Rolling Circles and Balls (Part 4)

So far in this series we’ve been looking at what happens when we roll circles on circles:

• In Part 1 we rolled a circle on a circle that’s the same size.

• In Part 2 we rolled a circle on a circle that’s twice as big.

• In Part 3 we rolled a circle inside a circle that was 2, 3, or 4 times as big.

In every case, we got lots of exciting math and pretty pictures. But all this pales in comparison to the marvels that occur when we roll a ball on another ball!

You’d never guess it, but the really amazing stuff happens when you roll a ball on another ball that’s exactly 3 times as big. In that case, the geometry of what’s going on turns out to be related to special relativity in a weird universe with 3 time dimensions and 4 space dimensions! Even more amazingly, it’s related to a strange number system called the split octonions.

The ordinary octonions are already strange enough. They’re an 8-dimensional number system where you can add, subtract, multiply and divide. They were invented in 1843 after the famous mathematician Hamilton invented a rather similar 4-dimensional number system called the quaternions. He told his college pal John Graves about it, since Graves was the one who got Hamilton interested in this stuff in the first place… though Graves had gone on to become a lawyer, not a mathematician. The day after Christmas that year, Graves sent Hamilton a letter saying he’d found an 8-dimensional number system with almost all the same properties! The one big missing property was the associative law for multiplication, namely:

The quaternions obey this, but the octonions don’t. For this and other reasons, they languished in obscurity for many years. But they eventually turned out to be the key to understanding some otherwise inexplicable symmetry groups called ‘exceptional groups’. Later still, they turned out to be important in string theory!

I’ve been fascinated by this stuff for a long time, in part because it starts out seeming crazy and impossible to understand… but eventually it makes sense. So, it’s a great example of how you can dramatically change your perspective by thinking for a long time. Also, it suggests that there could be patterns built into the structure of math, highly nonobvious patterns, which turn out to explain a lot about the universe.

About a decade ago I wrote a paper summarizing everything I’d learned so far:

• The octonions.

But I knew there was much more to understand. I wanted to work on this subject with a student. But I never dared until I met John Huerta, who, rather oddly, wanted to get a Ph.D. in math but work on physics. That’s generally not a good idea. But it’s exactly what I had wanted to do as a grad student, so I felt a certain sympathy for him.

And he seemed good at thinking about how algebra and particle physics fit together. So, I decided we should start by writing a paper on ‘grand unified theories’—theories of all the forces except gravity:

• The algebra of grand unified theories.

The arbitrary-looking collection of elementary particles we observe in nature turns out to contain secret patterns—patterns that jump into sharp focus using some modern algebra! Why do quarks have weird fractional charges like 2/3 and -1/3? Why does each generation of particles contain two quarks and two leptons? I can’t say we really know the answer to such questions, but the math of grand unified theories make these strange facts seem natural and inevitable.

The math turns out to involve rotations in 10 dimensions, and ‘spinors’: things that only come around back to the way they started after you turn them around twice. This turned out to be a great preparation for our later work.

As we wrote this article, I realized that John Huerta had a gift for mathematical prose. In fact, we recently won a prize for this paper! In two weeks we’ll meet at the big annual American Mathematical Society conference and pick it up.

John Huerta wound up becoming an expert on the octonions, and writing his thesis about how they make superstring theory possible in 10-dimensional spacetime:

• Division algebras, supersymmetry and higher gauge theory.

The wonderful fact is that string theory works well in 10 dimensions because the octonions are 8-dimensional! Suppose that at each moment in time, a string is like a closed loop. Then as time passes, it traces out a 2-dimensional sheet in spacetime, called a worldsheet:

In this picture, 'up' means 'forwards in time'. Unfortunately this picture is just 3-dimensional: the real story happens in 10 dimensions! Don't bother trying to visualize 10 dimensions, just count: in 10-dimensional spacetime there are 10 – 2 = 8 extra dimensions besides those of the string’s worldsheet. These are the directions in which the string can vibrate. Since the octonions are 8-dimensional, we can describe the string’s vibrations using octonions! The algebraic magic of this number system then lets us cook up a beautiful equation describing these vibrations: an equation that has ‘supersymmetry’.

For a full explanation, read John Huerta’s thesis. But for an easy overview, read this paper we published in Scientific American:

• The strangest numbers in string theory.

This got included in a collection called The Best Writing on Mathematics 2012, further confirming my opinion that collaborating with John Huerta was a good idea.

Anyway: string theory sounds fancy, but for many years I’d been tantalized by the relationship between the octonions and a much more prosaic physics problem: a ball rolling on another ball. I had a lot of clues saying these should be a nice relationship… though only if we work with a mutant version of the octonions called the ‘split’ octonions.

You probably know how we get the complex numbers by taking the ordinary real numbers and throwing in a square root of -1. But there’s also another number system, far less popular but still interesting, called the split complex numbers. Here we throw in a square root of 1 instead. Of course 1 already has two square roots, namely 1 and -1. But that doesn’t stop us from throwing in another!

