Name: Heeschの数字とタイル - Numberphile (Heesch Numbers and Tiling - Numberphile)
Uploaded: 2021-01-14T10:15:54.000Z
Duration: 9 min 20 s
Description: VoiceTubeの動画で発音を聞きながら英語表現を覚えよう！学べる英語：

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If you've got a shape, you want to stop putting it together.

So I've got just please the copies of the same shape, and I'm allowing myself to flip them over.

I have covered the boundary of my original shape with copies of the same.

So we might ask, What can I do that again?

If I had enough of these pieces, I could sort of.

Now I can put some things round and maybe I can I can sort of get myself into trouble, like, couldn't get stuck.

So But maybe if I work hard enough, I could put another move around and maybe another one.

This tile was discovered by someone called Casey Man.

He actually started knowing that this is a tile where you cannot go on forever.

On the way we describe that is this is an unbalanced tile.

So if you look at it this way one part of this tile, this piece here, we've got an arc of a circle where I can fit something on and it conserved.

But I can only fit the pieces of the tile that curve in onto the peace that comes out Now all these arcs are sort of in multiples of 1/6 of a circle.

And so if I look around this this yet here, I've got the positive piece, the convex piece, and that has four on.

Then here I have Kong Cave and that's actually two.

So if you notice the negatives 1234567 The positives.

And so as I keep on going, the amount off negative relative to positive is gonna grow and grow, and at some point it's gonna overwhelm things.

This is a time we're gonna have to stop at some point.

Andi, The time where we stop the number of rings we can do is something called the huge number.

So what's the Haitian number for that show?

So the huge number for this shape is with three on Dhere is something where we can show those rings.

And so no matter how hard you try, you can't go further than that.

With this particular type, we can ask what what are the numbers?

So when you surround it, it has to also be touched every edge.

So the circle I've just written it's his number.

This shape hairs lies somewhere between the circle, with zero on the square with infinity and obviously complete rings.

There's a fairly old example which looks like this.

So that means I can cover the boundary of this tile completely with copies off this time once.

We can have examples where you could do two rings.

Three rings like the example we started with on dhe four rings.

You could even get up to doing five whole rings on DDE.

You're seeing how this could be quite a complex operation here, that this is not the natural way necessarily.

You might feel the obvious way you can get yourself into trouble in all sorts of places as you put things on.

And in fact, this tile with huge number five also discovered by Casey Man is the world record holder.

And just to give you a sense, this is not an old problem.

Because even his number two was only found in 1991.

All whole numbers are the huge number for some time.

So if I said 4622 there could be a shape out there that will do it and then stop.

This is linked to some other tiling problems, but we sort of have no idea how you'd even start t prove that or find find examples.

This isn't like the Riemann hypothesis that where everyone wants to solve it.

This is a problem which now has a little bit of a knowledge about it.

So if you were toe, if you were to make significant progress on it, then people would look up on take note or those significant progress is probably something a bit more than saying, Oh, I've just found a tile with huge number six on DSO.

Both of the two examples from From Casey Man this example with with huge number three and the one with number five, they are both part off a sort of more general study that he did, looking at what you could do with Polly hexes, where you stick together, hexagons and then makes the markings on the side, it might be that you could get some rich structure from that.

You could then get a result where you say I can do every even number up to 50 and then that might be might be a more interesting result.

The other thing it's related to is a problem called the Einstein problem.

Einstein, as in one tile on that asks if there is a single tail.

I'm staying in German, so it's a bad pun which will tail the plane, but not periodically.

So all the infinite stylings we know so far a periodic no, we have things like the Penrose tiling, but that has two different shapes.

You put it together in all possible ways, and you can cover the whole plane.

But they're none of them, which are like the squares, the tiling of squares, which I could just pick up a shift and put down.

And it's the same everywhere when I do that.

You you would find a single tile that can create stylings of the plane, but not period.

You're asking for tidings that will only produce non periodic time.

There is a slight cheat with the Einstein problem.

Joan Taylor from Tasmania did her math degree and then raised a family, but always had an interest in mathematics and spent in particular this problem on DSO about 10 years ago.

Now she found a single tile has this property.

The only problem is that it's not a connected tile.

If I'm somewhere on this island, I can get anywhere else.

But imagine if I say, Well, my island is officially going to be this piece and this piece now I can only move them around together.

So that's a single shape, but with two pieces.

And so she found a particular shape that has this property.

You can see it there on this will only tile non periodically, but it's a slight cheap.

It's a really nice result because this was a problem that was like 50 or 60 years old.

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Heeschの数字とタイル - Numberphile (Heesch Numbers and Tiling - Numberphile)

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