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1/3 + 1/3 = 1/4 is how rotations link dimensions (What is Geometry?)

one-third plus one-third equals one-fourth

What is geometry? That is a very deep question, which brings us here, far below the surface. Up there, superficially, we have two-dimensional geometry. But geometry can have three, four, five dimensions or possibly millions of them. You may have heard of Lie theory, with powerful abstract tools for digging deep into the relations between dimensions. Well, with this series of videos, we're going to turn this all inside out, by working concretely with the main findings of Lie theory, and trying to understand from them, what is geometry? What is the nature of geometry? What are the essential questions, answers, ways of figuring things out? What kinds of geometry can there be?

So let's start with the main result. There are basically two situations and we have Dynkin diagrams for them. In this Dynkin diagram, all of the dimensions are independent. What that means is that they are at right angles to each other, which is a one-quarter turn, so-called 90 degrees. And in this Dynkin diagram, the dimensions form a chain. Each edge links together two dimensions with a rotation by one-third. A full rotation is 360 degrees so one-third of that is 120 degrees. What this diagram is saying is that this edge links these two dimensions by 120 degrees and this edge links these two dimensions by 120 degrees, however, these two dimensions are not connected by an edge and so they are separated by 90 degrees. In other words, 120 degrees plus 120 degrees equals 90 degrees. Strange, isn't it.

Now the key thing to appreciate is that basically this is the only way to link dimensions together into chains. We really need to build intuition on this. So let's think about it from a couple of different ways to make it very concrete. We'll have a Martian and a Human sort it out.

A rotation by one-third plus a rotation by one-third equals a rotation by one quarter.

In other words, 120 degrees + 120 degrees = 90 degrees

You could say, one-half of 90 degrees is 120 degrees. One-half of one-fourth is one-third. One eighth of a full rotation is one third of a full rotation.

This is a remarkable fact, whether you are from Earth or from Mars like me. But more than that, this may be the most important fact, the essence of geometry. This is precisely how n-dimensional space is linked together.

Dynkin diagrams

How do we know that? Because of the Dynkin diagrams. They are the treasure maps of geometry. They describe the structure of the root systems of the complex simple Lie algebras which are the complexifications of the real Lie algebras of the real connected compact Lie groups, notably the compact classical groups, which are fundamental to geometry, whether affine, projective, conformal or symplectic! Let us make sense of these treasure maps and thereby unify all of geometry!

Yes. Independent thinkers from Earth and interdependent thinkers from Mars are working together to explore what is geometry? How can we navigate our physical, mental, emotional, volitional universe? Ultimately, this will leads us to rotations in the real numbers, the complex numbers and the quaternions. Rotations link together dimensions in fascinating ways. We know from what you call Lie theory that the links between these dimensions are given by the Dynkin diagrams. Thinking abstractly, we ultimately arrive at the profound truth that one-third plus one-third equals one-fourth. In the spirit of working together towards a shared language, we're going to think concretely, start at the end, accept this conclusion, develop our intuition about it, and explore its significance.

So how does this equation relate to Dynkin diagrams?

A Dynkin diagram consists of nodes and edges. Each node signifies one dimension, in Euclidean space, with a direction, specified by a pair of vectors, positive and negative, which are called roots. In the Dynkin diagram, an ordinary edge between two nodes signifies that there are two dimensions separated by 120 degrees, in other words, two pairs of vectors, with the positive vectors separated by 120 degrees. If there is no edge between two nodes, then the dimensions are separated by 90 degrees, which means that they and their vectors are orthogonal, perpendicular, independent. From the Dynkin diagrams we see that a node is typically connected to one or two nodes, but is independent of all of the other nodes. If all of the nodes were disconnected, then they would all be independent of each other, which would not be interesting at all.

I say independent thinkers are the most interesting... but give those interdependent Martians a chance.

