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 โยฒ.

Let me say a little about that here.

We can think of the group SU(2) as the group of unit quaternions: namely, ๐ with |๐| = 1. We can think of the space of spinors, โยฒ, as the space of quaternions, โ. 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 ๐ in this story? It’s multiplying a quaternion on the *right* by the quaternion ๐. 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 ๐ *also* commutes with left multiplication by unit quaternions. But ๐ anticommutes with ๐:

๐๐ = โ๐๐

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 ๐. 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 ๐ 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 ๐ 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 ๐.

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.

]]>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*.

In this talk I explained the E_{8} root lattice and how it gives rise to the ‘octooctonionic projective plane’, a 128-dimensional manifold on which the compact Lie group called E_{8} 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*.

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 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 ‘E_{8} 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

together with everything obtained from these by even permutations of and where

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**

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

which form the corners of an octahedron, and

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 1 and

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 where the numbers are of the form with 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 And there is a way to think of this lattice as a rescaled copy of the famous E_{8} lattice! To do this, Conway and Sloane put a new norm on the icosians as follows. The usual quaternionic norm is

But for an icosian this norm is always of the form for some rationals and Conway and Sloane define a new norm on the icosians by setting

With this new norm, Conway and Sloane show the icosians are isomorphic to a rescaled version of the E_{8} lattice in

The 240 shortest nonzero vectors in this lattice are the vertices of an 8-dimensional convex polytope called the **E _{8} 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

In fact, there is an orthogonal projection from down to that maps the E_{8} 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 for real Since and are each of the form with and 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

For a pure imaginary icosian this is always of the form for some rationals and So, we can define a new norm on the pure imaginary icosians by

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

The 60 shortest nonzero vectors in the D_{6} lattice are called the **roots** of D_{6}, and they are the vertices of a 6-dimensional convex polytope called the **D _{6} root polytope**. There is an orthogonal projection from to that maps this polytope to an icosidodecahedron. In fact 30 vertices of the D

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

Let’s see some details! The usual coordinatization of the D_{6} lattice in Euclidean is

The roots of D_{6} are

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 given as a 3 × 6 matrix by

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 thus sends any D_{6} 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 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 E_{8}, see [B2].

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 E_{8} 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 that when suitably rescaled gives a projection mapping the E_{8} lattice in its usual coordinatization

to the icosians, and thus mapping the 240 E_{8} roots to two 600-cells. For completeness, here is that matrix:

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.

[B1] John Baez, Icosidodecahedron from D_{6}, *Visual Insight*, January 1, 2015.

[B2] John Baez, From the icosahedron to E_{8}, *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.

Sound is vibrations in air. Start with some note on the piano. Then play another note that vibrates 3/2 times as fast. Do this 12 times. Since

(3/2)ยนยฒ โ 128 = 2โท

when you’re done your note vibrates about 2โท times as fast as when you started!

Notes have letter names, and two notes whose frequencies differ by a power of 2 have the same letter name. So the notes you played form a 12-pointed star:

Each time you increase the frequency by a factor of 3/2 you move around the points of this star: from C to G to D to A, and so on. Each time you move about 7/12 of the way around the star, since

log(3/2) / log(2) โ 7/12

This is another way of stating the approximate equation I wrote before!

It’s great! It’s called the circle of fifths, for reasons that don’t need to concern us here.

But this pattern is just approximate! In reality

(3/2)ยนยฒ = 129.746…

not 128, and

log(3/2) / log(2) = 0.58496…

not 7/12 = 0.58333… So the circle of fifths does not precisely close:

The failure of it to precisely close is called the Pythagorean comma, and you can hear the problem here:

This video plays you notes that increase in frequency by a factor of 3/2 each time, and finally two notes that differ by the Pythagorean comma: they’re somewhat out of tune.

People have dealt with this in many, many ways. No solution makes everyone happy.

For example, the equal-tempered 12-tone scale now used on most pianos doesn’t have ‘perfect fifths’—that is, frequency ratios of 3/2. It has frequency ratios of

I have tried in this blog article to be understandable by people who don’t know standard music theory terminology—basic stuff like ‘octaves’ and ‘fifths’, or the letter names for notes. But the circle of fifths is very important for people who *do* know this terminology. It’s a very practical thing for musicians, for example if you want to remember how many sharps or flats there are in any key. Here’s a gentle introduction to it by Gracie Terzian:

Here she explains some things you can do with it:

Here’s another version of the circle of fifths made by “Just plain Bill”>—full of information used by actual musicians:

If you watch Terzian’s videos you’ll learn what all this stuff is about.

