Ready?

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.

That's quite nice.

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

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

Keeping going with the shape.

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.

Maybe I could go on forever.

Maybe not.

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.

There.

I have one.

They're here.

We have three.

And here we have one.

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?

Could be huge numbers.

What about circle?

What's the huge number of a circle?

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

But you really need to surround it.

Possible.

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

It's another simple example.

Is the square You could do that forever?

Yeah, so these numbers infinity.

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

They have to be whole numbers.

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

This is early example.

It has huge number one.

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

But then I can't go any further.

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 can see this shape does it on.

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.

The conjecture is all he sh numbers.

All whole numbers are the huge number for some time.

That's what we believe.

Yet we only know examples up to five.

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

Yep.

We believe that would be the case.

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.

How does a problem like this progress?

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

This is Yeah.

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.

So you take your shape.

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.

So they were the Einstein problem.

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.

I can draw some path we might set near.

This is a connected object.

I can.

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.

I can't move freely.

They they stay in the same places.

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.

But they often in mathematics, when you can't solve a problem for long enough, even a solution that's a little bit of a cheat can be a useful information.

It's a very valid cheat.

If that makes sense, because you do have one shape that you could move around together.

It's just the way it's defined.

I couldn't laser cut it, which is why, why I'm not so keen.

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