90 degrees means that nodes are independent whereas 120 degrees means that nodes are interdependent, interrelated, connected. The independence of 90 degrees and the interdependence of 120 degrees are two very different ways of looking at the world. This is a theme developed by the 20th century futurist Buckminister Fuller and furthered by his 21st century champion Kirby Urner, a shining star at Math 4 Wisdom. Kirby distinguishes Earthling math, based on 90 degrees, and Martian math, based on 120 degrees. Kirby's deepest value is clarity. He asks, Is there a way that humans can stop living in crisis mode? Our equation, one-half of 90 degrees is 120 degrees, should yield insights.

Visualizing with a cube

Whoa, whoa, whoa! We don't need your Martian fiddle faddle to see all that. Anything you can do, I can do with a cube. My tumbling cube. Look here. I set my cube on an edge. This edge above it is 90 degrees away. That's obvious. And how do I get there? I take two steps back around and then two steps upside down. And that gets me to what is obviously 90 degrees. And doing it this way I can say that the cube didn't rotate but instead it reflected. See that? That's why you should get yourself a cube.

Yes but it's not obvious to me that you are getting 120 degrees. A cube is square. A square rotates by one-fourth. How can you get one-third from a cube?

I'm from the Cubic Earth Society. Cubic means third power. If you set a cube on its vertex then you can see that three faces come together at a vertex. This is how Explaining the Turn and roll...

How do you get the equation? I don't understand your back around and upside down.

Turn and roll and roll and turn.

Would you agree that 120 degrees is super clear from this diddle and this daddle.

But how do I know that you end up with 90 degrees? Why don't we resolve it with mathematics?

Do you mean Math 4 Wisdom?

Why, sure, I believe I do. Let's lookie here at this Dynkin diagram for A3.

Interpreting the link

The Dynkin diagram {$A_3$} has three nodes which stand for three spatial dimensions. The middle node is linked by edges to the node on its left and the node on its right. This means that the middle dimension is 120 degrees from the dimension on the left and 120 degrees from the dimension on the right. But note that the leftmost node and the rightmost node are not directly linked. This means that the leftmost dimension and the rightmost dimension are independent, which is to say, they are separated by 90 degrees. That is what our equation says: 120 degrees plus 120 degrees is 90 degrees. We have this three-dimensional model that shows how that works.

Interpreting A2

Similarly, for {$A_2$} we can think of there being three states, 1, 2, 3. There are six roots because there are six actions that take us from one state to a different state. We are building the roots from three dimensions {$x_1$}, {$x_2$}, {$x_3$}, but the root system is two-dimensional and so we can draw it in two dimensions. A nice way to think about this is to let {$x_1$}, {$x_2$}, {$x_3$} refer to three adjacent faces of a cube. In coordinates, we can write (1, 0, 0), (0, 1, 0), (0, 0, 1) for the centers of these faces and (-1, 0, 0), (0, -1, 0), (0, 0, -1) for the centers of the opposite faces. We have six faces. Any two faces meet at an edge and there are twelve edges. The midpoints of the edges are given by pairs of coordinates plus or minus one as shown. Any three faces meet at a vertex, which is to say, a corner, and there are eight vertices given by triples of coordinates plus or minus one. Let us stand our cube on the vertex so that (-1,-1,-1) is at the very bottom and (1, 1, 1) is at the very top. Then we see that three edges come together at the bottom and they have pairs of negative coordinates. They come from three vertices, namely, those which have two negative coordinates and one positive coordinate. Going upwards, each vertex gives rise to two edges, so at this level there are six edges in all. The midpoints of these edges all lie in the same plane. The coordinates of these midpoints are those which have one positive coordinate and one negative coordinate. Thus these are the six roots {$x_1-x_2$}, {$x_1-x_3$}, {$x_2-x_3$} and also their negatives. This is the plane which cuts the cube in half. Going upwards further, we get the same structure but in the opposite order. Two edges come together in a vertex, and that vertex has two positive coordinates and one negative coordinate, and there are three ways of doing that. And those three vertices come together in the highest vertex, (1, 1, 1).