This ‘split’ game, which is a lot more profound than it sounds at first, also works for the quaternions and octonions. We get the octonions by starting with the real numbers and throwing in seven square roots of -1, for a total of 8 dimensions. For the split octonions, we start with the real numbers and throw in three square roots of -1 and four square roots of 1. The split octonions are surprisingly similar to the octonions. There are tricks to go back and forth between the two, so you should think of them as two forms of the same underlying thing.

Anyway: I really liked the idea of finding the split octonions lurking in a concrete physics problem like a ball rolling on another ball. I hoped maybe this could shed some new light on what the octonions are really all about.

James Dolan and I tried hard to get it to work. We made a lot of progress, but then we got stuck, because we didn’t realize it only works when one ball is 3 times as big as the other! That was just too crazy for us to guess.

In fact, some mathematicians had known about this for a long time. Things would have gone a lot faster if I’d read more papers early on. By the time we caught up with the experts, I’d left for Singapore, and John Huerta, still back in Riverside, was the one talking with James Dolan about this stuff. They figured out a lot more.

Then Huerta got his Ph.D. and took a job in Australia, which is as close to Singapore as it is to almost anything. I got a grant from the Foundational Questions Institute to bring John to Singapore and figure out more stuff about the octonions and physics… and we wound up writing a paper about the rolling ball problem:

• G₂ and the rolling ball.

Whoops! I haven’t introduced G₂ yet. It’s one of those ‘exceptional groups’ I mentioned: the smallest one, in fact. Like the octonions themselves, this group comes in a few different but closely related ‘forms’. The most famous form is the symmetry group of the octonions. But in our paper, we’re more interested in the ‘split’ form, which is the symmetry group of the split octonions. The reason is that this group is also the symmetry group of a ball rolling without slipping or twisting on another ball that’s exactly 3 times as big!

The fact that the same group shows up as the symmetries of these two different things is a huge clue that they’re deeply related. The challenge is to understand the relationship.

There are two parts to this challenge. One is to describe the rolling ball problem in terms of split octonions. The other is to reverse the story, and somehow get the split octonions to emerge naturally from the study of a rolling ball!

In our paper we tackled both parts. Describing the rolling ball problem using split octonions had already been done by other mathematicians, for example here:

• Robert Bryant and Lucas Hsu, Rigidity of integral curves of rank 2 distributions.

• Gil Bor and Richard Montgomery, G₂ and the “rolling distribution”.

• Andrei Agrachev, Rolling balls and octonions.

• Aroldo Kaplan, Quaternions and octonions in mechanics.

We do however give a simpler explanation of why this description only works when one ball is 3 times as big as the other.

The other part, getting the split octonions to show up starting from the rolling ball problem, seems to be new to us. We show that in a certain sense, quantizing the rolling ball gives the split octonions! Very roughly, split octonions can been as quantum states of the rolling ball.

At this point I’ve gone almost as far as I can without laying on some heavy math. In theory I could show you pretty animations of a little ball rolling on a big one, and use these to illustrate the special thing that happens when the big one is 3 times as big. In theory I might be able to explain the whole story without many equations or much math jargon. That would be lots of fun…

… for you. But it would be a huge amount of work for me. So at this point, to make my job easier, I want to turn up the math level a notch or two. And this is a good point for both of us take a little break.

In the next and final post in this series, I’ll sketch how the problem of a little ball rolling on a big stationary ball can be described using split octonions… and why the symmetries of this problem give a group that’s the split form of G₂… if the big ball has a radius that’s 3 times the radius of the little one!

I will not quantize the rolling ball problem—for that, you’ll need to read our paper.

Teaching the Math of Climate Science

When you’re just getting started on simulating the weather, it’s good to start with an aqua-planet. That’s a planet like our Earth, but with no land!

Click on this picture to see an aqua-planet created by H. Miura:

Of course, it’s important to include land, because it has huge effects. Click on this to see what I mean:

This simulation is supposed to illustrate a Madden–Julian oscillation: the largest form of variability in the tropical atmosphere on time scales of 30-90 days! It’s a pulse that moves east across the Indian Ocean and Pacific ocean at 4-8 meters/second. It manifests itself as patches of anomalously high rainfall… but also patches of anomalously low rainfall. Strong Madden-Julian Oscillations are often, but not always, seen 6-12 months before an El Niño starts.

Wouldn’t it be cool if math majors could learn to do simulations like these? If not of the full-fledged Earth, at least of an aqua-planet?

Soon they will.

Climate science at Cal State Northridge

At the huge fall meeting of the American Geophysical Union, I met Helen Steele Cox from the geography department at Cal State Northridge. She was standing in front of a poster describing their new Climate Science Program. They got a ‘NICE’ grant from NASA to develop new courses—where ‘NICE’ means NASA Innovations in Climate Education. This grant also helps them run a seminar every other week where they invite climate scientists and the like from JPL and other nearby places to talk about their work.

What really excited me about this program is that it includes courses designed to teach math majors—and others—the skills needed to go into climate science. Since I’m supposed to be developing the syllabus for an undergraduate ‘Mathematics of the Environment’ course, I’m eager to hear about such things.

She told me to talk to David Klein in the math department there. He used to work on general relativity, but now—like me—he’s gotten interested in climate issues. I emailed him, and he told me what’s going on.

