This Week’s Finds – Lecture 10

30 November, 2022

This Thursday is the last of my lectures on This Week’s Finds… until they resume next September. As usual we’ll meet in Room 6206 of the James Clerk Maxwell Building, home of the Department of Mathematics of the University of Edinburgh. And as usual you can attend via Zoom:

https://ed-ac-uk.zoom.us/j/82270325098
Meeting ID: 822 7032 5098
Passcode: Yoneda36

Later a video of the talk will appear on my YouTube channel.

I’ll talk about quaternions in physics and Dyson’s ‘three-fold way’: the way the real numbers, complex numbers and quaternions interact. For details, try my paper Division algebras and quantum theory.

One cute fact is how an electron is like a quaternion! More precisely: how quaternions show up in the spin-1/2 representation of SU(2) on β„‚Β².

Let me say a little about that here.

We can think of the group SU(2) as the group of unit quaternions: namely, π‘ž with |π‘ž| = 1. We can think of the space of spinors, β„‚Β², as the space of quaternions, ℍ. Then acting on a spinor by an element of SU(2) becomes multiplying a quaternion on the left by a unit quaternion!

But what does it mean to multiply a spinor by 𝑖 in this story? It’s multiplying a quaternion on the right by the quaternion 𝑖. Note: this commutes with left multiplications by all unit quaternions.

But there are some subtleties here. For example: multiplying a quaternion on the right by 𝑗 also commutes with left multiplication by unit quaternions. But 𝑗 anticommutes with 𝑖:

𝑖𝑗 = βˆ’π‘—π‘–

So there must be an ‘antilinear’ operator on spinors which commutes with the action of SU(2): that is, an operator that anticommutes with multiplication by 𝑖. Moreover this operator squares to -1.

In physics this operator is usually called ‘time reversal’. It reverses angular momentum.

You should have noticed something else, too. Our choice of right multiplication by 𝑖 to make the quaternions into a complex vector space was arbitrary: any unit imaginary quaternion would do! There was also arbitrariness in our choice of 𝑗 to be the time reversal operator.

So there’s a whole 2-sphere of different complex structures on the space of spinors, all preserved by the action of SU(2). And after we pick one, there’s a circle of different possible time reversal operators!

So far, all I’m saying is that quaternions help clarify some facts about the spin-1/2 particle that would otherwise seem a bit mysterious or weird.

For example, I was always struck by the arbitrariness of the choice of time reversal operator. Physicists usually just pick one! But now I know it corresponds to a choice of a second square root of -1 in the quaternions, one that anticommutes with our first choice: the one we call 𝑖.

At the very least, it’s entertaining. And it might even suggest some new things we could try: like ‘gauging’ time reversal symmetry (changing its definition in a way that depends on where we are), or even gauging the complex structure on spinors.


This Week’s Finds – Lecture 9

29 November, 2022

 

In this talk I explained the quaternions and octonions, and showed how to multiply them using the dot product and cross product of vectors.

For more details, including a proof that octonion multiplication obeys |ab|=|a||b|, go here:

Octonions and the Standard Model (Part 2).

This was one of a series of lectures based on my column This Week’s Finds.

 

 


This Week’s Finds – Lecture 8

29 November, 2022

 

In this talk I explained the E8 root lattice and how it gives rise to the ‘octooctonionic projective plane’, a 128-dimensional manifold on which the compact Lie group called E8 acts as symmetries. I also discussed how some special root lattices give various notions of ‘integer’ for the real numbers, complex numbers, quaternions and octonions.

For more, read my paper Coxeter and Dynkin diagrams.

This was one of a series of lectures based on my column This Week’s Finds.


Mathematics for Humanity

27 November, 2022

I’m working with an organization that may eventually fund proposals to fund workshops for research groups working on “mathematics for humanity”. This would include math related to climate change, health, democracy, economics, AI, etc.

I can’t give details unless and until it solidifies.

However, it would help me to know a bunch of possible good proposals. Can you help me imagine some?

A good proposal needs:

• a clearly well-defined subject where mathematics is already helping humanity but could help more, together with

• a specific group of people who already have a track record of doing good work on this subject, and

• some evidence that having a workshop, maybe as long as 3 months, bringing together this group and other people, would help them do good things.

I’m saying this because I don’t want vague ideas like “oh it would be cool if a bunch of category theorists could figure out how to make social media better”.

I asked for suggestions on Mathstodon and got these so far:

figuring out how to better communicate risks and other statistical information,

developing ways to measure and combat gerrymandering,

improving machine learning to get more reliable, safe and clearly understandable systems,

studying tipping points and ‘tipping elements’ in the Earth’s climate system,

creating higher-quality open-access climate simulation software,

using operations research to disrupt human trafficking.

