[PROF SEQUIN]: What do you make of this? 5, 6, 3, 3. What do you think comes next?
[BRADY]: I'm gonna go... 7!
[PROF SEQUIN]: 7. No, actually, it is another 3. Now you have a guess at what's coming next.
[PROF SEQUIN]: Good! Yes. As a matter of fact it continues to be 3, 3, 3.
And you wonder, this is really a strange sequence, you know, what are we going to do with this?
Well, this is the number of regular polytopes that exist in 3, 4, 5, 6, 7, 8, 9, and all higher dimensions.
And that's what we are going to talk about today.
Regular polytopes in N dimensions. A polytope is just a more general term that generalizes for
2-dimensional polygons, 3-dimensional polyhedra, and everything higher we call a regular polytope.
That sounds a little scary, so let's start with something much simpler; the platonic solids.
So here is a tetrahedron, that is the first platonic solid.
Next we have a simple cube.
Third one — octahedron, made from eight equilateral triangles.
Number four, twelve faces — the dodecahedron. This one has twelve pentagons.
And finally, the real jewel... —
Ahhh.... the icosahedron, made out of twenty equilateral triangles, and this is probably a Swiss crystal.
[BRADY]: Of course.
[PROF SEQUIN]: And here is another view of an icosahedron — this is a pattern by M.C. Escher.
Could you show to a non-mathematician, in an intuitive way, why there are exactly five platonic solids —
not more, and not less...?
Well, one way to do it is to look at the surface of these objects.
So this dodecahedron, for instance, is made out of regular pentagons.
And so all of these platonic solids are made out of regular polygons.
So probably we should step down a dimension and look at what regular polygons are possible.
Simplest one would be a regular equilateral triangle. Then we can do a square. Then we can do a pentagon.
Then we can do an hexagon, and so on, and if you go all the way to infinity, we end up with something that
looks like a circle.
So there are infinitely many of these regular polygons in two dimensions.
So really, this sequence up here, it has another character before the five, if you go to 2 dimensions.
And what's that?
It's not an eight—it's infinity.
And if you really wanna go to the extreme, what about 1 dimension?
Well, it's just a line segment — nothing else.
What about 0 dimension? Well, it's just a point; nothing else. But now, 1, 1, ∞, 5, 6, 3, 3, 3, 3, 3, 3 -
that's a really weird sequence, you know, and, um, I guess that would really confuse people.
Okay, all of the platonic solids are made from one of these regular polygons.
And so, how many different ways can we use these regular polygons to make one of these solids?
Start with the equilateral triangle, and ask 'how can we make a platonic solid out of that one?'
Well, we'll need a minimum of three of those to be put around the corner in order to make a
valid corner in three dimensions.
And if we put three of these triangles around a shared vertex, then the bottom is also an equilateral triangle.
That would give us a tetrahedron.
Okay, so next we can try to put four equilateral triangles around the shared vertex.
And we can bring them together and make a four-sided pyramid. And, remember, to make it regular
all of the dihedral angles here have to be exactly the same.
So that means that the bottom now is a perfect square.
We can take this square pyramid and match it with an identical square pyramid.
So that makes the second platonic solid; the octahedron.
Now we can try to take five of those and put them around a vertex.
And this makes this five-sided pyramid — so it's completely symmetrical.
If we continue forming these kinds of corners, and try to sort of wrap it around,
will this actually get a closed surface?
Will this actually end up in something, something useful?
Well, we know we have this icosahedron, okay? But if you have to really figure out whether it works,
it takes a little... thinking.
You can see on top here — we will have this five-sided pyramid.
So we can take one of these five-sided pyramids —
we cannot put this one just up on a mirror — we can take another one, but the two have to be rotated
against one another by 36°. And if we do that,
then in between we can see there's a triangle strip, with triangles going up–down, up–down, up–down...
essentially forming a five-sided anti-prism. Adding ten more triangles,
it's kind of beautiful that indeed this surface does close and result in number three platonic solid.
[BRADY]: And what about six triangles? [PROF SEQUIN]: Oh, good question! Glad you asked that.
So, let's try to take six triangles, and put them together,
and we get a wonderful nice hexagon, here.
So, unfortunately, it's totally flat.
And so this doesn't really curve and it doesn't help you to make some kind of closed object.
You can make a kitchen floor with that one.
Now if we go to seven triangles, they don't even fit into the plane –
it's kind of get the warped chip. And that is no good. So we're done with triangles.
For now we have to go to squares.