They’ve taught this course twice:

• Phys 595 CL. Mathematics and Physics of Climate Change. Atmospheric dynamics and thermodynamics, radiation and radiative transfer, green-house effect, mathematics of remote sounding, introduction to atmospheric and climate modeling. Syllabus here.

They’ve just finished teaching this one:

• Math 396 CL. Introduction to Mathematical Climate Science. This course in applied mathematics will introduce students to applications of vector calculus and differential equations to the study of global climate. Fundamental equations governing atmospheric dynamics will be derived and solved for a variety of situations. Topics include: thermodynamics of the atmosphere, potential temperature, parcel concepts, hydrostatic balance, dynamics of air motion and wind flows, energy balance, an introduction to radiative transfer, and elementary mathematical climate models. Syllabus here.

In some ways, the most intriguing is the one they haven’t taught yet:

• Math 483 CL. Mathematical Modeling. Possible topics include fundamental principles of atmospheric radiation and convection, two dimensional models, varying parameters within models, numerical simulation of atmospheric fluid flow from both a theoretical and applied setting.

There’s no syllabus it yet, but they want to focus the course on four projects:

1. Modeling a Lorenz dynamical system, using the trajectories as analogies to weather and the attractor as an analogy to climate.

2. Modeling a land-sea breeze.

3. Creating a 2d model of an aqua-planet: that is, one with no land.

4. Doing some projects with EdGCM, a proprietary ‘educational general climate model’.

It would be great to take student-made software and add it to the Azimuth Code Project. If they were well-documented, future generations of students could go ahead and improve on them. And an open-source GCM would be a wonderful thing.

As more and more schools teach climate science—not just to Earth scientists, but also to math and computer science students—this sort of ‘open-source climate modeling software’ should become more and more common.

Some questions:

Do you know other schools that are teaching climate modeling in the math department?

Do you know of efforts to formalize the sharing of open-source climate software for educational purposes?

Game Theory for Undergraduates

Starting in January I’m teaching an introduction to game theory to students who have taken a year of calculus, a quarter of multivariable calculus, but in general nothing else. The syllabus says this course:

Covers two-person zero-sum games, minimax theorem, and relation to linear programming. Includes nonzero-sum games, Nash equilibrium theorem, bargaining, the core, and the Shapley value. Addresses economic market games.

However, I can do what I want, and I’d like to include some evolutionary game theory. Right now I’m rounding up resources to help me teach this course.

Here are three books that are not really suitable as texts for a course of this sort, but useful nonetheless:

• Andrew M. Colman, Game Theory and its Applications in the Social and Biological Sciences, Routledge, London, 1995.

This describes 2-person and multi-person zero-sum and nonzero-sum games, concepts like ‘Nash equilibrium’, ‘core’ and ‘Shapley value’, their applications, and—especially refreshing—empirical evidence comparing the theory of ideal rational players to what actual organisms (including people) do in the real world.

• J. D. Williams, The Compleat Strategyst, McGraw–Hill, New York, 1966.

This old-fashioned book is chatty and personable. It features tons of zero-sum games described by 2×2, 3×3 and 4×4 matrices, analyzed in loving detail. It’s very limited in scope, but a good supply of examples.

• Lynne Pepall, Dan Richards and George Norman, Industrial Organization: Contemporary Theory and Empirical Applications, Blackwell, Oxford, 2008.

This is a book about industrial organizations, antitrust law, monopolies and oligopolies. But it uses a hefty portion of game theory, especially in the chapters on ‘Static games and Cournot competition’, ‘Price competition’, and ‘Dynamic games and first and second movers’. So, I think I can squeeze some nice examples and ideas out of this book to use in my course.

Over on Google+

I also got a lot of help from a discussion on Google+. (If ever it seems a bit too quiet here, visit me over there!)

I want the students to play games in class, and Lee Worden had some great advice on how to do that effectively:

For actually playing in class, I like the black-card, red-card system:

• Charles A. Holt and Monica Capra, Classroom games: a prisoner’s dilemma.

Students can keep their cards at their seats and use them for a whole series of 2-person or n-person games.

I like the double entendre in the title of Holt and Capra’s paper! I also like their suggestion of letting students play for small amounts of money: this would grab their attention and also make it easier to explain what their objective should be.

I’m also considering letting them play for points that improve their grade. But this might be controversial! Maybe if it only has a small effect on their grade?

(Students often ask, when they do badly in a course, what they can do to improve their grade. Usually I just say “learn the material and get good at solving problems!” But now I could say “let’s play a game. If you win, I’ll add 5 points to your class score. If you lose….”)

Vincent Knight gave a nice long reply:

I teach game theory in our MSc program and can suggest two “games” that can be played in class:

• Your comic suggests it already, the 2/3rds of the average game. I use that in class and play it twice, once before rationalising it and once after. In the meantime I get TAs to put the results in to a google spreadsheet and show the distribution of guesses to the students. The immediate question is: “what would happen if we played again and again”. This brings up ideas of convergence to equilibria.

• The second game I play with students is an Iterated Prisoner’s dilemma. I separate the whole class (40 students) in to 4 teams and play a round robin tournament of 5 rounds. Specifying that the goal is to minimise total “years in prison” (and not the number of duels won). This often throws up a coalition or two at the end which is quite cool.

