

Posts: 1,574   +1,133
Ok…. what’s it for, apart from the plane explanation, what would be a practical use for that??

You could probably use it for encryption - and verification
Like when a bank asks you for your 5 and 8 letter - Maybe if you have a lot of variation from initial set up - the bank asks for details of say any 12 points for your PC - - no one intercepting those points will be able to recreate easily the overall image- that is highly encrypted on your PC and the banks

Plus some weird geometries etc , proofs can lead to a new mathematics system that may better explain the universe or a more efficient algorithm - not saying that applies her.

A more philosophical answer what is the purpose of a lot of stuff outside those the fulfill our needs like shether , food, companionship etc - Probably this brings a lot of contentment to a lot of people for little negative cost
 

colemar

Posts: 17   +6
I see two prototiles, not one. For example, blue hats mirror red hats.
It seems they are a single shape only up to axial symmetry.
Usually in geometry two shapes are equal when they are congruent, that is, translations and rotations are enough to make them match.
You can't match blue hats with red hats using only translations and rotations, but only if you flip over one.
If they were identical actual tiles, they would need rough surfaces on both sides to be glued on the floor.
 
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yannus

Posts: 260   +253
Ok…. what’s it for, apart from the plane explanation, what would be a practical use for that??
I remember working on a gamma rays telescope that needed a unique pattern with sensors, both on the "lens" sensors and on the "sensor" sensors and then combine the two "pictures" and deduce the origin of the gamma rays. I imagine that it could apply to that kind of challenges.
 

daffy duck

Posts: 123   +89
I see two prototiles, not one. For example, blue hats mirror red hats.
It seems they are a single shape only up to axial symmetry.
Usually in geometry two shapes are equal when they are congruent, that is, translations and rotations are enough to make them match.
You can't match blue hats with red hats using only translations and rotations, but only if you flip over one.
If they were identical actual tiles, they would need rough surfaces on both sides to be glued on the floor.


There is only ONE tile, the red and blue tiles are the exact same, they are coloured to make a point. All sorts of weird tiling can be done if you allow two different tiles, including mirror imaged tiles.
 

colemar

Posts: 17   +6
There is only ONE tile, the red and blue tiles are the exact same, they are coloured to make a point. All sorts of weird tiling can be done if you allow two different tiles, including mirror imaged tiles.
Well, look again, red and blue tiles are specular.

Here is proof. Above is a small area taken from the original image, below is the same area rotated 180 degrees.
IQfWwwTs_o.jpg

Can you move the top blue "hat" over the bottom red "hat" and match it?
No, because they are not congruent.

Also, you cannot tile the plane with blue hats only, nor with red hats only.
It is not known if there exists a shape which can aperiodically tile the plane without flipping.