JOHN HUNTON: Well, as far as I'm concerned, the important
thing about five is that it's not three, four, or six.
And three, four, or six are each numbers that have really
special properties in relation to tiling the plane--
so creating repeating patterns of tiles.
So I brought some props.
And here they are squares.
These are the sort of squares that we all put on our walls.
And the pattern is very simple.
And anyone can carry on putting down square tiles, and
the pattern repeats itself on, and on, and on.
You don't need instructions because
you know how it continues.
Now, suppose you want something a bit
more exotic than that.
Let's say you wanted triangle tiles.
Say your tiles were triangles--
like that.
Well, you could put the next one upside down, and the third
one back the same way.
And then you can continue, and then you put the next row in.
And you can continue this for as long as you want.
So that would carry on for as long as you want, using the
same rule, again and again and again, just so you know how to
put down the tiles-- one up, one down.
So that's gonna be the three.
So these are tiles with three sides, three corners.
Our original squares had four sides, four corners.
But maybe you want something more exotic still, and you
want to use hexagonal tiles.
So I can use this pattern again.
I'll just draw over it.
So there was a hexagon.
You see this red.
And I could have another hexagon.
It would come in, and that one, and my third
hexagon, and so.
And I can continue to put hexagons down, as well.
This is what the bees know.
And all the bees have to know is where to put the next
little cell of their honeycomb.
So here's the question--
can we use 5-sided?
So the answer is, no, we can't.
five sides, equal sides, equal angles.
So I might put the next one down here.
And now you begin to see your problem.
Because the bit I'd have to fill in around here is a
bigger angle than my pentagon has at its angle.
I'd get something like that, and there'd be this little bit
of extra space.
Now you see, it didn't happen with the hexagons, because the
hexagons had just that little bit extra angle.
And so they filled in these gaps.
So here you'll have another.
And you get another gap.
Triangles, 3-sided figures, squares, 4-sided figures, and
hexagons, 6-sided figures, are all good for tiling your wall.
But five, pentagons--
they're no good.
Six is the last one you can do.
As soon as you start getting even bigger figures, again,
there with many sides, then the angles become too shallow.
So that the next one would only have a little bit-- if I
put in another polygon out here to match that one there--
I'd end up with only a little sliver, and I'd never get
something in there.
So you can't do 7-sided figures,
8-sided, 9-sided, whatever.
The realization there were problems with pentagons goes
back a very, very long way.
Kepler knew that there were problems, and no doubt the
ancient Greeks did, as well.
The ancient Greeks had a lot of these
interesting tiling patterns.
Kepler made a real concerted effort to get interesting
patterns using as small a number of
different shapes as possible.
And he had something that was pretty nice.
It contained a lot of structure.
But it wasn't as good as what Roger Penrose managed to come
up with in the 1970s, which was a tiling using these, plus
three other shapes--
shapes appropriate for filling in these gaps and rules again
to tell you when to put down a pentagon and when to put in a
filling-in shape.
And he was able to come up with a pattern
that went on forever.
It had a curious property that it never repeated itself.
It kept on changing little bits as you go along.
What Penrose had was he had a systematic way of filling in
the gaps, and coming up with a pattern where the local
configurations of all the tilings kept on appearing
again and again-- just in a different order.
So there were constant slight changes, but the local patches
all looked more or less the same as each other.
And this was the beginning of the realization that there
were a lot of different patterns that had this strange
property of non-repeating and yet highly, highly structured.
How do we know patterns like this never repeat themselves?
That really depends on the rules that they're given by.
So each one of these has to come with some kind of
description as to how you make them.
One of the most common ways in which patterns like this are
found are by taking a very regular pattern, but in a much
higher dimensional space, and then cutting through that
space at an irrational angle.
And the irrationality of the way you've cut it will mean
that you'll never get complete repetition.
Penrose tilings have been the origin of a lot of excitement
for some time-- really since the mid-1980s
when this story started.
Penrose discovered his tilings in the '70s, and 10 years
later they were discovered to be out there in nature.
And Dan Schlectman found that he could make an alloy of
various different types of metal.
And the atomic arrangements that went into this alloy, the
way the atoms sat, arranged themselves on a
three-dimensional version of a Penrose Tiling.
It took quite a while for it to be fully established, but I
guess it now really is definitively established that
this is so as he got the Nobel Prize for
Chemistry this last year.
BRADY HARAN: Why had people not found it in nature before,
if there's a natural tendency towards it?
JOHN HUNTON: Well, this is a very interesting, perhaps
philosophic, question about the nature of language.
Because one good answer--
I do believe this is a good answer--
is we didn't see it, because we didn't look for it.
I think the story to tell at this point is the way in which
some years ago--
I think it was the 1930s--
material was found that, with hindsight, we realized was
based on similar sort of non-repeating patterns.
But we didn't have the mathematics, the pure
mathematics--
the geometry had not be discovered--
and we didn't have the language to be able to either
talk about it or see it.
So I find it a great example of how science can't see what
it doesn't have the language to describe.