I don’t only use the above on our MSc students but also at outreach events and I’ve written a series of blog posts about it:

• School kids: http://goo.gl/5u6Ic

• PhD students: http://goo.gl/6rkOt

• Conference delegates: http://goo.gl/JGWM7

• MSc students: http://goo.gl/oHoz0

The slides I use for the outreach event are available here: http://goo.gl/vJVWV. They include some cool videos (that have certainly made the rounds). I use some of that in the class itself.

I’m also in the middle of a teaching certification process called pcutl and my first module portfolio is available here: http://goo.gl/NhJYg. There’s a lot more stuff then you might care about in there but towards the end is a lesson plan as well as a reflection about how the session went with the students. There are some pics of the session (with the students up and playing the game) here: http://goo.gl/wBZwC.

The notes that I use are in the above portfolio but here is my page on game theory which contains the notes I use on the MSc course (which only has the time to go in to normal form games) and also some videos and Sage Mathematical Software System code: http://goo.gl/RXr1k.

Here are 3 videos I put together that I get my students to watch:

• Normal form games and mixed equilibria: http://goo.gl/dBtDK

• Routing games (Pigou’s example): http://goo.gl/807G4

• Cooperative games (Shapley Value): http://goo.gl/Pzf1F

Finally (I really do apologise for the length of this comment), here are some books I recommend:

• Webb’s Game Theory (in my opinion written for mathematicians): http://goo.gl/2M83l

• Osborne’s Introduction to Game Theory (a very nice and easy to read text): http://goo.gl/FXbcd

• Rosenthal’s A Complete Idiot’s Guide to Game Theory (this is more of a bedside read, that could serve as an introduction to game theory for a non mathematician): http://goo.gl/PCs76

I’m actually going to be writing a new game theory course for final year undergraduates next year and will be sharing any resources I put together for that if it’s of interest to anybody :)

And here are some other suggestions I got:

• Peter Morris, Introduction to Game Theory, Springer, Berlin, 1994.

Over on Google+, Joerg Fliege said this “is an excellent book for undergraduate students to start with. I used it myself a couple of years ago for a course in game theory. It is a bit outdated, though, and does not cover repeat games to any depth.”

• K. G. Binmore, Playing for Real: a Text on Game Theory, Oxford U. Press, Oxford, 2007.

Benjamin McKay said: “It has almost no prerequisites, but gets into some serious stuff. I taught game theory once from my own lecture notes, but then I found Binmore’s book and I wish I had used it instead.” A summary says:

This new book is a replacement for Binmore’s previous game theory textbook, Fun and Games. It is a lighthearted introduction to game theory suitable for advanced undergraduate students or beginning graduate students. It aims to answer three questions: What is game theory? How is game theory applied? Why is game theory right? It is the only book that tackles all three questions seriously without getting heavily mathematical.

• Herbert Gintis, Game Theory Evolving: a Problem-Centered Introduction to Modeling Strategic Behavior, Princeton U. Press, Princeton, 2000.

A summary says this book

exposes students to the techniques and applications of game theory through a problems involving human (and even animal) behaviour. This book shows students how to apply game theory to model how people behave in ways that reflect the nature of human sociality and individuality.

Finally, this one is mostly too advanced for my course, but it’s 750 pages and it’s free.

• Noam Nisan, Tim Roughgarde, Eva Tardos and Vijay V. Vazirani, editors, Algorithmic Game Theory, Cambridge U. Press, Cambridge, 2007.

It’s about:

• algorithms for computing equilibria in games and markets,

• auction algorithms,

• mechanism design (also known
as ‘reverse game theory’, this is the art of designing a game that coaxes the players into becoming good at doing something you want),

• the price of anarchy: how the efficiency of a system degrades due to selfish behavior of its agents.

Over on Google+, Adam Smith said:

One suggestion is to get some mechanism design into the course (auctions, VCG, …) and from there into matching. Reasons to do this:

1) Teaching the stable marriage theorem is very fun.

2) This year’s Nobel prize in economics went to two game theorists for their work on matchings and markets.

3) Interesting auctions are everywhere—on Ebay, Google’s ad auctioning system, spectrum distribution, …

I’m Looking For Good Math Grad Students

I am looking for hardworking math grad students who:

1) know some category theory and ideally a bit of 2-category theory,

2) know some mathematical physics, stochastic processes and/or Bayesian network theory, and

3) want to apply these ideas to subjects like chemistry, biology, ecology and climate science.

If this is you, please email me and/or apply to the math Ph.D. program at U.C. Riverside. To apply, follow the directions here. For more information, go here. The deadline is January 5th.

We have very little money for foreign students, so this advertisement is mainly for students from the US and especially California. If you want to work with me, mention my name in your application.

I can’t promise to work with you, of course, until you’re accepted and I get to know you and decide we can work well together! Luckily there are other good professors in the department doing other interesting things.

I urge would-be students to come to my seminar, which meets once a week, and also my special sessions where we work on projects, which currently also occur once a week. I’ll pick students from among people who do these things. Right now there are 6 candidates. I can’t take this many new students every year, so I’ll pick the ones who show the most initiative and promise.

