We're
looking for the, the
math
for the graphic design for the for the next ICM,
there were tried many ideas, and I was personally very inspired by the work of El Lissitzky with his three-dimensional
object, curved object. A lot of interesting geometry.
But what eventually was selected is more inspired by Kazimir Malevich and his very simple shapes, namely squares,
solid colours. So, uh, it took me a while to warm up to this ideas,
but then I realised there's some cool math about squares too. So I'm going to talk about one today. Particularly about square tilings.
So, like, for example here you've uh - here you can tile the whole, uh, plane with just two kinds of squares.
So this is, maybe imagine this two squares and if I just periodically repeat it
this is going to tile the whole, the whole plane.
This is the simplest tiling you can make out of squares which is not all equal, and it illustrates something with the Pythagorean theorem.
So the way it works is that - think about so you can either - so
you can you can hit, take this figure and uh
maybe, maybe anchor it this, at this red dot and so then we move it,
we can be, you can anchor all the other red dots here in this picture. But if you connect all this red dots,
let's try my best,
and so then this is - that should have been a square, I did this
You can imagine it's a square! And, and so what this illustrates that you can,
you can either tile the plane with this.
With these two square or with this one square repeated. And so it has to be the case that the area of these two squares,
equals to the area of this big square.
And so this is really Pythagorean theorem because to, to see this you can, you can think that this piece is the same as -
so if you, you want to take this two squares and move it to inside the yellow square, then this piece will go here,
that piece
will go there, and maybe this piece will go over there. And so this illustrates that out of these pieces
you can construct either one big square or two small squares, and that's the a squared plus b squared equals c squared.
So that's a, that's a nice illustration.
(Brady: And this overall tiling is the simplest tiling you can do because - other than just squares?) - Just - other than just square grid
yeah, that's the simplest one. - (Brady: Those two different sizes)
(is there anything interesting about their proportion to each other?)
No,
they can be arbitrary. So this,
because they in, in, in a squared plus b square plus equals c squared you can choose a and b,
arbitrary, right? - (Brady: And same with the tiling then? It could be a giant square and a small square?) - Exactly what changes in the video
is that the size of the smaller square changes. And so they all - the tiling starts, starts kind of waltzing around.
There's something really neat math involved about how you generally think about,
about square tilings, and so maybe I'll explain it here.
So this is a tiling, so it's a piece of a tiling of a plane by squares. We're going to represent it like this,
there's a lot of notation here. I'll explain it in steps.
So, so what we see here is our squares and squares are numbered 1 through 15,
so this is numbered in white. And then there are horizontal lines and I - in the horizontal lines if it,
if it, if several squares share the same horizontal lin I consider it as one horizontal line. I drew them in red and I
labelled them from the beginning of the alphabet.
So this is A, then this is B, this line is C and so forth, up to down to H. And there are vertical lines.
The same thing which just, if some squares
share the same vertical line I consider it as one line and I labelled it from the end of the alphabet.
So this line is Z, this is Y, this is X, W, V, and U.
And so now we want to represent this picture as a graph, and what will be in the graph?
Here, now you can make a choice and the fact that you can
make one choice or the other choice will be interesting later. So but now we will choose -
(Brady: One quick question.)
(Where, where did this tiling come from? Like - ) - No, this is just - I just imagined. You -
(Brady: You just made up lines?) - Exactly, exactly. I just made it up.
- (Brady: This wasn't a special thing?) - No. No. No just a random one.
Just a random one. And so now we're going to think, now we're going to
discuss how you think about tilings in general,
so in particular you'll see how you can generate a random tiling or some - something more interesting than this.
And so, graph is the graph in the plane. It's a bunch of vertices connected by a bunch of edges.
And since it's in the plane, it also has these faces. So this is like a face here.
And so what will be the vertices? The vertices will be the red lines.
Actually,
in New York City, A is blue and it's also a line with subway, but
anyway.
This didn't work out. So the red lines will be the vertices,
so this is A and I put it in the - at the same height as A here.
So then, then there's B, C, D and so forth. And what are the edges? The edges is when the square,
say square five, and it has a line at the top and a line the bottom.
So here the five is between A and C, and so five is this H here that connects A to C.
And so maybe this square 7
this is, goes between C and F and so here's this H7, C and F.
So the, the vertices are the horizontal lines, the squares are the edges,
and then the,
the vertical lines, they are the faces. Like,
like look here. So there's a vertical line. It's surrounded by squares 7, 11, 8, and 12.
So we'll look at this X here. 7 and 11. There are 7 and 11 here.
And 8 and 12 they're on this, on the other side of this vertical line. And now the most important thing,
how do you remember the actual size and positions of the squares if you look at this graph?
