Submitted by 13artzklauser t3_127fx27 in explainlikeimfive
Can somebody explain what's exactly special and unique about The Hat 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.
Thank you for your explanation. What do you mean by "which bit of the tiling you look at?"
Imagine the tiling is on a wall outside, and goes on for miles.
Now imagine you wanted to take a photo of the tiling.
With squares, or triangles, you could take two different photos of the wall in two different places, but the photos would be identical. You wouldn't be able to tell them apart.
With 'the hat' the two different photos would be different.
In reality, eventually, 'the hat' tiling does repeat, but not anywhere near enough for it to be noticeable.
> In reality, eventually, 'the hat' tiling does repeat, but not anywhere near enough for it to be noticeable.
To clarrifiy (for OP): that's only because your photo only shows so much of the thing, and there are is only a finite (but very huge) number of possibilities your photo could show. If your photo zooms out more when needed, every part of the wall would be truly unique.
Wow you explained it so well!! I got it now. Thank you so much!
#ELI5
I'd say that's ... three white birds high, and three white birds wide. You agree? There's like, nine complete white birds you can see. Right?
Now I tell you, that's a snapshot of a much bigger mural, painted on the side of a building.
Can you find out where that 3x3 birds section IS on that huge mural?
You probably can't, because no matter which section you focus in on in the mural, all you see are 3x3 birds.
Make sense so far?
The patterns made by the hats ... they don't repeat like that.
Take a look at JUST the darkest blue ones.. They don't really repeat like the white birds do.
And the math people who figured this out, realized that no matter how big the mural is, the dark blue hats in one section, do not resemble the same pattern of dark blue hats in any other section.
That should blow your mind.
Mind=Blown
Your linked illustrations were very helpful. My understanding of the subject has become better. Thank you.
Sometimes it baffles me why more people don't use illustrations or pictures to explain things.
I guess I'm not really getting it. I starred a couple of the dark blue hats here. It looks to me like these are the exact same orientation and every other shape around them is the same too. So this pattern has already repeated itself just within a roughly 10x5 block of hats. Am I missing something?
With the birds, if you slid it over even one row of the pattern it would be the same shape, with the hats you’ve got to go over 10 rows and down 2 to get the pattern to even be similar that means how far would you have to go straight over for the pattern to repeat on the same row, and that’s using one shape.
It's a lot more complicated than that, and I'm not really the best to explain that. I don't really understand the math myself, but I do understand the basic idea they're explaining.
But I think of of the keys is, you have to look at a much larger patch. Like, everything in that image, plus a few more equally sized blocks, together, don't repeat.
Looks like a pile of T-Shirts. Also thanks for the explanation
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).
Thank you for your relatively easier explanation. Cool fun fact at the end, too!
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togtogtog t1_jee0bl7 wrote
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.