I wanted to actually start uh with a challenge, a little puzzle. People are probably familiar
with the tetrahedron, and here's a square based pyramid, so the triangular faces are both the same
on these shapes. The challenge for you, and we'll perhaps solve it at the end of the video, is if I
put these two shapes together tell me how many faces there are on the shape that I would make
by fusing two of the triangular faces together. Okay, there's your challenge. You might think, oh,
it doesn't sound terribly difficult for us Numberphile- so but we'll look at that at
the end. What I wanted to talk about in this video was an interesting thing I discovered working with
a sculptor that lives nearby my house here; he's very famous actually, Conrad Shawross,
and he loves using mathematical shapes in the sculptures that he makes. And one of the early
challenges he faced, he realised that using a bit of mathematics actually cut down the amount of
work that he had to do as a sculptor. So he's been really obsessed with this shape the tetrahedron;
he sort of thought this was like one of the great building blocks of nature, and indeed
he's right. The ancient Greeks of course thought the platonic solids were the kind of atoms that
made up the universe. So at that time we thought earth, wind, fire and water were the atoms of
the universe; and each of those would have a shape. Earth, that was the shape of the cube;
water, the icosahedron, made out of 20 equilateral triangles; air was the octahedron made out of
eight equilateral triangles; and fire was this shape here, the tetrahedron, the kind of spikiest
of the shapes. Of course we've got one platonic solid left over which is the dodecahedron,
and Plato left that to be the shape of the universe, which is kind of interesting.
But Conrad Shawcross was- was interested as a sculptor in, yeah, well what about
taking this atomic shape the tetrahedron and and seeing what he can build out of it. So one of his
very first projects, he built 2,000 tetrahedrons and set himself the task of trying to see
what he could build out of these tetrahedrons. And actually he had a deadline and suddenly the
deadline looms and he's got to try and find how to put these 2,000 tetrahedron together; but then
he talked to a mathematician who said, well you realise you don't actually have 2,000 objects
you've got to find how to put them together? In fact you have a much smaller number of building
blocks. Because if you put two tetrahedron together for example there's really only
one shape you can make out of that, there isn't another way I can put these tetrahedron together
and get a different sort of shape. So already actually you've halved the problem;
and and this is the power of symmetry, you know, and that's my area of expertise, symmetry.
Symmetry often is a clever way of cutting down a problem which at first sight seems like there are,
you know, 2,000 different things you've got to put together. But then he realised, no actually,
you can halve that. There are only 1,000 objects that you need to consider, I've just got to put
these together. But the mathematician said, no in fact you can cut it down even more because
if I have three tetrahedron there's only one way I can put three tetrahedron together.
So here are my little paper tetrahedron I made earlier for you. Here's our two put together;
but actually there's only one way I can attach a third object, and what I make is this kind of
weird shape almost little like a boat isn't it? So you might say, okay well that's one way to
put the tetrahedron together but maybe I can fuse this on a different edge - and look I've
got something different. But of course the point is that symmetry means that I can turn this and
I've always got that little boat structure. So it doesn't matter how many different ways I try, the
symmetry means I'll always get the same object. So Conrad used this as a shortcut in a way,
the symmetry of how you put these together, to say I don't need to consider 2,000 I
only need to consider a third of that and use this as my basic building block; and
that's what he used to make some of his uh first interesting structures out of this tetrahedron.
(Brady: But if you go up to four then it- then you lose it? Four you can start-)
Yeah then you start to um have uh different directions that you're coming out, and so this is
really the basic building block for his sculptures is this particular one here - but then there are
new decisions you've got to make. And what he found was there were interesting tendrils;
quite unexpected shapes that emerged out of this. If you look at Conrad's work
you see just amazing kind of weird directions that he's been taken on by this particular object. And
recently he's discovered a kind of new sort of shape which I actually had not been aware of;
because one of the interesting thing is with platonic solids, which of them can you pack
and fill space with? A cube? Clearly you can stack cubes and and fill the whole of space, but
do any of the other objects actually fill space? You can take a a slightly non-symmetrical
tetrahedron, squash it a bit and fill space with that. But with the actual symmetrical tetrahedron,
if you try and fill space it doesn't work, and there's always kind of a little bit left over.
And Conrad's made this very interesting shape that looks like it's trying to be a
an icosahedron, it's sort of building up lots of triangles, but they don't quite fit together
and there's kind of an interesting gap. I think that's very interesting, an artist
using these shapes to sort of discover very a asymmetrical shape made out of these tetrahedrons.
It looks like it's trying to be something symmetrical but doesn't quite work, it's kind of
interesting, the sweet spot between something that is trying to be perfect but it isn't quite. (Does
that tetrahedral boat have a name?) - It really should do doesn't it? I mean- but now you've
challenged me with that I don't know what that would be called - perhaps people in the comments
down below, either we can christen it, give it a name, or maybe somebody does know a name for that.
Perhaps it could be called the Shawcross shape
or something. - (It is boat-y.)
It is a little bit boat-y. So I think it's interesting
how you can use symmetry very often to to cut down the hard work you've got to do;
and and very often in when I'm looking at a particular mathematical problem,
sometimes you're looking at an equation um and it can be solved in many different ways,
uh but spotting that actually all of the solutions are kind of symmetrical versions of each other
is often a way of cutting down the work. You just have to find one solution and then use the
symmetry to find all of the others. So that's a very common trick in mathematics and in science.
But it was intriguing, it was the first time I'd seen it used by a sculptor as a- you know,
oh yeah use the symmetry and cut down the work. - (What work was he cutting down Professor?
Was it- was it- was)
(he having to cast less pieces or something like that?)
- Well he'd actually made all of these 2,000 cubes out of wood; but what he'd cut down was
just- what he needed to put together was these three and then they were the building blocks
from which everything else- you know, there was nothing new that would emerge if he hadn't have
done that. At first I think there may be many more possibilities but the symmetry showed, no, all of
these are actually examples of the same thing. So maybe we should come back to our little challenge
because that was about putting shapes together and understanding what shape emerges by putting two
together. So at first sight you may say this is pretty obvious - this shape has 5 faces, this one
has 4 faces, when I put these together I lose two faces. So 5 plus 4 is 9, so I should get 7 faces.
So what's the big uh problem? But something rather curious happens
because when I put these together I lose two more faces! Because look at this - these two
line up, and so these two actually become one face, so this object actually has 5 faces: 1, 2,
3, 4, 5 faces. So that's a little bit
of a surprise that comes out of that. I hope you perhaps experienced an 'oh, that's unexpected'.
And what's curious about this is this was set in one of the SAT exams uh in America, just to see
how good you are at conceptual visualisation. And they didn't realise that um- this kind of rather
curious answer. So the the people who said 7 got the score and those who said 5 um were given
no marks at all, but actually it turns out rather curiously that when you put these together
um this shape actually only has 5 faces. In this shape you've got um 2 faces on this side
and 2 that faces on this side that fuse. So so actually this - which you might have first thought
counted as uh 2 different faces - you've got 1, 2, 3, the square on the bottom,
and the last triangle there, 5. - (Totally got me.)
I know! - (I would
have got the mark in the SATs though.) - Exactly! That was what was so crazy.
See the links in the description for more videos with Marcus du Sautoy
plus links to his books and theatre productions - other things he's up to. ...so for example
you might want to know - well is this particular statement provable from the axioms?