5 Platonic Solids - Numberphile


Uploaded by numberphile on 06.11.2012

Transcript:
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 20 triangles.
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--
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
324 degrees.
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--
120 degrees--
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.