Each topic already has people already working on it, so these are good examples. Can you think of more, and point me to groups of people working on these things?


The Icosidodecahedron

14 November, 2022

The icosidodecahedron can be built by truncating either a regular icosahedron or a regular dodecahedron. It has 30 vertices, one at the center of each edge of the icosahedron—or equivalently, one at the center of each edge of a dodecahedron. It is a beautiful, highly symmetrical shape. But it is just a shadow of a more symmetrical shape with twice as many vertices, which lives in a space with twice as many dimensions! Namely, it is a projection down to 3d space of a 6-dimensional polytope with 60 vertices.

Even better, it is also a slice of a more symmetrical 4d polytope with 120 vertices, which in turn is the projection down to 4d space of an even more symmetrical 8-dimensional polytope with 240 vertices: the so-called ‘E8 root polytope’. Note how the numbers keep doubling: 30, 60, 120 and 240.

To understand all this, start with the group of rotational symmetries of the icosahedron. This is a 60-element subgroup of the rotation group SO(3), so it has a double cover, called the binary icosahedral group, consisting of 120 unit quaternions. With a suitable choice of coordinates, we can take these to be

\displaystyle{ \pm 1 , \quad \frac{\pm 1 \pm i \pm j \pm k}{2}, \quad \frac{\pm i \pm \phi j \pm \Phi k}{2} }

together with everything obtained from these by even permutations of 1, i, j, and k, where

\displaystyle{ \phi = \frac{\sqrt{5} - 1}{2}, \quad \Phi = \frac{\sqrt{5} + 1}{2} }

are the ‘little’ and ‘big’ golden ratios, respectively. These 120 unit quaternions are the vertices of a convex polytope in 4 dimensions. In fact this is a regular polytope, called the 600-cell since it has 600 regular tetrahedra as faces.

If we slice the 600-cell with halfway between two of its opposite vertices, we get an icosidodecahedron. This is easiest to see by intersecting the 600-cell with the space of purely imaginary quaternions

\{ ai + bj + ck : \; a,b,c \in \mathbb{R} \}

Of the 600-cell’s vertices, those that lie in this 3-dimensional space are

\pm i, \pm j, \pm k

which form the corners of an octahedron, and

\displaystyle{ \frac{\pm i \pm \phi j \pm \Phi k}{2} ,  \quad  \frac{\pm j \pm \phi k \pm \Phi i}{2} , \quad  \frac{\pm k \pm \phi i \pm \Phi j}{2}   }

which form the corners of three ‘golden boxes’. A golden box is the 3d analogue of a golden rectangle: its three sides are in the proportions \phi, 1 and \Phi.

It is well-known that these points are the vertices of an icosidodecahedron. Here are the three golden boxes and octahedron inscribed in an icosidodecahedron, as drawn by Rahul Narain:

But we are not done with the binary icosahedral group—far from it!

Integer linear combinations of these 120 elements of the quaternions form a subring of the quaternions, which Conway and Sloane [CS] call the icosians. Since any icosian can be written as a + bi + cj + dk where the numbers a,b,c,d \in \mathbb{R} are of the form x + y \sqrt{5} with x,y rational, any icosian gives an 8-tuple of rational numbers. However, we do not get all 8-tuples of rationals this way, only those lying in a certain lattice in \mathbb{R}^8. And there is a way to think of this lattice as a rescaled copy of the famous E8 lattice! To do this, Conway and Sloane put a new norm on the icosians as follows. The usual quaternionic norm is

\|a + bi + cj + dk\|^2 = a^2 + b^2 + c^2 + d^2

But for an icosian this norm is always of the form x + \sqrt{5} y for some rationals x and y. Conway and Sloane define a new norm on the icosians by setting

|a + bi + cj + dk|^2 = x + y

With this new norm, Conway and Sloane show the icosians are isomorphic to a rescaled version of the E8 lattice in \mathbb{R}^8.

The 240 shortest nonzero vectors in this lattice are the vertices of an 8-dimensional convex polytope called the E8 root polytope:

However, if we remember that each of these 240 vectors came from a quaternion, we can also think of them as 240 quaternions. These turn out to be the vertices of two 600-cells in the quaternions! In the usual quaternionic norm, one of these 600-cells is larger than the other by a factor of \Phi.

In fact, there is an orthogonal projection from \mathbb{R}^8 down to \mathbb{R}^4 that maps the E8 root polytope to the 600-cell. So, in a very real sense, the 600-cell is the ‘shadow’ of a polytope with twice as many vertices, living in a space whose dimension is twice as large. And as a spinoff, this fact gives the same sort of relationship between the icosidodecahedron and a 6-dimensional polytope.