]]>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*.

The most fundamental of these is the **natural minor scale**. The C major scale goes

C D E F G A B C

The C natural minor scale goes

C D Eโญ F G Aโญ Bโญ C

As you can see the 3rd, 6th and 7th notes of the scale are ‘flatted’: moved down a half-tone compared to the major scale. This gives the natural minor scale a darker, even ‘sadder’ quality compared to the major scale.

I prefer to work with note numbers instead of note names, not because I’m a mathematician so I love numbers, but because then we can simultaneously talk about different keys at once, not just the key of C. In this approach we call the notes of the major scale

1 2 3 4 5 6 7 8

and then the natural minor scale is

1 2 โญ3 4 5 โญ6 โญ7 8

Don’t ask me why the flats are written in front of the numbers now instead of after them—it’s just a convention.

Now, one thing about ‘common practice’ western harmony is the 7th tone plays a special role. It’s just a half-step below the 8, and we act like that dissonance makes it want very strongly to go up to the 8. The 8 is one octave above the 1, twice the frequency. Either the 1 or 8 instantly serves as a home base: we feel like a piece or passage is done, or momentarily at peace, when we play these notes. We say the 7 wants to ‘resolve’ to the 8, and we call it the ‘leading-tone’ for this reason: it suggests that we’ve almost reached the tonic, and makes us want to get there!

There’s much more we could say here, but it all combines to make people want a scale that’s like minor but contains the 7 instead of the โญ7. And since this scale is motivated by reasons of harmony theory, it’s called the harmonic minor scale. It goes like this:

1 2 โญ3 4 5 โญ6 7 8

However, now people singing this scale find it mildly awkward to jump up from โญ6 to the 7 because the distance between them is larger. In fact it’s 3 half-tones, larger than any step in the major or natural minor scale! One way to shrink this gap is to raise the โญ6 to a 6 as well. This gives the melodic minor scale:

1 2 โญ3 4 5 6 7 8

By now we’re almost back to the major scale! The only difference is the flatted 3. However, that’s still a lot: the โญ3 is considered the true hallmark of minorness. There are reasons for this, like the massive importance of the 1 3 5 chord, which serves to pound home the message “we’re back to 1, and this is the major scale, so we are very happy”. Playing 1 โญ3 5 says “we’re back to 1, but this is minor, so we are done but we are sad”.

However, singing *up* the scale is different from singing *down* the scale. When we sing *up* the melodic major scale we are very happy to sing the 7 right before the 8, because it’s the leading-tone: it tells us we’re almost home. But when we sing *down* we don’t so much mind plunging from the 8 down to โญ7, and then it’s not so far down to โญ6: these are both steps of a whole tone. If we do this we are singing in the natural minor scale. So what I called ‘melodic minor’ is also called **melodic minor ascending**, while natural minor is also called **melodic minor descending**.

Here I should admit that while this is an oft-told pedagogical story, the actual reality is more complex. Good composers or improvisers use whatever form of minor they want at any given moment! However, most western musicians have heard some version of the story I just told, and that does affect what they do.

To listen to these various forms of the minor scale, and hear them explained more eloquently than I just did, try this:

Grazie Terzian is the patient teacher of music theory I wish I’d had much earlier. You may feel a bit impatient listening to her carefully working through various scales, but that’s because she’s giving you enough time for the information to really sink into your brain!

Anyway: we’ve seen one form of major scale and three forms of minor, one of which has two names. All these scales differ solely in whether or not we flat the 3, 6 or 7. So, we can act like mathematicians and fit them into a cube where the operations of flatting the 3, 6 or 7 are drawn as arrows:

Here to save space I’ve written flatted notes with little superscripts like instead of โญ3: it makes no difference to the meaning.

This chart shows that flatting the 3 pushes our scale into minor territory, while flatting the 6 and then the 7th are ways to further intensify the darkness of the scale. But you’ll also see that we’re just using a few of the available options!

In part 1 I showed you another way to modify the major scale, namely by starting it at various different notes to get different ‘modes’. If we list them in order of the starting note—1, 2, 3, etc.—they look like this:

For example, Ionian is just major. But we saw that it is also very nice to list the modes from the ‘brightest’ to the ‘darkest’. Rob van Hal made a nice chart showing how this works:

Skipping over Lydian, which is a bit of an exception, we start with major—that is, Ionian—and then start flatting more and more notes. When we reach the Phrygian and Locrian we flat the 2 and then the 5, which are very drastic things to do. So these modes have a downright sinister quality. But before we reach these, we pass through various modes that fit into my cube!