(Brady: By the way, everyone I did try to get the Professor to use brown paper, but
he won a fields metal - he gets to do what he wants.)
The main thing is that we have to record, to record the actual shape of this picture and we need to know
what is the height?
What is the horizontal, what's the height of each, of each horizontal line? So we need to - there will be some equation on.
So we need to know
the height of each - like for example the height of each C. It's just the vertical coordinate of this horizontal line.
If I know this vertical coordinate, we know we know, this, this whole picture. And what is the equation on this, on this height?
If I look at the, at the length from here to here,
it's, on the one hand, is the size of this square - of the square 4 - plus the size of the square 5.
And on the other hand the size of the square 7 plus the size of the square 8.
So the equation is the size of 4 plus the size 5, that is equal the size
of 7 plus the size of 8. But those sizes, what are they?
So the size of the square 5 is the difference between the height of the,
of the line A and the height of the line C.
So we get the height
of A
minus the height of C,
plus the same now for five so the height of B,
plu- minus, ah, this is when the ban on paper is good!
Minus the height of C.
And that is equal to the height of C
minus the height of, so now this, this F plus the height of C minus the height of, so the other thing is D.
And so when we rewrite this we get that the height of C
is one quarter
of the height of the sum of heights of all its neighbours. - (Brady: Where did that come from Professor? Is that, that just - ) - That's just, just
rearranging this, thar's just rearranging this equality. So the equality, the equality that this, the - what comes from the,
the sizes of squares that come from the top, add up to the same amount as the sizes of square that come from the bottom.
That becomes the equation that the height, the height of this line is the average of the height of these four lines.
Or if you look at this graph, the value of the function at this vertex is the average
of the value of this function and all of its neighbours.
So that is, that is called -
such a function is called harmonic. This is this a very important notion in math. The result of this discussion
is that a square tiling is is a graph on the plane
together with a harmonic function on this graph. A harmonic function is, is the one that
the value of it at each vertex, like the value here,
should be the average of the of the value of all of its neighbours. It's like it's like if you think of uh,
maybe people think of uh, some social analogy
of when the the society maybe, there's a harmony in society if what the person
has is the average of all the - [laughs]
all the people he's uh, he's connected with and then maybe the society is in harmony? Or, anyways,
harmonic functions are very important in um in everything that has to do with two dimensions.
So like like, uh, any kind of mathematical physics that has to do with two dimensions, they're important in in
electromagnetism and fluid mechanics and in all,
all, they also just - so harmonic functions you can study in the discrete situation like here,
so it's a kind of a graph and a, and, and something and a function on the vertices of the graph.
Or you can study in the continuum. So maybe the continuum we'll do on the brown paper. Before we go there
there is a, there's a very important notion of
of which is illustrated in this picture. There's a very nice notion of if you have a graph in the plane
then there's a notion of a dual graph.
And if you have a harmonic function, then there is a notion of a conjugate harmonic function,
maybe we'll explain and we'll explain with this picture. And so what is the dual graph? Here we made the choice.
We made a choice, we decided the vertices of the graph are the horizontal lines
and
edges are the squares and the faces are the, are the vertical line.
But of course this, the picture is completely symmetric between the horizontal and the vertical if I just look sideways,
what was horizontal then becomes vertical. And so in this picture in terms of graph, that
means the following duality for graphs. If you have a graph in the plane
then you can, you can introduce a newer graph called dual. The vertices of the dual graph are the faces of your original graph,
that would be the vertex of the dual graph and will be another vertex of the dual graph. And they're connected by
an edge if these two faces were sharing an edge to begin with. Here's an example.
So what was a face before becomes a vertex, what was an edge stays an edge,
and what was a vertex now becomes a face.
So this, see this is going around the face. And if you had the harmonic function on the graph
then there's a related function on the dual graph, called the conjugate harmonic, which in this case is just the - so this is again
very simple to explain. Instead of recording the height of the horizontal lines you record the
x-coordinates of the vertical lines. That will be your your conjugate harmonic. So say this is the edge number seven,
so it's the one at here. And since this is a square
what we have here is that the change of the function, of the height function between these two vertices,
equal to the change of the height function between the dual vertices because that was a square. And that's the general definition of uh -
well, maybe we'll do the rest on the brown paper.
(Brady: So Professor, if I, if I got out a ruler and measured all the distances of everything here, like)
(you know to the to the millimeter, you're saying - and I could feed them into these equations - )
Yeah, that's exactly yeah, that'll be exactly the -
(Brady: So, is basically what you've done here is you did a tiling and then you've just shown me other ways to represent it?)