The key is to look at pure imaginary icosians: those of the form a i + b j + c k for real a,b,c. Since a,b and c are each of the form x + \sqrt{5}y with x and y rational, any pure imaginary icosian gives a 6-tuple of rational numbers. We do not get all 6-tuples of rationals this way, but only those lying in a certain lattice. We have

\|ai + bj + ck\|^2 = a^2 + b^2 + c^2

For a pure imaginary icosian this is always of the form x + \sqrt{5} y for some rationals x and y. So, we can define a new norm on the pure imaginary icosians by

|ai + bj + ck|^2 = x + y

With this new norm, the pure imaginary icosians are isomorphic to a rescaled version of a familiar lattice in \mathbb{R}^6, called the ‘D6 lattice’.

The 60 shortest nonzero vectors in the D6 lattice are called the roots of D6, and they are the vertices of a 6-dimensional convex polytope called the D6 root polytope. There is an orthogonal projection from \mathbb{R}^6 to \mathbb{R}^3 that maps this polytope to an icosidodecahedron. In fact 30 vertices of the D6 root polytope map to the vertices of this icosidodecahedron, while the other 30 map to vertices of a second, smaller icosidodecahedron.

Here is an image of the setup, created by Greg Egan:

Let’s see some details! The usual coordinatization of the D6 lattice in Euclidean \mathbb{R}^6 is

\mathrm{D}_6 = \left\{ (x_1, \dots, x_6) : \; x_i  \in \mathbb{Z}, \; \sum_i x_i \in 2\mathbb{Z} \right\} \subset \mathbb{R}^6

The roots of D6 are

(\pm 1, \pm 1, 0, 0, 0, 0)

and all vectors obtained by permuting the six coordinates. We shall see that these vectors are sent to the vertices of an icosidodecahedron by the linear map T \colon  \mathbb{R}^6 \to \mathbb{R}^3 given as a 3 × 6 matrix by

\left( \begin{array}{cccccc}  \Phi &  \Phi  & -1 & -1 & 0 &  0 \\  0 &  0  & \Phi &  -\Phi & -1 & 1 \\  -1 &  1 &  0 &  0 &  \Phi  & \Phi  \end{array} \right)

The rows of this matrix are orthogonal, all with the same norm, so after rescaling it by a constant factor we obtain an orthogonal projection. The columns of this matrix are six vertices of an icosahedron, chosen so that we never have a vertex and its opposite. For any pair of columns, they are either neighboring vertices of the icosahedron, or a vertex and the opposite of a neighboring vertex.

The map T thus sends any D6 root to either the sum or the difference of two neighboring icosahedron vertices. In this way we obtain all possible sums and differences of neighboring vertices of the icosahedron. It is easy to see that the sums of neighboring vertices give the vertices of an icosidodecahedron, since by definition the icosidodecahedron has vertices at the midpoints of the edges of a regular icosahedron. It is less obvious that the differences of neighboring vertices of the icosahedron give the vertices of a second, smaller icosidodecahedron. But thanks to the symmetry of the situation, we can check this by considering just one example. In fact the vectors defining the vertices of the larger icosidodecahedron turn out to be precisely \Phi times the vectors defining the vertices of the smaller one!

The beauties we have just seen are part of an even larger pattern relating all the non-crystallographic Coxeter groups to crystallographic Coxeter groups. For more, see the work of Fring and Korff [FK1,FK2], Boehm, Dechant and Twarock [BDT] and the many papers they refer to. Fring and Korff apply these ideas to integrable systems in physics, while the latter authors explore connections to affine Dynkin diagrams. For more relations between the icosahedron and E8, see [B2].

Acknowledgements

I thank Greg Egan for help with developing these ideas. The spinning icosidodecahedron was created by Cyp and was put on Wikicommons with a Creative Commons Attribution-Share Alike 3.0 Unported license. The 600-cell was made using Robert Webb’s Stella software and is on Wikicommons. The icosidodecahedron with three golden boxes and an octahedron inscribed in it was created by Rahul Narain on Mathstodon. The projection of the 240 E8 roots to the plane was created by Claudio Rocchini and put on Wikicommons with a Creative Commons Attribution 3.0 Unported license. The spinning pair of icosidodecahedra was created by Greg Egan and appears in an earlier blog article on this subject [B1]. The article here is an expanded version of that earlier article: the only thing I left out is the matrix describing a linear map S \colon \mathbb{R}^8 \to \mathbb{R}^4 that when suitably rescaled gives a projection mapping the E8 lattice in its usual coordinatization

\{ x \in \mathbb{R}^8: \, \textrm{all } x_i \in \mathbb{Z} \textrm{ or all } x_i \in \mathbb{Z} + \frac{1}{2} \textrm{ and } \sum_i x_i \in 2\mathbb{Z} \}

to the icosians, and thus mapping the 240 E8 roots to two 600-cells. For completeness, here is that matrix:

