Maybe it's best to start with the Oilers formula.

Well, this formula tells something about trying, really the spheres.

So maybe I have to tell you what the trying latest furious Well, intuitively it is a sphere.

But it iss made combinatorial so that every face is a triangle.

So you want to attached bunch of triangles in such a way that the entire shape is essentially identical to a sphere.

You can continuously move it to form a make it round.

So maybe the easiest example a ce a simplex.

So simplex has four vortices one too 34 And if you just look at the boundary of it the tetrahedron, then it is essentially a sphere can continuously deform into a sphere.

Every face of it is a triangle.

In fact, you see four of them 123 and four.

So, as you can imagine, there are many, many different ways to triangulate this fear.

Another example, maybe, is the second easiest example is the boundary of octahedron is essentially a sphere and I can go on.

Maybe the third formula example is the boundary of Ike was a hydrant where I glued together 20 triangles to make us feel e So you're less formula says something about trying latest fears.

He noticed that if you stare each one of these trying lady spheres say the two dimensional spheres.

That's as we're considering, though, and if you first count the number of vour titties.

So if I do the tetrahedron, I see four over to seize 1234 And then I'm going to count the edges.

123456 And then I count the triangles.

There are four triangles in the texture hatred.

So I'm counting zero dimensional faces, one dimensional faces and two dimensional faces.

So let me call them F zero left one left too.

And what Wheeler did waas to compute the alternating some at zero minus f Juan plus f too, which is, in our case for minus six plus four, which is equal to two.

So let me do the same computation for the octahedron, the alternating some off the number of faces of various dimension.

Then I get six minus 12 plus eight, which is equal to two.

As you have guessed, this is a general pattern.

No matter how you triangulate your two dimensional sphere.

The alternating some of the number of faces is always equal to two.

Okay, so that was always observation.

So let's just imagine how this, uh, wonderful but seemingly accidental pattern will continue in higher dimensions.

So you can certainly think about three dimensional spheres instead of two dimensional spheres.

And certainly you can triangulate them, and there are infinitely many different ways of triangulating them, and you can do the same.

Now you have four numbers to think about.

Instead of three.

You have purchases, just triangles and many, many tetra hydrants.

So the triangle latest fears in the next dimension will be a way to make a sphere by gluing together many of the tetrahedron.

It's the next dimension is always hard to imagine.

So before making the correct guess, let's let's go to even lowered major.

So that was the picture for two dimensional spheres.

But let's this time imagine one dimensional spheres triangulated one dimensional spears, so triangles in one dimensions are supposed to be just straight lines.

We know very well how typical triangulated one dimensional spheres look like this, this one triangulated one sphere where I used five are to cease and five budgets.

So my numbers will be F zero equals five and F one equals five and the oil ist relation.

In this case, we'll look like this.

And again, the number zero that we see here, ISS independent off the triangle ization off the one dimensional sphere that I have chosen.

For example, if you have used seven vortices, you are forced to use seven edges too, because you want the entire thing to be a one dimensional sphere which is a circle.

So with the always relation in one dimension is simply saying that ab zero minus f 10 So let's go even lower even and think about zero dimensional spears.

So this is a little bit tricky if you have never thought about zero dimensional spheres.

But what is really a sphere sphere is supposed to be that something that lives in our space, which consists of a ll points that are off the same distance from the origin or whatever point that you have chosen.

So the zero dimensional sphere should live in a one dimensional space, which is the line.

And if I collect all points that are off, say distance one from Burrage in then I would get exactly 2.0 dimensional spear is two point, and that's well, in my sense, that's a triangle with this fear.

And there is really one number F zero The number of earth is is and that number of artists should be always too.

So, in this case, the olders formula will look like AP zero because two it's going up again.

That's right.

So I had to in Dimension zero zero in Dimension One and two in Dimension to so we can imagine what would be the next number and say in Dimension three.

What is it gonna show?

I guess you should guess, Is this gonna also like, Is it gonna go back to zero?

Yes, that is a correct guess.

And in fact, although it's hard to imagine three dimensional spheres for this particular problem, we can actually test it for our favorite trying to triangulate the three spheres because there is, ah, higher dimensional analog up the tetrahedron.

So there is called the simplex and there is a simplex in any dimension.

So if you'll take four points in general position in three space, then you get the tetrahedron and analogous Lee.

If you take five general points in four dimensional space, then you will get a simplex, and we can kind of imagine how it would look like, especially if we force it to live in our familiar three dimensional space so it will have five vortices.

And because the five points, which lives in four space, are in general position, every pair of the burgesses will be connected by an edge.

So that's how the set of edges are.

Four dimensional simplex will look like a five dry it in this piece of paper, and similarly, every triple up points that you see in this picture will form a triangle because the five points in four space or internal position and every four collection off the five points will form a Tetra Hydro.

Face off that simplex so we can compute these numbers F.

So f zero That's five and F one will be five.

Choose to.

That's one of the binomial coefficients, which is 10 and f, too.

We'll be 5 to 3 next entry in Pascal's triangle and F three will be five choose for, which is five.

So let's test our prediction if you start from five sub struck 10 at 10 substrate five and you got zero.

It alternated like we said, it's right and you can imagine how it will work.

And the amazing fact is that something like this happens for all Triangle a two spheres in any dimension.

You just see whether you're living in a even dimensional space or ought dimensional space.

And then you know that no matter how you translate the sphere, if you perform this Sultanate ing some computation, you get either two or zero.

The number either zero or two, depending on the parity of the dimension, is called the Oiler characteristic.

It's they're all a characteristic of the sphere and this independent up the Trianon ization.

So that's amazing fact, which took many, many years for mathematicians to rigorously justify.

