I've got a shape for you, Brady. If you've got a shape you want to start putting it together.
So I've got just- these are 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,
well, 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.
I could get stuck. So - but maybe if I work hard enough, I could put another loop round, and maybe another one.
Maybe I could go on forever, maybe not. This tile was discovered by someone called Casey Mann.
He actually started knowing that this is a tile where you cannot go on forever. And the way we describe that is,
this is an unbalanced tile. So if you look at it, there's one part of this tile, this piece here, where you've got an arc of a circle,
where I can fit something on, and it can sort of - but I can only fit the pieces of the tile that curve in onto the piece that curves out.
Now all these arcs are sort of in
multiples of a sixth of a circle.
And so if I sort of look round this shape, here I've got the positive piece, the convex piece, and that has four.
And then here I have concave, and that's actually two there,
I have one there.
Here we have three,
and here we have one. So if you notice the negatives: one, two, three, four, five, six, seven;
the positives are only four. And so as I keep on going, the amount of negative, relative to positive,
is going to grow and grow. And at some point it's going to overwhelm things.
This is a tile we're gonna have to stop at some point and
the time where we stop, the number of rings we can do,
is something called the Heesch number. (Brady: So what's the Heesch number for that shape?)
So the Heesch number for this shape is, is 3. And here 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 tile.
We can ask, well, what other numbers can be Heesch numbers? What about a circle?
What's the Heesch number of a circle?
(Brady: So when you surround it, it has to also be touched at every edge? Right.) - Yep, you really need to surround it.
So the circle, I've just written its Heesch number. It's zero.
Another simple example
is
the square. - (Brady: You could do that forever)
Yeah
So the Heesch numbers's infinity. This shape here lies somewhere between
the circle, with zero, and the square with infinity. And obviously complete rings,
they have to be whole numbers. There's a fairly old
example which looks like this. And this is an early example and it has Heesch number 1.
So that means I can cover the boundary of this tile completely with copies of
this tile once, but then I can't go any further. We can have examples where you can do two rings,
three rings like the example we started with, and four rings.
You can even get up to doing five whole rings and
you can see this shape does it. And you're seeing how this could be quite a complex operation here.
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 Heesch number five, also
discovered by Casey Mann, is the world record holder. And just to give you a sense,
this is not an old problem because even Heesch number two was only found in 1991.
The conjecture is all Heesch numbers, all whole numbers, are the Heesch number for some tile.
That's what we believe, yet we only know examples up to five.
(Brady: So if I said 4622)
(there could be a shape out there that will do it and then stop?) - Yeah, we believe that would be the case,
this is linked to some other tiling problems.
But we sort of had no idea how you'd even start to prove that or find examples.
(Brady: How does a problem like this progress? This isn't like the Riemann hypothesis, is it, 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 to make significant progress on it then
people would look up and take note. Although significant progress is probably something a bit more than saying
oh, I've just found a tile with Heesch number six.
And so both of the two examples from, from Casey Mann - this example with Heesch number three and the one with Heesch number five,
they are both part of a sort of more general study that he did
looking at what you could do with polyhexes where you stick together
hexagons and then make some markings on the side. It might be that you could get some rich structure from that,
you know, 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. Ein-stein as in one tile.
And that asks if there is a single tile,
Ein stein in German, so it's a bad pun, which will tile the plane, but not periodically.
(Brady: So all the infinite tilings we know so far are 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 then none of them which are like the squares,
the tiling of squares, which I can just pick up, shift and put down, and it's the same everywhere when I do that.
So the - with the Einstein problem, you would find a single tile
that can create tilings of the plane, but not periodically. You're asking for tilings that will only produce non-periodic tiling.
There is a slight cheat with the Einstein problem. Joan Taylor in Tasmania did her math degree and then raised a family,
but always had an interest in mathematics, and spent in particular this problem.
And so about ten years ago now, she found a single tile that has this property.
The only problem is that it's not a connected tile. I can draw some path we might set - you know,
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 them freely. 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, and this will only tile non periodically.
But yeah, it's a slight cheat. It's, it's a really nice result because this was a problem that was like fifty or sixty years old.
I think 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.
And it's a very valid cheat, if that makes sense,
because you do have one shape that you can move around together. It's just the way it's defined
I couldn't laser cut it, which is why I'm not so keen.
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