Right, but it's it's the - it's - in reality it's the other way. There's a, they're very, they have, you can write an interesting graph
and then you, there'll be uh, there'll be;
so then, then, then you can study harmonic functions on this graph.
And harmonic functions on the graph, if you if you're trying to tile a whole plane
you can imagine that the very, very large distances - that your graph
may look something like, well like a plane. And so your function will look some - you know,
maybe some continuous harmonic function in the, at the infinity.
So this this relates to uh, it relates to the notions of continuous harmonic function,
which I will discuss on the brown paper in a second. And so this is, this is, this is a fun -
this is a fun combinatorial geometric way to think about harmonic functions. Also a way to parameterise the,
parameterise the square tilings by harmonic functions.
(Brady: So do people who are trying to come up with tilings - because sometimes you'll see)
(oh someone's come up with a clever new tiling. Are they using - ) - Yeah of course, this is very classical.
This is, this is, this is very classical. This was known to mathematicians for the longest time.
There's a lot of very serious mathematics goes into figuring out how,
if you study some discrete structures some - like such as a square tiling. If - or some maybe some other
combinatorial random structure which are discrete. And uh, how in a very large distance on a very large scale
they will, they will uh
they will behave as something we know from the continuum. From - the harmonic functions were first introduced in the continuum and they decided,
in like electrostatics,
electrostatic potential is an example of harmonic function - people people studied it for, for what it is.
It's very interesting how this large scale continuous thing emerged from,
from small
scale discrete combinatorial things. And that is, and that is actually relates to,
to some um, to some very concrete problems in mathematical physics. One, one thing mathematical physics
try to do is try to connect different scales. Like for example, we know that solids or liquids,
both of them are are really discrete in microscopic scale. So there's something really something happens on the microscopic scale. At the large scale
we see, we see equations of say elasticity of hydrodynamics or other equations of continuous mathematical physics,
and it's a challenge sometimes. Sometimes an easy,
sometimes very hard one for mathematicians to, starting from microscopic description, justify the the macroscopic behaviour. And one thing which is really
which really helps if you can somehow reduce your microscopic problem to a problem about, say, harmonic functions.
Because harmonic functions, they have a really nice -
so if you have harmonic functions on graphs and the graphs are,
and the graph kind of approximates something continues by making, becoming denser and denser. Then, then, you know
the harmonic function, discrete one, will converge to a harmonic function the continuous one. Now suppose you have a continuous harmonic function.
And so we have a function of of two variables, some function f of x,y and why's it called harmonic?
What's the continuous analog of the property we discussed? That the value of the function at any point is actually the average
of its value over all of its, maybe some distance neighbours. Or if you can,
in fact average over the disk. If I take the value at any point (x,y) this is equal to the average of f
over this whole disk.
That's the continuous version. And the, the basic property about
harmonic functions in math - it's something you prove in complex analysis - that in fact they come in pairs. They come in pairs
as the real and imaginary part of, in fact a complex function. Like for example
if I take the function of the form x plus i which is, i is the square root of of minus one.
So this is the square root of minus 1.
x plus i y
I square it.
And then then by the law of algebras I'm going to get
x squared
minus y squared. This is, this is the real part.
Plus this i times 2xy. And so now
it's a theorem, though not very easy to prove, that both of these things are harmonic and moreover
they're conjugate in exactly the same sense as we discussed this with the graph. The level set of these functions,
this is my function f and this is my function g, and what are the level sets of g? The level sets of g,
they're hyperbolas like that. So this is, this is the level sets of g.
This is where (x,y) equals a constant. And what are the level sets of f? The level sets of f,
they're, they go exactly perpendicular.
And so then if I look at the, at the gradient, the gradient is something which goes perpendicular to the level set.
So maybe this is the gradient of g.
And this is the gradient of f. They're exactly different so - but this I, this I messed up.
This is supposed to be exactly the same size and they're perpendicular.
This is a continuous version,
what we had here is the discrete. Here in the discrete the two conjugate functions where the heights of the
horizontal lines and the x-coordinate of the vertical lines, This function changes
vertically by a certain amount, and the other function changes horizontally by exactly the same amount so the gradients go exactly perpendicular.
So that's a discrete version of that, of that phenomenon over there.
[Preview of more videos] Let's imagine a game with the following rules:
we start with two
numbers, the sum of these numbers will be more than 50 trillion. Great mathematical accomplishment, you really
do things that you would not believe yourself are possible. And you have to find it within you. - No, I did not get it right
(Brady: How do you feel about that?) - I uh
Well, you know you win some you lose some.
Ah!
Did I make a big blunder?
Turns out he did not realise.