I’m working on network theory and information theory, and I’m also getting started on climate physics, especially glacial cycles. You can decide if these topics interest you by clicking on the links. I’m not taking students who want to do thesis work on my old interests (quantum gravity and n-categories).

The U.C.R. math building looks 2-dimensional in this shot, but I promise you’ll get a well-rounded education if you work with me.

Mathematics of the Environment (Part 10)

There’s a lot more to say, but just one more class to say it! Next quarter I’ll be busy teaching an undergraduate course on evolutionary game theory and a grad course on Lagrangian methods in classical mechanics, together with this seminar and weekly meetings with my students. So, to keep from burning out, I’m going to temporarily switch this seminar to a different topic, where I have a textbook all lined up:

• John Baez and Jacob Biamonte, A Course on Quantum Techniques in Stochastic Mechanics.

I will stop putting up online notes. I’ll also teach the classical mechanics using a book I helped write:

• John Baez and Derek Wise, Lectures on Classical Mechanics.

This should make my job a bit easier: explaining climate physics is a lot more work, since I’m just an amateur! But I hope to come back to this topic someday.

In this final class let’s talk a bit about recent work on glacial cycles and changes in the Earth’s orbit. To keep my job manageable, I’ll just talk about one paper.

The work of Didier Paillard

We’ve seen a few puzzles about how Milankovich cycles are related to the glacial cycles. There are many more I haven’t even gotten around to explaining:

• Milankovich cycles: problems, Wikipedia.

But let’s dive in and look at a model that tries to solve some:

• Didier Paillard, The timing of Pleistocene glaciations from a simple multiple-state climate model, Nature 391 (1998), 378–391.

Paillard starts by telling us the good news:

The Earth’s climate over the past million years has been characterized by a succession of cold and warm periods, known as glacial–interglacial cycles, with periodicities corresponding to those of the Earth’s main orbital parameters; precession (23 kyr), obliquity (41 kyr) and eccentricity (100 kyr). The astronomical theory of climate, in which the orbital variations are taken to drive the climate changes, has been very successful in explaining many features of the palaeoclimate records.

I’m not including reference numbers, but here he cites a famous paper which we discussed in Part 8:

• J. D. Hays, J. Imbrie, and N. J. Shackleton, Variations in the earth’s orbit: pacemaker of the Ice Ages, Science 194 (1976), 1121–1132.

The main result of this paper was to find peaks in the power spectrum of various temperature proxies that match some of the periods of the Milankovitch cycles. This has repeatedly been confirmed. In fact, one of the students in this course, Blake Pollard, has already checked this. I want to pressure him to write a blog article including the nice graphs he’s generated.

But then comes the bad news:

Nevertheless, the timing of the main glacial and interglacial periods remains puzzling in many respects. In particular, the main glacial–interglacial switches occur approximately every 100 kyr, but the changes in insolation forcing are very small in this frequency band.

Here’s an article on the first problem:

• 100,000-year problem, Wikipedia.

The basic idea is that during the last million years, the glacial cycles seem to happening roughly every 100 thousand years:

The Milankovich cycles that most closely match this are two cycles in the eccentricity of the Earth’s orbit which have periods of 95 and 123 thousand years. But as we saw last time, these have very tiny effects on the average solar energy hitting the Earth year round. The obliquity and precession cycles have no effect on the average solar energy hitting the Earth, but they have a noticeable effect on how much hits it in a given latitude in a given season!

Alas, we didn’t get around to calculating that yet. But this gives you a sense of it:

As common in paleontology, time here goes from right to left. The yellow curve shows the amount of solar power hitting the Earth at a latitude of 65° N at the summer solstice. This quantity is often called simply the insolation, though that term also means other things. The insolation curve most closely resembles the red curve showing precession cycles, which have periods near 20 thousand years. But during this stretch of time, ice ages have been happening roughly once every 100 thousand years! Why? That’s the 100,000 year problem.

Continuing the quotation:

Similarly, an especially warm interglacial episode, about 400,000 years ago, occurred at a time when insolation variations were minimal.

If you look at the graph above, you’ll see what he means.

Next, he sketches what he’ll do:

Here I propose that multiple equilibria in the climate system can provide a resolution of these problems
within the framework of astronomical theory. I present two simple models that successfully simulate each glacial–interglacial cycle over the late Pleistocene epoch at the correct time and with approximately the correct amplitude. Moreover, in a simulation over the past 2 million years, the onset of the observed prominent 100-kyr cycles around 0.8 to 1 million years ago is correctly reproduced.

Paillard’s model

I’ll just talk about his first, simpler model. It assumes the Earth can be in three different states:

• i: interglacial

• g: mild glacial

• G: full glacial

In this model:

• The Earth goes from i to g as soon as the insolation goes below some level

• The Earth then goes from g to G as soon as the volume of ice goes above some level

• The Earth then goes from G to i as soon as the insolation goes above some level

Only the transitions i → g and g → G are allowed! The reverse transitions G → g and g → i are forbidden. Paillard draws a schematic picture of the model, like this:

Of course, he also most specify how the ice volume grows when the Earth is in its mild glacial g state. He says:

I assume that the ice sheet needs some minimal time in order to grow and exceed the volume […] and that the insolation maxima preceding the g–G transition must remain below the level The g–G transition then can occur at the next insolation decrease, when it falls below .