Thus we see the relationship between the two-dimensional root system {$A_2$}, which is built out of the actions {$x_1-x_2$}, {$x_1-x_3$}, {$x_2-x_3$}, and the three-dimensional cube system, which is built out of the states {$x_1$}, {$x_2$}, {$x_3$}. We can think of the action {$x_1-x_3$} as composed from two basic actions, {$x_1-x_2$} and {$x_2-x_3$}, known as the simple roots. We also include their negatives, their inverse actions, to get six roots in all. If we allow for unlimited combinations of these actions, then we will get an infinite hexagonal lattice. The root system is like the rule set for generating that lattice. It is highly symmetric. The crucial thing for us to note is that there is a 120 degree angle between the roots {$x_1-x_2$} and {$x_2-x_3$}.

We can calculate this angle using the coordinates, which are vectors. The inner product is negative one. Each length is the square root of two. Multiplying the lengths together gives two. Thus the cos of theta is negative one over two. Theta is 120 degrees. This is the angle that links together the two adjacent nodes in a Dynkin diagram. We can imagine how this proceeds. The simple root {$x_2-x_3$} can link with a new simple root {$x_3-x_4$} and the angle between them will likewise be 120 degrees. But we will have trouble imagining this because the new simple root is now taking part in a fourth dimension.

Nevertheless, we can imagine this in two dimensions if we keep relabeling our root system {$A_2$} as we rotate it by 120 degrees. This is actually a helpful way to think about it because it emphasizes the simplicity. The simple roots are linked by 120 degrees in a chain and related to each other by these root systems {$A_2$}.

Imagining A3

However, we can also imagine this in three dimensions to see more concretely how {$x_1-x_2$} and {$x_2-x_3$} are separated by 120 degrees and {$x_2-x_3$} and {$x_3-x_4$} are related by 120 degrees but the roots {$x_1-x_2$} and {$x_3-x_4$} are related by 90 degrees because they are not adjacent. The latter fact is clear from the angle formula. The inner product is 0. Cos theta equals 0 means that theta is 90 degrees.

{$x_1-x_2$}, {$x_2-x_3$}, {$x_3-x_4$} are the simple roots of a three-dimensional root system. How can we draw that in three dimensions? Let's call them {$a$}, {$b$}, {$c$}. We know that {$a$} and {$c$} are orthogonal, so let's assign them coordinates (1, 0, 0) and (0, 0, 1). Let's solve for the possible values of {$b=(b_1, b_2, b_3)$}. We know that angle between {$a$} and {$b$} is 120 degrees and likewise between {$b$} and {$c$}. Applying the angle formula, cos 120 degrees is negative one half. The inner product of a and b is {$b_1$}. So negative one half equals {$b_1$} over the length of B. {$b_1$} equals negative one half times the length of B. Similarly, {$b_3$} equals negative one half times the length of B. We just need to get {$b_2$} and we can do that by computing the length squared which is the length squared over 4 plus {$b_2$} squared plus the length squared over 4. Combining we see that {$b_2$} squared is the total length squared over 2. That means that {$b_2$} is plus or minus the square root of 2 over 2 times the length of B. And let's say the length of B is 1.

Then let's take a look at these lengths. If the length from the origin to the midpoint of an edge is one, then the length of the edge itself is square root of 2. And half that length is the square root of 2 over 2, which takes us from the origin to the center of a face. So our root is taking us to the center of a face. And the diagonal across the face has length 2. So one quarter of that is one-half. If we go one-half up and one-half across then we end up at the midpoint of this edge, indeed.

Let's check out what angle this root makes with the plane. We simply knock out the vertical component and use the angle formula. We get that cosine of theta is square root of three over two, which means that the angle is sixty degrees. Yes, that root gets pretty steep.

Yes, I see that also when I stand my doddle on the third root then the second root is steep like that, too.

Harmony

You twirl your diddle and you twirl your daddle. I go two step back around, two step upside down with my tumbling dice.

Who says God doesn't play with dice!

I like your diddle and I like your daddle. I am sorry! I admit I was wrong. Diddle daddle ain't no fiddle faddle. I say from now on we call the {$A_2$} root system the diddle. And we extend it with the daddle. So we make chains of diddle daddle diddle daddle diddle daddle. And we call the {$A_3$} root system the Doddle.