Let’s look at them:

We’re now tracing out a different path from top to bottom. **Ionian** has no notes flatted. In **Mixolydian** we flat the 7. In **Dorian** we also flat the 3. Then in **Aeolian** we also flat the 6.

I mentioned that the โญ3 is considered the true hallmark of minorness. Thus, in the classification of modes, those with a flatted 3 are considered ‘minor’ while those without are considered ‘major’. So in our new path from the cube’s top to its bottom, we switch from major to minor modes when we pass from Mixolydian to Dorian.

Note that Ionian is just our old friend the major scale, and Aeolian is our friend the natural minor. We can combine the two cubes I’ve showed you, and see how they fit together:

Now we can get from the top to Dorian following two paths that pass only through scales or modes we’ve seen! Similarly we can get from melodic minor ascending to the bottom following two paths through scales or modes we’ve seen. In general, moving around this cube through the course of a piece provides a lot of interesting ways to subtly change the mood.

But two corners of our cube don’t have names yet! These are more exotic! But of course they *exist*, and are sometimes used in music. The mode

1 2 3 4 5 โญ6 7

is called **harmonic major**, and it’s used in the Beatles’ ‘Blackbird’. The mode

1 2 3 4 5 โญ6 โญ7

is called the **melodic major** scale, or also **Mixolydian flat 6** or **Aeolian dominant**. It’s used in the theme song of the movie *The Mask of Zorro*, called ‘I Want to Spend My Lifetime Loving You’.

So, let’s add these two modes to our cube:

This is the whole enchilada: a ‘commuting cube’, meaning that regardless of which path we take from any point to any other point, we get the same mode in the end. We can also strip it of all the musical names and think of it in a purely mathematical way:

We could go further and study a 5-dimensional hypercube where we also consider the results of flatting the 2 and 5. That would let us include darker and scarier modes like Phrygian, Phrygian dominant and Locrian—but it would be tougher to draw!

]]>It feels a lot like the internet of the late 80’s or early 90’s, with people pitching in to build things they themselves use, not serving as cogs in the giant machine of surveillance capitalism. I invite you to join us!

I plan to write things there, polish them up a bit and put them here. An example is my post about modes. Today I had fun solving a puzzle about sundials. My goal was to use the minimum amount of math.

Colin Beveridge posed this puzzle:

“Suppose I planted a metre-long straight stick vertically in the ground and traced the locus of the end of its shadow. What shape would it make? Happy to assume a locally flat Earth if it makes things easier.”

Equivalently: what curve is traced out by the shadow of the tip of a sundial during one day, if the shadow lands on flat ground?

The answer is: a hyperbola—or in one very special case a straight line!

To see this, work in Earth-centered coordinates and treat the Sun as a point S moving in a circle over the course of a day. Treat the ground as a plane P. Sunlight traces out a line L going from S to the sundial’s tip T and hitting this plane P at some point X.

As S goes around in a circle, what curve does X trace out?

That’s the math question I’m solving.

To solve it, we need an obvious math fact: as a point S goes around a circle, the line going through S and any point T traces out a cone.

And another less obvious but very famous fact: when we intersect a cone with a plane P we get a curve called a ‘conic section’, which can be a circle, ellipse, parabola, hyperbola, or a line.

So, the only question is which of these curves we can actually get!

As the Sun sets, the shadow of our sundial gets arbitrarily long—so we can only get a circle or ellipse if the Sun never sets.

We only get a parabola if the Sun sets in the exact same place on the horizon that it rises—since the two ‘ends’ of a parabola go off to infinity in the same direction.

All these cases are a bit unusual. In most circumstances the curve we get will be a hyperbola or a straight line.

We get a straight line only when the Sun rises at one point on the horizon, is straight overhead at noon, and sets at the opposite point of the horizon. This would happen every day if you lived at the equator and the Earth’s axis wasn’t tilted. But in reality this situation is rare.

So, the shadow traced out by a sundial’s tip is usually a hyperbola!

You can play around with these hyperbola-shaped shadows here:

• Intellegenti Pauca, Hyperbola shadows, Geogebra.

There’s a lot more one can say about this: for example, what happens with the change of seasons? But I wanted to keep this simple!

Click on this picture for some details about a nice sundial that shows off its hyperbolae:

]]>• 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.

]]>