Being a mathematician rather than a climate scientist, I can think of more than one way to interpret this. I think it means:

1. If the Earth is in its g state and the insolation stays below some value for a time then the Earth jumps into the G state.

2. If the Earth is in its g state and the insolation rises above we wait until it drops below some value and then the Earth jumps into its G state.

An alternative interpretation is:

2′. If the Earth is in its g state and the insolation rises above we wait until it drops below some value Then we ‘reset the clock’ and proceed according to rule 1.

I’ll try to sort this out. Now, the insolation as a function of time is known—you can compute it using the formula and the data here:

• Insolation, Azimuth Project.

So, the only thing required to complete Paillard’s model are choices of these numbers:

He likes to measure insolation in terms of its standard deviation from its mean value. With this normalization he takes:

and

Then his model gives these results:

(Click to enlarge.) The bottom graph shows temperature as measured by the extra amount of oxygen-18 in some geological records. So, we can see that the Earth often pops rather suddenly into a warm interglacial state and cools a bit more slowly into a glacial state. In the model, this ‘popping into a warm state’ happens instantaneously in the middle graph. The main thing is to compare this to the bottom graph!

It looks quite good! So, I want to think about this type of model more.

The way the models pops suddenly into the very cold G state does not look quite so good. But still, it’s exciting how such a simple model fits the overall profile of the glacial cycle—at least for the last million years.

Paillard says his model is fairly robust, too:

This model is not very sensitive to parameter changes. Different threshold values will slightly offset the transitions by a few hundred years, but the overall shape will remain the same for a broad range of values. There is no significant changes when is between -0.97 and -0.64, between -0.23 and 0.32, between -0.30 and 0.13, between 0.97 and 1.16, and between 27 kyr and 60 kyr. Even when the parameters are out of these bounds, the changes are minor: when is between -0.63 and -0.09, the succession of regimes remains the same except for present time, which becomes a g regime. When is chosen between 0.33 and 0.87, only the duration of stage 11.3 changes to become more comparable to other interglacial stages.

Marine isotope stages

There’s a lot more to say. For example, what does the model say about the time more than a million years ago, when the glacial cycles happened roughly every 41 thousand years, instead of every 100? I won’t answer this. Instead, I’ll conclude by explaining something very basic—but worth knowing.

What’s ‘stage 11.3’? This refers to the numbers down at the bottom of Paillard’s chart: these numbers are Marine Isotope Stages. 11.3 is a ‘substage’, not shown on the chart.

Marine Isotopes Stages are official periods of time used by people who study glacial cycles. The even-numbered ones roughly correspond to glacial periods, and the odd-numbered ones to interglacials. By now over a hundred stages have been identified, going back 6 million years!

Just to give you a little sense of what’s going on, here are the start dates of the last 11 stages, with hot ones in red and the cold ones in blue:

• MIS 1: 11 thousand years ago. This marks the end of the last glacial cycle. More precisely, this is about 500 years after the end of the Younger Dryas event.

• MIS 2: 24 thousand years ago. The Last Glacial Maximum occurred between 26.5 and 19 thousand years ago. At that time we had ice sheets down to the Great Lakes, the mouth of the Rhine, and covering the British Isles. Homo sapiens arrived in the Americas later, around 18 thousand years ago.

• MIS 3: 60 thousand years ago. For comparison, Homo sapiens arrived in central Asia around 50 thousand years ago. About 35 thousand years ago the calendar was invented, Homo sapiens arrived in Europe, and Homo neanderthalensis. went extinct.

• MIS 4: 71 (or maybe 74) thousand years ago.

• MIS 5: 130 thousand years ago. The Eemian, the last really warm interglacial period before ours, began at this time and ended about 114 thousand years ago. If you look at this chart, you’ll see MIS 3 was a much less warm interglacial:

(Now time is going to the right again. Click for more details.)

• MIS 6: 190 thousand years ago.

• MIS 7: 244 thousand years ago. The first known Homo sapiens date back to 250 thousand years ago.

• MIS 8: 301 thousand years ago.

• MIS 9: 334 thousand years ago.

• MIS 10: 364 thousand years ago. The first known Homo neanderthalensis date back to about 350 thousand years ago.

• MIS 11: 427 thousand years ago. This stage is supposedly the most similar to MIS 1, and looking at the graph above you can see why people say that.

I hope you agree that it’s worth understanding the glacial cycles, not just because we need to understand how the Earth will respond to the big boost of carbon dioxide that we’re dosing it with now, but because it’s a fascinating physics problem—and because glaciation has been a powerful force in Earth’s recent history, and the history of our species.

Table of Contents

For your convenience, here are links to all the notes for this course:

• Part 1 – The mathematics of planet Earth.
• Part 2 – Simple estimates of the Earth’s temperature.
• Part 3 – The greenhouse effect.
• Part 4 – History of the Earth’s climate.
• Part 5 – A model showing bistability of the Earth’s climate due to the ice albedo effect: statics.
• Part 6 – A model showing bistability of the Earth’s climate due to the ice albedo effect: dynamics.
• Part 7 – Stochastic differential equations and stochastic resonance.
• Part 8 – A stochastic energy balance model and Milankovitch cycles.
• Part 9 – Changes in insolation due to changes in the eccentricity of the Earth’s orbit.
• Part 10 – Didier Paillard’s model of the glacial cycles.

