KATIE STECKLES: I would
like to talk to you
about the number 5.
And the thing that I like most
about the number 5 is that
there are only five
platonic solids.
JAMES GRIME: This is
a regular polygon.
It's a regular triangle.
And it has the same number
of triangles
meeting at each corner.
KATIE STECKLES: The definition
of a platonic solid is a shape
where you've got sort of
different sides to the shape,
but all of the sides
are the same shape.
That shape is a regular kind of
polygon-- so like a square
where all the sides are the same
length and the angles are
the same and every corner
looks the same.
So those are the conditions.
JAMES GRIME: And this is, in
fact, called a tetrahedron.
It's like a pyramid.
But it's got a triangular
base, not a square base.
KATIE STECKLES: So a tetrahedron
is four triangles.
JAMES GRIME: This is
the octahedron.
It has eight--
oct--
eight triangular faces.
KATIE STECKLES: So if I get two
of those and stick them
together, you'll see that I
also get the same corner
everywhere else.
And this is an octahedron.
JAMES GRIME: So unlike our
tetrahedron, this has four
triangles meeting
at each corner.
KATIE STECKLES: And
this is one of my
favorite platonic solids.
This is called an icosahedron.
You can see every single corner
has got five triangles
meeting at it.
And all of the faces
are triangles.
And in fact, you get
So this is an icosahedron.
JAMES GRIME: This
one over here--
my marvelous props-- well,
Brady, what's that?
BRADY HARAN: That is a cube.
JAMES GRIME: Yeah.
It's a hexahedron.
Well done.
So a hexahedron has
six square faces.
KATIE STECKLES: So I'm
happy with that.
And I'll put a cube
on my list.
And that is six squares.
At this point, I would
jump to pentagons.
I don't have any pentagons,
unfortunately.
But you can fit three pentagons
around a point.
JAMES GRIME: This is
the dodecahedron.
Dodec means 12.
It has 12 pentagon faces.
And at each vertex, we
have three pentagons.
OK, so we're going to show
that there are only five
possible regular solids.
So let's imagine we're looking
at a shape that
has triangular faces.
OK, so it has to have
at least three.
Let's think of a corner.
There we go.
Let's try to get that right.
So there you have it.
Let's imagine this
corner here.
And around this corner are
three regular triangles.
Now, we've got this gap here.
Here's a big gap--
a space.
If you glue this side to this
side, it's going to be a
concave shape.
And that will actually be three
triangles around a point.
That will be your tetrahedron.
Let's add another triangle.
Let's make it like this.
So now we've got
four triangles.
We've got this gap again.
Now, if we glue these
sides together--
this gap-- four triangles
around a point--
well, that will be
your octahedron.
That will make a octahedron--
fine.
Let's draw another
triangle in here.
So now we've got five triangles
around a point.
Glue it together.
Here's a little missing
gap here.
Glue it together.
And this will make your
icosahedron--
five triangles around a point.
If you tried to keep going--
if you try six triangles
around a point, though, all
you get is a flat object.
It's just a flat
piece of paper.
It's not concave.
It's not a shape.
So that's as far as you
can go-- seven--
eight.
You can't add any
more triangles.
All right, let's
do the squares.
Let's have three squares
around a point.
There we go.
And these are all 90 degrees--
There's a missing gap here.
Glue it together--
this side glued to this side.
And you'll get your cube.
If you try and add a fourth
square to it,
you'll get 360 degrees.
You'll get a flat
shape again--
no point.
Let's try the pentagons.
That's all we have to do.
Pentagons--
I will do my best.
BRADY HARAN: Pentagons are hard
to draw, aren't they?
JAMES GRIME: Yes.
I'm a mathematician rather than
an artist, so I may just
leave it there.
I think I'm going to have to--
I'm going to struggle
around this.
But if I draw pentagons,
a pentagon has an
angle of 108 degrees.
You can fit three
around a point.
If you fit three around a point,
that would add up to
And you have a little gap.
And you can glue it together.
And if you do it, you'll
end up with three
pentagons around a point.
That's a dodecahedron.
You can't fit four
about a point.
The angles are too big.
So let's move on.
Let's try the hexagons.
Hexagons have an angle
of 120 degrees.
If you try three hexagons
around a point, that
will add up to 360.
BRADY HARAN: You
can draw them.
JAMES GRIME: OK, I'll
draw those--
he says.
Let's draw three hexagons
around a point.
Each one has 120 degrees--
It adds up to 360.
It's flat.
It makes a flat shape.
It's not a concave shape.
It's not a solid.
After that, if you're looking at
seven sided shapes-- eight
sided shapes--
the angles are too big.
And they don't making regular
solids at all.
BRADY HARAN: Katie, I think the
viewers are going to have
two questions.
Here's the first question.
Where does a sphere fit
into all of this?
KATIE STECKLES: A sphere isn't
a platonic solid because it
doesn't technically
have faces.
I guess in some sense the sphere
is kind of the limit of
having more and more faces.
Like if you get enough faces,
eventually the faces are all
of size zero.
But it's not counted as a solid
of any shape because
it's not got kind of individual
faces that are a shape.
BRADY HARAN: OK.
And the other question they're
going to have is, where can I
get some of those cool
magnet toys.
KATIE STECKLES: These
are amazing.
It's called Polydron.
They do like a clippy
together version.
But this is the magnetic
version, which is like the
Ferrari of Polydron.
And while we're using
brand names, I think
this is really nice.
They do sell pentagons.
And if anyone wants to get me
a very nice present, some
pentagons would be
much appreciated.
But as it stands, I own
squares and triangles.