Fold them up, so basically three squares around the shared vertex leaves off an opening of 90 degrees.
And we forcefully try to close this opening.
And now we have a valid 3D corner.
If you take a contraption like that, and match it on the backside with exactly the same contraption,
then you get a cube made out of six squares.
Now I know what you're gonna ask next...
'What about four squares?'
Well... glad you asked.
—Same problem here. So we can put four squares together — and they make either a complete flat.
Or if we were to somehow bend it in weird ways, then
(it'll) unknot all the dihedral angles and all the edges would be the same;
and so it not would be a regular corner — that's unuseful.
So we're done with squares.
We can put three pentagons together, and form a nice 3D corner.
And again the question arises: does that really close if we continue the process?
And the easiest way to see it is to take one pentagon and put five pentagons around it,
completing five of these corners.
And this makes this nice kind of salad bowl.
And now if we put another salad bowl like that on top of it,
— again rotate by 36 degrees so it fits nicely together —
then, indeed, we can convince ourselves very quickly that that leads to yet another regular platonic solid.
Trying to put four of these pentagons around the corner exceeds a total of 360 degrees,
so you already get something that's warped like a potato chip — and it's no good.
We're done with pentagons!
—You know what happens with three hexagons, of course: three flat tiling. That's no good.
What about heptagons?
Well, they don't even verily fit into a plane. It's too much.
So, from there on up, we're sort of done.
Nothing new can happen.
Now you know exactly why there are five platonic solids — and exactly five platonic solids.
—Professor, is a sphere a platonic solid?
—No, it isn't. Because even if you think about a very fine tessellation, you could not find a tessellation on a sphere
where all the faces would be exactly the same,
and all the edges are all the same, and all the vertices are the same...
So we're basically done with three dimensions...
And so, I must admit that for many of us there is indeed a sixth platonic solid.
And it is ... the Utah teapot!
For the people in the computer-graphics community, this is such a famous object
which has been used again and again to test our rendering software and solid-modeling software.
And it just shows up... it's just in about every single graphic paper in some form or another.
So we like to think of it as the sixth platonic solid.
Now these regular polytopes are made out of regular polytopes one dimension lower.
In the same way that we made platonic solids out of 2D regular polygons,
we can now try to make four-dimensional regular polytopes out of three-dimensional platonic solids.
Each one of these four-dimensional regular polytopes is of an object that has, as a surface,
a thick crust of three-dimensional platonic solids.
And just like we made a cube out of squares, we're now trying to make a hypercube out of cubes in its surface.
We used to take three squares, and then forcefully try to close this gap of ninety degrees.
And by closing it, we force this corner to pop out of two dimension —
become a three-dimensional valid polyhedral corner.
And that's the corner of a cube.
Now, we're doing exactly the same thing, except —
think that behind each one of these squares, there is a complete cube.
And these three cubes now are sharing a joint edge, perpendicular to this plane in that particular corner.
So there's a ninety degree wedge of open space
and we're trying to forcefully close that.
And we cannot close that in three dimensions if there are cubes behind each one of these four visible squares.
But trust me, if we really do that and we can pop out in the fourth dimension, we can close up, you know,
these cubes — and we get a hypercube.
[BRADY]: Those cubes behind these that I was being asked to imagine...
they've kind of overlapped with each other now...
[PROF SEQUIN]: No, because they pop out into four dimensional space,
in the same way that when we took these two dimensional squares — and we force them to close.
They didn't overlap; they just popped out of two dimensions into three dimensions.
And four-dimensional space is really much bigger than three dimensional space,
so there's ample room for these cubes to simply pop out and form a true four dimensional corner.
[BRADY]: But to my three-dimensional eyes, he is sitting in 3D space... I'm like...
how did that happen? But if I had four-dimensional eyes, I'd be thinking...
[PROF SEQUIN]: That's correct. And now you're hitting on the real problem.
So we can figure out that this must be possible, but then... can we visualize that?
I mean, how do we know what these thing looks like...?
Well... for that, we have to use some kind of a shadow, or a projection of a four-dimensional object
down into three-dimensional space.
And I'm sort of torn between, you know, should we use a shadow or should we use
what I would call a 'wireframe'?
And you see... the trade-off between those two options.
Just like if I showed you the wireframe of the dodecahedron —
you can really see through it, and it tells you much more than if you just had a faceless shadow of this object.
You can eventually start to get a feeling by looking at three-dimensional wireframes
or projection of such wireframes —
what the four-dimensional object might indeed look like.
And — doing this projection — we still have an option.