Symmetry and the Fourth Dimension (Part 8)

Surprise!

I bet you thought this series had died. But it was only snoozing.

I start projects at a rate faster than I can finish them. I wish I could do what a character in Greg Egan’s Permutation City could do, and have my ‘exoself’ adjust my personality so I would stick with just one project for an arbitrarily long time:

The workshop abutted a warehouse full of table legs—one hundred and sixty-two thousand, three hundred and twenty-nine, so far. Peer could imagine nothing more satisfying than reaching the two hundred thousand mark—although he knew that he would probably change his mind and abandon the workshop before that happened; new vocations were imposed by his exoself at random intervals, but statistically, the next one was overdue. Before taking up woodworking, he’d passionately devoured all the higher mathematics texts in the central library, run all the tutorial software, and then contributed several important new results to group theory—unconcerned by the fact that the Elysian mathematicians would never be aware of his work. Before that, he had written over three hundred comic operas, with librettos in Italian, English and French—and staged most of them, with puppet performers and audiences. Before that, he had patiently studied the structure and biochemistry of the brain for sixty-seven years; towards the end he had fully grasped, to his own satisfaction, the nature of the process of consciousness. Every one of these pursuits had been utterly engrossing, and satisfying, at the time.

But since I can’t do this, I’ve been trying to develop enough discipline to make sure I eventually come around back and finish most of what I start.

The missing solids

If you were paying attention, you should have noticed something funny when we worked our way from a Platonic solid to its dual by chopping off its corners more and more. For example, in Part 5 we started with the cube:

cube		•—4—o—3—o
truncated cube		•—4—•—3—o
cuboctahedron		o—4—•—3—o
truncated octahedron		o—4—•—3—•
octahedron		o—4—o—3—•

See what’s funny? We get 5 shapes as we go from our Platonic solid to its dual… but there are 2³ = 8 ways to mark the 3 dots in the Coxeter diagram either black (•) or white (o).

It makes sense to leave out the diagram where all dots are white, for reasons that should become clear. But that leaves two more diagrams missing from our chart!

The missing diagrams are the one with the end dots black:

•—4—o—3—•

and the one with all dots black:

•—4—•—3—•

Are there shapes corresponding to these diagrams?

YES!

And in fact, these will be some of the most complex and beautiful shapes we’ve met so far!

Coxeter diagrams and polyhedra

To get our hands on them, we have to remember the rules of the game. We’ve been dealing with Coxeter diagrams with three dots, and these dots stand for vertex, edge and face, in this order:

V—4—E—3—F

The polyhedra we’re playing with have corners that arise from the dots that are blackened. Let me remind you how, with some examples:

• a cube obviously has one corner for each vertex of the cube, so we blacken the V dot:

•—4—o—3—o

• a truncated cube has one corner for each vertex-edge flag of the cube, meaning a pair consisting of a vertex and an edge it lies on:

So, we blacken the V and E dots:

•—4—•—3—o

• a cuboctahedron has one corner for each edge of the cube:

So, we blacken just the E dot:

o—4—•—3—o

• a truncated octahedron has one corner for each edge-face flag of the cube, meaning a pair consisting of an edge and face containing it. So, we blacken the E and F dots:

o—4—•—3—•

• an octahedron has one corner for each face of the cube:

So, we blacken the F dot:

o—4—o—3—•

How to find the missing solids

Now let’s look at the two diagrams that are missing from this list! This one:

•—4—o—3—•

has the dots for ‘vertex’ and ‘face’ blackened. So, following the idea that’s worked so far, it should stand for a polyhedron that has one corner for each vertex-face flag of the cube: that is, each pair consisting of a vertex and a face that it lies on.

The cube has 6 faces and each face has 4 vertices lying on it:

So, it has 6 × 4 = 24 vertex-face flags. And if we make a shape with one corner for each of these, we get this:

This is called the rhombicuboctahedron. The corners of each red square here correspond to the 4 vertices lying on a given face of the cube. So indeed, this thing has one corner for each vertex-face flag of the cube!

Similarly, in this diagram:

•—4—•—3—•

the dots for ‘vertex’, ‘edge’ and ‘face’ are all blackened. So it should stand for a polyhedron that has one corner for each complete flag of the cube. Remember, a complete flag consists of a vertex, an edge and a face, where the vertex lies on the edge and the edge lies on the face.

Now the cube has 6 faces, each with 4 edges, each with 2 vertices. So, it has 6 × 4 × 2 = 48 complete flags. And if we make a shape with one corner for each of these, we get this:

This is called the truncated cuboctahedron, because you can also get it from truncating an cuboctahedron.

Puzzle 1. Why does that happen? Why should snipping off the corners of an cuboctahedron give a shape with one corner for each complete flag of the cube?

Next let’s go through all three families of shapes:

• the tetrahedron family,
• the cube/octahedron family, and
• the dodecahedron/icosahedron family

and list their two ‘missing members’.