I will study your tumbling dice. There are two ways, left and right, that we can two step back around, two step upside down. Can this yield an eight-fold Bott periodicity?

I don't know about that. But we need to understand the Dynkin diagram treasure map. Don't you see! They say X marks the spot. But really Diddle marks the spot. And the edge is a Daddle that takes us from one Diddle to another Diddle. And together they make up a Doddle.

So we need to understand what the Diddle and the Daddle and the Doddle mean as regards Lie algebras and Lie groups and geometry.

Then we'll understand Math 4 Wisdom. Did you like this exploration? Well then, like iŧ! Do you want to see us again? Why don't you hit the Subscribe button? You want homework? Make a model of a doddle. Do you have an attitude? Leave your comments, speak your mind! You're a supportive person! Support us through Patreon. Thank you from Math 4 Wisdom!

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Earthling explains in terms of {$e_1$}.

Martian explains in terms of {$f_1 = e_1-e_2$}. What would all the formulas look like in terms of quadrays?

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Lie theory is the study of groups that are also differentiable manifolds. A group is a system of actions which has four rules. The first rule is that combining two actions, one after the other, yields a third action. The order in which you combine them may or may not matter depending on the group. The second rule is that there is a unique action which does nothing, which is to say, combining it with any other action just keeps that action the same. The third rule is that every action can be undone by its inverse action. Combining an action with its inverse action yields the do nothing action. The fourth rule is that in combining three actions, it doesn't matter if we start by combining the first and the second, or start by combining the second and the third, as it will yield the same result. We don't need parentheses. Groups are great for expressing symmetries. A Lie group can express continuous symmetries. A Lie group is a group whose actions can be understood as laid out in a differentiable manifold, which is to say, a smooth surface, such as a line or circle or plane or sphere. For example, we can associate each point of a circle with a rotation which takes us to that point. We can associate with each point on a line with a translation that moves us to that point. That's the basic idea. The group of translations may go off to infinity, as with a line, in which case it is not compact. The group of rotations, as for a circle, turns back on itself and does not go to infinity and so is called a compact group. A Lie group may be an n-dimensional vector space over the real numbers, but also over the complex numbers, and even the quaternions, if we are careful to explain what we mean by that. It may be a vector space over the reals even though we may use complex numbers to define the elements, which is to say, it gets tricky.

But let's study our model further and make some related calculations. For that, we should start learning about Lie theory, which is named after the Norwegian mathematician Sophus Lie.

Lie theory tells us what n-dimensional space can look like, as regards geometry. Basically, there are four infinite families, but the one that matters today is the most basic, the plain vanilla, which is the special linear group and, in its compact real form, is the special unitary group. The Dynkin diagram {$A_1$} has one node which signifies one dimension where there is one pair of positive and negative vectors, called roots. The Dynkin diagram {$A_2$} has two nodes which signify two dimensions in which there are three pairs of roots, for a total of six roots. The Dynkin diagram {$A_3$} has three nodes which signify three dimensions in which there are six pairs of roots, for a total of twelve roots. We can keep going but these three little ones are plenty for us to imagine the whole series {$A_n$} and practically all the other nodes in all the other Dynkin diagrams to boot.

There is an elegant way of writing out the roots by which we can think of them as encoding actions. For example, the two roots of {$A_1$} can be written as {$x_1-x_2$} and {$x_2-x_1$}. We can think of 1 and 2 as states. Then the action {$x_1-x_2$} takes us from state 2 to state 1 and the reverse action {$x_2-x_1$} takes us from state 1 to state 2. We can draw this one-dimensional root system using the two-dimensional plane with axes {$x_1$} and {$x_2$}.

It is the pinnacle of Lie theory, which dates back to

This is where we start if we want to integrate all of geometry, if we want to understand, what kind of geometry is possible and what it all means.

Starting in 18

Integrate geometry

X marks the spot

Asterisk marks the spot

Treasure map