Can somebody explain what’s exactly special and unique about [The Hat](https://www.sciencealert.com/this-surprisingly-simple-shape-solves-a-longstanding-mathematical-problem) to me like I’m five?
I understand that the shape’s unique feature can be explained by knowing the difference between periodic and aperiodic tiling. I’ve read about it online but I think I still need further explanation in layman’s terms.
Thank you.
In: 18
For some shapes, you can tile them in a simple, repetative way.
For example, you can put a load of squares together, and no matter which bit of the tiling you look at, it always has the same pattern.
The same thing happens with triangles.
With ‘the hat’, whichever bit of tiling you look at, the pattern is always different.
It’s the first time this has been done using only a single shape of tile.
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For some shapes, you can tile them in a simple, repetative way.
For example, you can put a load of squares together, and no matter which bit of the tiling you look at, it always has the same pattern.
The same thing happens with triangles.
With ‘the hat’, whichever bit of tiling you look at, the pattern is always different.
It’s the first time this has been done using only a single shape of tile.
For some shapes, you can tile them in a simple, repetative way.
For example, you can put a load of squares together, and no matter which bit of the tiling you look at, it always has the same pattern.
The same thing happens with triangles.
With ‘the hat’, whichever bit of tiling you look at, the pattern is always different.
It’s the first time this has been done using only a single shape of tile.
Let’s take a number like 3 divided by 7.
That would be 0.428571428571428571 with 428571 repeating infinitely.
That would be analogous to periodic tiling.
You put shapes in a certain order and you can tile infinity by just repeating those shapes in that order.
It might not be 1 shape, the arrangement might be quite complicated, but it still repeats.
Now think of a number like Pi.
The first few digits of pi are:
3.141592653589793238462643383279502884197… but it continues on forever.
You will never find a series of digits that’s finite in length that makes up Pi by just repeating it infinitely.
This is like aperiodic tiling, you will never find a finite group of shapes that makes up the whole thing by repeating it infinitely.
The arrangement of any part is always at least slightly off.
Now math guys have known about aperiodic tiling for a while, but they found it worked with only a few different shapes put together.
They did more math and didn’t find anything that said they couldn’t do it with just 1 shape.
That’s why the hat is exciting to the math people. Knowing something can exist and actually finding the thing are two very different things in math. So they were really excited when they found that it was a relatively simple shape.
It could have been some monstrous shape that takes up 200 GB, or something that would only be found in the year 2723 or something.
They’re just glad we have it in the here and now and on a regular screen.
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Let’s take a number like 3 divided by 7.
That would be 0.428571428571428571 with 428571 repeating infinitely.
That would be analogous to periodic tiling.
You put shapes in a certain order and you can tile infinity by just repeating those shapes in that order.
It might not be 1 shape, the arrangement might be quite complicated, but it still repeats.
Now think of a number like Pi.
The first few digits of pi are:
3.141592653589793238462643383279502884197… but it continues on forever.
You will never find a series of digits that’s finite in length that makes up Pi by just repeating it infinitely.
This is like aperiodic tiling, you will never find a finite group of shapes that makes up the whole thing by repeating it infinitely.
The arrangement of any part is always at least slightly off.
Now math guys have known about aperiodic tiling for a while, but they found it worked with only a few different shapes put together.
They did more math and didn’t find anything that said they couldn’t do it with just 1 shape.
That’s why the hat is exciting to the math people. Knowing something can exist and actually finding the thing are two very different things in math. So they were really excited when they found that it was a relatively simple shape.
It could have been some monstrous shape that takes up 200 GB, or something that would only be found in the year 2723 or something.
They’re just glad we have it in the here and now and on a regular screen.
Let’s take a number like 3 divided by 7.
That would be 0.428571428571428571 with 428571 repeating infinitely.
That would be analogous to periodic tiling.
You put shapes in a certain order and you can tile infinity by just repeating those shapes in that order.
It might not be 1 shape, the arrangement might be quite complicated, but it still repeats.
Now think of a number like Pi.
The first few digits of pi are:
3.141592653589793238462643383279502884197… but it continues on forever.
You will never find a series of digits that’s finite in length that makes up Pi by just repeating it infinitely.
This is like aperiodic tiling, you will never find a finite group of shapes that makes up the whole thing by repeating it infinitely.
The arrangement of any part is always at least slightly off.
Now math guys have known about aperiodic tiling for a while, but they found it worked with only a few different shapes put together.
They did more math and didn’t find anything that said they couldn’t do it with just 1 shape.
That’s why the hat is exciting to the math people. Knowing something can exist and actually finding the thing are two very different things in math. So they were really excited when they found that it was a relatively simple shape.
It could have been some monstrous shape that takes up 200 GB, or something that would only be found in the year 2723 or something.
They’re just glad we have it in the here and now and on a regular screen.
In a periodic tiling you can take a copy of the original tiling and slide it some combination of left/right and up/down such that it will look exactly like the original tiling. The most basic example is to imagine an infinite sheet of grid paper. If you slide it around you wouldn’t be able to tell whether you are looking at the original or the copy. This is called *translational symmetry*.
In an Aperiodic tiling you can’t do this. Such filings do not have translational symmetry. You might be able to find sub patterns that look like other sub patterns but if you try to map them to each other you’d find that the tiles around them eventually don’t line up. There are many such tilings.
What’s unique about “the hat” is that it’s the first time an aperiodic tiling has been made using a single shape. The first aperiodic tiling used thousands of different shapes. And before now we’d been able to find multiple aperiodic tilings using as few as two shapes. But this is the first one which uses just a single shape. And to make it even more interesting it was found not by a professional mathematician but a hobbyist (though he did work with two universities to prove the shape he used was, in fact, capable of aperiodic tiling).
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