We can project in various different ways.
We're going from a two-dimensional square to a three-dimensional object.
And in an oblique projection, I start with the red square in front
and then the blue square is the ones off in the back,
and the green lines essentially show me the depth.
We can do the same thing by starting with a complete cube — the red cube is in front
the blue cube is in the back —
and then the green lines show essentially the extrusion of this cube in this oblique direction.
Alternatively, we can use a perspective projection — everything in the back would appear smaller.
And we can look straight at the face.
And by doing that, you know, the back face will be a smaller square (that's off) really behind the red square;
and the green lines show kind of the depth going from the front to the back.
And the same we can do with the complete cube.
And we get, you know, in the back a smaller cube shown in blue,
and then the edges that go from the front to the back are shown in green.
So here is the oblique projection.
You can see the fatter cubes being the ones in front, and then the thinner cube in the back...
and then the slightly conical edges leading from the front to the back.
So that's a valid depiction of a wireframe of a hypercube.
And here is the alternative model — a perspective projection.
So the bluish cube is the front cube, and the yellow cube is the one in the back — which is here much smaller.
The red edges that go from the front to the back.
And this hypercube has a total of eight cubes:
the front cube in blue, the back cube in yellow,
and then some squashed cubes here showing one on each of six faces.
So there's a total of eight cubes making the surface — or the thick crust — of the hypercube.
[BRADY]: So that yellow cube isn't inside the blue cube...
[PROF SEQUIN]: No. In four dimensions, it would be up in the fourth dimension (at) a certain distance.
Because of perspective projection, what's further up there gets projected into the back...
and appears smaller in this projection.
Now, we're going to systematically look at all the platonic solids;
see how many we can group around the shared edge, how much empty space there is,
and then we push that one out and push that into a valid 4D corner.
The simplest object to start with is the tetrahedron.
The tetrahedron has a dihedral angle of 70 and a half degrees.
We need at least three of these tetrahedrons around the edge to make a valid corner.
Three, of course, fits very easily.
Four fits. Even five fits, but just barely: we just have a few degrees left.
And so, when we try to fit five tetrahedrons around this, we just get a little bit bending in four-dimensional space.
And we will need — as we will see — a whole lot of these tetrahedra (to) make it actually work out.
So we can start out with three tetrahedra
and ask what happens if we forcefully, you know, bend that into a corner. Can we repeat it?
And the answer is we will get the 'simplex', or the '5 Cell'.
So, because of the projection, you know, it becomes just a three-dimensional object;
and it loses some of its symmetry.
So we have five vertices, but the fifth one, — to me, to make it asymmetric as possible —
I put it here right in the middle.
So we get the outer tetrahedron, and then we get the temples here on each side.
They all represent additional four tetrahedra.
And they don't look very regular, and that's because of the projection.
But, in four dimensional space, all five of these tetrahedra are completely regular.
And so that's the 5 Cell or the simplex.
That's what happens when you take three tetrahedra around a shared edge
and essentially force out the empty space and bend it into a true four-dimensional corner.
Four tetrahedra around the corner, you know, we get this object here, okay?
It's called the 'cross-polytope'.
And it's actually the dual of the hypercube that we've seen before, okay?...
So that's the cross-polytope — the second regular polytope in four dimensions.
Five tetrahedra put around a shared edge — you get something that has a lot of tetrahedra.
As a matter of fact, this thing has 600 tetrahedra.
Most of them are so crunched up in the middle here because of the projection...
and you just see one giant tetrahedra on the outside.
And then, adjacent to each faces, you see smaller... fairly much flattened-out tetrahedra...
And you have to just believe me that, in there, there are essentially — in addition to this one out here —
another 599 tetrahedra.
This is called the '600 Cell'.
Six tetrahedra would exceed the dihedral angle of three-hundred sixty degrees...
And that's like, you know, something warped that, uh... is not regular. So that's of no use...
We have seen cubes; they have dihedral angles of 90°.
Three of them will fit around an axis till it'll leave an open gap ...
and then we fold this one up — we've seen we get the hypercube.
If we try to put four cubes around the joint axis — you can readily visualize —
that would readily fill the space without any bending.
And so we can tile all of three-dimensional space with that, but it will never bend and make a regular
polytope in a higher dimension.
So we're done with the cubes.
The next platonic solid you may want to try is the octahedron.
It has a dihedral angle of 109 and a half degrees.
So that's less than a hundred and twenty degrees here.
And that means we can indeed fit three around a joint edge, and
still leave a little bit open space, which we can squeeze out and make this then pop into the fourth dimension.