\left( \begin{array}{cccccccc}  \Phi+1 & \Phi -1 & 0  & 0 &  0 &  0 &   0  & 0 \\  0 & 0 & \Phi &  \Phi  & -1 & -1 & 0 & 0 \\  0 & 0  & 0 &  0  & \Phi &  -\Phi & -1 & 1   \\  0 & 0 & -1 &  1 &  0 &  0 &  \Phi  & \Phi  \end{array} \right)

The first image at the bottom of this post was also created by Greg Egan, on Mathstodon. The second shows an icosahedron and 3 golden rectangles morphing to an icosidodecahedron with 3 golden boxes, with an octahedron present at every stage. It was created by Vincent Pantaloni, on Geogrebra.

References

[B1] John Baez, Icosidodecahedron from D6, Visual Insight, January 1, 2015.

[B2] John Baez, From the icosahedron to E8, London Math. Soc. Newsletter 476 (2018), 18–23.

[BDT] Celine Boehm, Pierre-Philippe Dechant and Reidun Twarock, Affine extensions of non-crystallographic Coxeter groups induced by projection, J. Math. Phys. 54, 093508 (2013).

[CS] John H. C. Conway and Neil J. A. Sloane, Sphere Packings, Lattices and Groups, Springer, Berlin, 2013.

[FK1] Andreas Fring and Christian Korff, Affine Toda field theories related to Coxeter groups of non-crystallographic type, Nucl. Phys. B729 (2005), 361–386.

[FK2] Andreas Fring and Christian Korff, Non-crystallographic reduction of generalized Calogero–Moser models, J. Phys. A39 (2006) 1115–1132.


This Week’s Finds – Lecture 7

10 November, 2022

 

Coxeter and Dynkin diagrams classify some of the most beautiful objects in mathematics. In this talk I went through all the connected Dynkin diagrams and say how they correspond to compact simple Lie group— which happen to be act as symmetries of geometrical structures built using the real numbers, complex numbers, quaternions and octonions!

For more, read my paper Coxeter and Dynkin diagrams.

This was one of a series of lectures based on my column This Week’s Finds.


Categories and Epidemiology

1 November, 2022

I gave a talk about my work using category theory to help design software for epidemic modeling:

Category theory and epidemiology, African Mathematics Seminar, Wednesday November 2, 2022, 3 pm Nairobi time or noon UTC. Organized by Layla Sorkatti and Jared Ongaro.

This talk was a lot less technical than previous ones I’ve given on this subject, which were aimed mainly at category theorists! You can see it here:

Abstract. Category theory provides a general framework for building models of dynamical systems. We explain this framework and illustrate it with the example of “stock and flow diagrams”. These diagrams are widely used for simulations in epidemiology. Although tools already exist for drawing these diagrams and solving the systems of differential equations they describe, we have created a new software package called StockFlow which uses ideas from category theory to overcome some limitations of existing software. We illustrate this with code in StockFlow that implements a simplified version of a COVID-19 model used in Canada. This is joint work with Xiaoyan Li, Sophie Libkind, Nathaniel Osgood and Evan Patterson.

Check out these papers for more:

• John Baez, Xiaoyan Li, Sophie Libkind, Nathaniel Osgood and Evan Patterson, Compositional modeling with stock and flow diagrams.

• Andrew Baas, James Fairbanks, Micah Halter, Sophie Libkind and Evan Patterson, An algebraic framework for structured epidemic modeling.

For some more mathematical talks on the same subject, go here.


This Week’s Finds – Lecture 6

28 October, 2022

 

Coxeter and Dynkin diagrams classify some of the most beautiful objects in mathematics. Here I use Dynkin diagrams to classify compact Lie groups—and especially compact semisimple Lie groups.

This is one of a series of lectures at the University of Edinburgh on topics drawn from my column This Week’s Finds:

http://math.ucr.edu/home/baez/twf/

Cover image by Tom Ruen, CC BY-SA 4.0


This Week’s Finds – Lecture 5

24 October, 2022

 

Coxeter and Dynkin diagrams classify some of the most beautiful objects in mathematics. Here I explain how Dynkin diagrams classify root lattices.

For more, read my paper “Coxeter and Dynkin diagrams” here:

http://math.ucr.edu/home/baez/twf/

To attend the talks on Zoom go here.


This Week’s Finds – Lecture 4

20 October, 2022

 

Coxeter and Dynkin diagrams classify some of the most beautiful objects in mathematics. Here I explain how Coxeter groups classify finite reflection groups: that is, finite groups of transformations of \mathbb{R}^n generated by reflections.

For more, read my paper “Coxeter and Dynkin diagrams” here:

http://math.ucr.edu/home/baez/twf/

To attend the talks on Zoom go here.