But we know now as a fact that no matter how you try ng late this fear, you always get either zero or two.

That was great, but you may not like this Ah, alternating pattern 0 to 0 to zero to.

But there is another way off expressing this pattern that does not care about the parity of the dimension and this is our starting point.

So let me express the oil a characteristic in another way.

So, for example, let me start from our hoped a heater, an example 6 30 C's 12 edges and eight triangles in the octahedron.

And what are going to do next?

ISS to attach a string off one's.

So that's to make it a triangle shape.

And next I'm going to do something like Paschal's rule for constructing his triangle but using substructure ins instead of addition.

So what do I mean?

So I see one and six here, So I'm going to sub strapped one from six and write down five below and I do the same.

I see one in five.

I subtract one from five and write down four here.

And then I look at five and 12 and so struck five from 12.

I have a seven here.

I do the same.

I got three.

I got three and I get one.

This sequence, which you get in the end of this process, is called the H vector off the triangulate this fear and as you see there is ah, some pattern here.

Is this palindrome me?

This 1331 But if you read it from the other direction, ists again.

1331 And this was the case for the octahedron.

So let's try to see whether this was an accident or not.

So let's maybe try the I cause I hear 12 Arti C's 30 edges 20 triangles Attach a string of ones on the other side, off triangle and ice.

Obstruct and complete this triangle 10 here, 19 9 Here, get nine here at one.

Here again, you see the same pattern.

If you keep the bottom row, it is palindrome e 1991 Okay, Seems like such an arbitrary game to apply.

But that's right.

But at least you can see what's going on on one of these numbers.

So let me just focus on the last one.

So the 1st 1 sort up this is built in.

I just wrote down one here, but the fact that one appeared and the last spot was, ah, miracle, sort of.

But what was happening?

So I started with F zero and F one and f too.

And how did I get one in here?

I subtract one from zero and I have substructure F zero minus one from F one and ice obstructed this from that.

And the assertion is that this this equal to one.

So if I write the same thing again, it is what we have seen before the Oilers, at least for two dimensional spheres.

Willis formula is part of a broader pattern which says that you have this palindrome IQ sequence in the bottom of this strange Paschal like triangle.

Well, one thing that people have noticed ISS, that if you view Willis formula as a part of this broader symmetry, then this dependency on the parity of the dimension really goes away.

So let's try to do ah, similar game in one dimension.

Maybe we do the simplex with five averted season four space, and we already have computed the number of faces it had five purty seas, tan edges, 10 triangles and five trade routes.

Gain I at the long screen of ones from the other side, and I do this obstruction is subtract one from five, subtract one from four sub strapped for from 10 and continue.

This is what I get.

11111 There's a palindrome IQ sequence of very special kind.

That's just duel.

Maybe one or two more examples just to make sure that the pattern we're seeing here is not an accident.

So maybe let's do ah, higher dimensional version off the octahedron number over to seize eight.

How many edges need to be a little bit clever to actually compute this number and the answer iss actually 24 F too.

If you count the number of triangles in this higher dimensional patron, you get 32 if you count the number of Petra hydrants, you get 60.

Let's do the same game this time, little bit faster.

111111 and 76 17 5 11 15 4641 And that's my end.

Result in this sequence that, you see is what is known as the h vector off the this particular trying to disappear.

But if you think about at least very small part of the symmetry which predicts that the last century in your H factor should be one.

Then again, if this oil s formula for this three dimensional sphere, because think about how you got this one starting from the original data, you have your F zero.

You have your F zero minus one and this number is F one minus F zero plus one.

This number is F two minus one plus F zero in Swan and this number here ISS F three minus F to F one minus zero plus one and the symmetry of the eight Director dictates that this should be one.

I can write this again in a triple inform, saying that the alternating some of the number of faces of various dimension zero for this triangulated, three dimensional sphere way, we're almost there.

So if you think off all its formula in any dimension as a small part of this larger symmetry that you see in the H vector in the bottom row off this substructure in Paschal Triangle, then the dependence on the parity of the dimension goes away.

All this formal really assess that in any dimension.

The last entry of this triangle that you see should be one which is just a very tiny part off the entire symmetry.

It's a highly non obvious, but in a certain sense is natural symmetry.

So the other numbers are also interesting, and they are betting numbers off some higher dimensional space, which is not a sphere but is constructed from a sphere.

And the fact that the sequences palindrome E, for example, this four here is equal to the other four that, you see gives you another formula on the number of faces off various dimensions, which is different from all this.

Wilma.

It's is something like or less formula.

But is this really knew that the fact that the H vector of trying Lee to sphere is always pulling draw make it was discovered then somewhat justified about 100 years ago by mathematicians Knocks then and Duncan Somerville.

They didn't do this.

They had, ah, somewhat more complicated way of expressing this symmetry off what we now call H vector.

But later mathematicians have realized that this is the best way to understand, and I think this particular way off viewing this Metreon is due to Richard standing.

It's also now in this building the sequence that you started with the sequence that you put on one side of the triangle, which comes the number of faces of various mention.

They are called the F victor because we usually call F C o F.

One f 23 f or the face and say What and where do we jump to age?

Well, there is something called the G Victor, too, and it is really essentially related to the unsold problem that I promise you to introduce.

So this is a fact we know several different ways up justifying the fact that this H vector is palindrome IQ in whatever dimension for whatever trying latest fear that you start with.

If you produce h vector in terms of the number of faces, then you always see this meh tree.

But from all these examples that we have executed so far, you see another pattern other than Smith Tree.

So, for example, think off this particular H vector one for six for one.

Another thing that's apparent in the example is that the sequence increases up the middle one and for his larger and 60 see from Russia.

And, of course, you see the same sequence going the other way.

So this pattern is called the union modality.