And the result is what I believe to be the most beautiful four-dimensional regular polytope:
the '24 Cell', made out of twenty-four octahedra.
You can see the outermost octahedron, because I chose that to be, you know, preserved in the projection.
But, then, you see these rather flattened octahedra: this one triangle here...
another triangle, this small one here..., and sort of a...
distorted antiprism in between...
and then there are more on the inside.
I really like this particular object the most, because it's not too complex —
you can still see what's going on, you can still look in the inside,
see the innermost, very tiny little octahedron at the center.
Now, it turns out this object has 1152 symmetries in four-dimensional space.
And then you simply get that by multiplying the symmetries of an octahedron — which are 48 —
with the number of octahedra — twenty-four.
You multiply this out — you get to 1152.
Because you can take — in four-dimensional space — any one of the octahedron
and put it in the place — by suitable rotation — of any one of the other 24 octahedra,
in any one of the 48 possible positions.
I live at the street address 11-52. And then I found out that this object has 1152 symmetries.
I thought that somehow fate really meant me to be, you know, into geometry.
The next one on the list is the dodecahedron,
with a 116 and a half degrees of a dihedral angle.
That is still less than 120 degrees, so we can still force three of them around an edge
and then join it into a valid four-dimensional corner.
It's not bending all that much,
And so we will need a hundred and twenty of those objects in the crust
to form a valid four-dimensional regular polytope.
So this is a model of the 120 Cell.
Of course you can see the outermost dodecahedron.
You can see kind of these flattened pancake-like, you know, dodecahedra on top here.
And each one of the faces has one of those.
And then stacked on the inside there are a few more, adding up to a total of a hundred and twenty of those.
We still have the icosahedron.
Unfortunately it has a very shallow dihedral angle.
It is more than 120 degrees; it is actually 138 degrees, roughly.
And so even three of them would not fit around this edge without overlapping.
And so we cannot form a valid corner.
And, unfortunately, this is useless, as far as making a four-dimensional regular polytope.
You basically can figure it out yourself, right?
You know how to make five-dimensional regular polytopes.
You look at the four-dimensional regular polytopes, look at their dihedral angles, and figure out, you know,
can I fit at least three of them around an edge so I can make a valid corner?
Most of these regular polytopes in four dimensions are very round, and they're not very useful
There are really only two of them that actually have some hope of generating a new regular polytope in higher dimension.
One of them is the simplex, or the 5 Cell; and the other one is the hypercube.
[Brady]: The other ones have got too shallow ... ?
[PROF SEQUIN]: Yeah, the other ones have too shallow of dihedral angles,
and they just don't make valid corners in the next higher dimension.
And so, from five dimensions onwards,
there are really only just three regular polytopes in each one of these dimensions.
One is the simplex series — you can always make a simplex.
By taking a simplex in a particular dimension — I'm starting with the tetrahedron —
and then I put another vertex at the center of gravity.
And now I use the fourth dimension, and essentially raise this vertex up in the fourth dimension
until it has exactly the same distance from all the other four vertices. That makes a new simplex.
And then I can take that particular simplex, put a new vertex at its center,
raise it up in the next higher dimension until it is exactly the same distance from all the previously existing vertices
and make the next simplex.
That always works.
So I can work my way up to infinity by always making an additional simplex in exactly that manner.
The problem now is how do we depict those things.
'Cause clearly projecting it down — you could do anything.
So, it's better to figure out how do we make a model that has the right connectivity, even though
the geometry really sort of bogus at this point because it's so much distorted from what we originally have.
And I have make a few more simplices in higher dimensions.
You just need to always add one more vertex, and then figure out how to make a nice little graph
that will get the right connectivity.
So, this one has six vertices. I claim this is a projection of the 5D simplex.
'Cause the simplex also is the complete graph; every vertex is connected directly to every other vertex.
And you can see in this case — you start at any one vertex, there're five edges going off,
one to each of the other vertices.
So we want to make a 3D model, that has six corners, is reasonably symmetric, but not too symmetric.
So we could start with an octahedron. But in the octahedron then, when you connect opposite vertices,
those three edges will all intersect in the middle.
That would be so nice. You wouldn't see what's going on.
So you'd have to deliberately distort this octahedron, warp it a little bit, so that those three spaced diagonals
that go through the middle of the regular octahedron —
they have separated out. And now they do not intersect with each other anymore.
So the trick was really just finding six vertices in a relatively symmetrical arrangement,
so we can connect every one with every other one without any intersections.
And that's the game that we have to play for every one of the dimensions that we want to build a simplex model.
We already had this object that had six vertices completely connected —
the warped octahedron — and that actually has an extra free space in the center.
It's so easy therefore to put a seventh vertex right here in the center.
This one can again be connected to all the other vertices.
And so now we have a complete graph of seven vertices.
And that would be a nice model of a six-dimensional simplex.
There is a second sequence that works for all dimensions.
And it's called the 'measure polytope' or 'hypercube sequence'.
The cube is the object with which we measure the volume of three-dimensional space.
Just like the square is the geometrical element that we use to measure two-dimensional space.
Similarly, the hypercube would measure four-dimensional space.
So the measure polytopes are always the equivalent of a cube in the higher dimensions.
And the best way of visualizing what's going on is essentially going in stepwise extrusions.
So, we start with a line. And then we take this line, and extrude it basically into a square.
And then we sort of take that square and extrude it perpendicular to itself into a cube.
And then, we're doing one more extrusion of that cube into a hypercube,
to an extrusion of the hypercube into the 5D measure polytope.
And you could continue that way.
Or you can change the scale.
So, what I prefer — maybe in six dimensions — would be to take a regular cube
and essentially sweep that in three different dimensions to make yourself a thickened cube frames
which is showing every edge as the 'sweep path' of one of those cubes.
So this particular diagram would be a depiction of a sixth-dimensional hypercube,
which you get by extrusion.
Another projection of the sixth-dimensional hypercube results in this 'Rhombic Triacontahedron',
which has thirty rhombic faces on the outside.
But, on the inside, there's a whole lot of intersecting edges going on. And you may not like that.
So it all depends on what you really want to get out of your model.
In some sense, this makes a very nice sort of climbing structure for children.
[Brady]: That's not very cuby though, is it? It's hard to see the cube.
[PROF SEQUIN]: Yeah, for some angles, if you look at... here it looks like an oblique kind of squashed cube
that starts with one of the cube face.
Or a very flattened cube, you know, we have just like three faces here.
And then on the inside there will be another three flat faces,
but they of course already intersect with edges from some of the other cubes.
—We can go cubes all the way to infinity. —That is correct.
So, you know, we started out with five platonic solids.
We went to the six regular polytopes in four dimensions.
And after that I said 3, 3, 3, 3, 3...
I have shown you two of those three series;
the simplex series, and then the measure polytopes series.
So there must be another one.
We've seen that we can make a series of measure polytopes through all the dimensions.
In each one of these dimensions we can also form the 'dual'.
Just like in three dimensions we have the cube, and then we have each dual which is the octahedron.
By holding it that way, you can see the top and bottom vertex in my right hand
correspond to the top and bottom face in this particular cube.
For each of the six faces in the cube, I now have a vertex.
And for each of the vertices in the cube, I now have a face.
[BRADY]: Like its evil twin... [PROF SEQUIN]: Yeah. Oh, nice twins actually...
[PROF SEQUIN]: So they're duals of one another.
We also have the models that show the same relationship — not quite as obvious — in 4 dimensions.
So here we have the measure polytope — the hypercube.
And, if we take the eight cubes in the crust of the hypercube, and replace them by eight vertices,
we get the corresponding measure polytope, which is the dual to this hypercube.
[BRADY]: Evil twin!
[PROF SEQUIN]: And the same principle goes on.
It gets, you know, near impossible to kind of visualize in higher dimensions, but again,
every measure polytope in d-dimension is made out of measure polytopes of d-minus-one-dimension.
You replace each one of those cells with a vertex at its center, and connect them properly,
and you get the 'cross polytope', which is the dual to the measure polytope in any one of these dimensions.
And that's the third series that we can form.
Um, here is again our sequence.
You know, we have infinitely many polygons in two dimensions.
We have five platonic solids in three dimensions.
Six polytopes in four dimensions.
And from there on, it's just always only three regular polytopes.
So, in conclusion, I think I can say we're sort of lucky that we live in one of the two dimensions
where it's really interesting — and we have a variety of different regular polytopes.
[BRADY]: We don't live in the most interesting dimension though. We live in the second most interesting...
[PROF SEQUIN]: Well, if you think that we live in space-time continuum, which is four-dimensional, so we live in that one too.
So, what, I'm inside, a four-dimensional polytope right now?
Yes, you're inside the 120 cell space is entirely filled with dodecahedra because
in this spherical space, dodecahedra tiled space. It's non-Euclidean...