Tetrahedron family

Cuboctahedron: •—3—o—3—•

 

Truncated octahedron: •—3—•—3—•

 

Cube/octahedron family

Rhombicuboctahedron: •—4—o—3—•

 

Truncated cuboctahedron: •—4—•—3—•

 

Dodecahedron/icosahedron family

Rhombicosidodecahedron: •—5—o—3—•

 

Truncated icosidodecahedron: •—5—•—3—•

 

These last two are my favorites, since they’re the fanciest. Let’s explore them a bit further.

In this diagram:

•—5—o—3—•

the dots for ‘vertex’ and ‘face’ are blackened. So, this gives a solid with one corner for each vertex-face flag of the dodecahedron.

How many flags of this sort are there? As its name suggests, the dodecahedron has 12 faces, and each face has 5 vertices lying on it:

So, it has 12 × 5 = 60 vertex-face flags. And if we make a shape with one corner for each of these, we get the rhombicosidodecahedron:

The corners of each red pentagon here correspond to the 5 vertices lying on a given face of the dodecahedron. So indeed, this thing has one corner for each vertex-face flag of the dodecahedron.

In this diagram:

•—5—•—3—•

the dots for ‘vertex’, ‘edge’ and ‘face’ are all blackened. So it should stand for a polyhedron that has one corner for each complete flag of the dodecahedron. Now the dodecahedron has 12 faces, each with 5 edges, each with 2 vertices. So, it has 12 × 5 × 2 = 120 complete flags. And if we make a shape with one corner for each of these, we get this:

This is called the truncated icosidodecahedron, because you can also get it from truncating an icosidodecahedron.

Puzzle 2. Why does that happen? Why should snipping off the corners of an icosidodecahedron give a shape with one corner for each complete flag of the dodecahedron?

Afterword

As usual, the pretty pictures of solids with brass balls at the vertices were made by Tom Ruen using Robert Webb’s Stella software.

You can see the previous episodes here:

• Part 1: Platonic solids and Coxeter complexes.

• Part 2: Coxeter groups.

• Part 3: Coxeter diagrams.

• Part 4: duals of Platonic solids.

• Part 5: Interpolating between a Platonic solid and its dual, and how to describe this using Coxeter diagrams. Example: the cube/octahedron family.

• Part 6: Interpolating between a Platonic solid and its dual. Example: the dodecahedron/icosahedron family.

• Part 7: Interpolating between a Platonic solid and its dual. Example: the tetrahedron family.

Mathematics of the Environment (Part 9)

I didn’t manage to cover everything I intended last time, so I’m moving the stuff about the eccentricity of the Earth’s orbit to this week, and expanding it.

Sunshine and the Earth’s orbit

I bet some of you are hungry for some math. As I mentioned, it takes some work to see how changes in the eccentricity of the Earth’s orbit affect the annual average of sunlight hitting the top of the Earth’s atmosphere. Luckily Greg Egan has done this work for us. While the result is surely not new, his approach makes nice use of the fact that both gravity and solar radiation obey an inverse-square law. That’s pretty cool.

Here is his calculation with some details filled in.

Let’s think of the Earth as moving around an ellipse with one focus at the origin. Its angular momentum is then

where is its mass, and are its polar coordinates, and is the angular component of its velocity:

So,

and

Since the brightness of a distant object goes like , the solar energy hitting the Earth per unit time is

for some constant It follows that the energy delivered per unit of angular progress around the orbit is

Thus, the total energy delivered in one period will be

So far we haven’t used the the fact that the Earth’s orbit is elliptical. Next we’ll do that. Our goal will be to show that depends only very slightly on the eccentricity of the Earth’s orbit. But we need to review a bit of geometry first.

The geometry of ellipses

If the Earth is moving in an ellipse with one focus at the origin, its equation in polar coordinates is

where is the eccentricity and is the somewhat dirty-sounding semi-latus rectum. You can think of as a kind of average radius of the ellipse—more on that in a minute.

Let’s think of the origin in this coordinate system as the Sun—that’s close to true, though the Sun moves a little. Then the Earth gets closest to the Sun when is as big as possible. So, the Earth is closest to the Sun when , and then its distance is

Similarly, the Earth is farthest from the Sun happens when , and then its distance is

We call the perihelion and the aphelion.

The semi-major axis is half the distance between the opposite points on the Earth’s orbit that are farthest from each other. This is denoted These points occur at and , so the distance between these points is , and

So, the semi-major axis is the arithmetic mean of the perihelion and aphelion.

The semi-minor axis is half the distance between the opposite points on the Earth’s orbit that are closest to each other. This is denoted

Puzzle 1. Show that the semi-minor axis is the geometric mean of the perihelion and aphelion:

I said the semi-latus rectum is also a kind of average radius of the ellipse. Just to make that precise, try this:

Puzzle 2. Show that the semi-latus rectum is the harmonic mean of the perihelion and aphelion:

This puzzle is just for fun: the Greeks loved arithmetic, geometric and harmonic means, and the Greek mathematician Apollonius wrote a book on conic sections, so he must have known these facts and loved them. The conventional wisdom is that the Greeks never realized that the planets move in elliptical orbits. However, the wonderful movie Agora presents a great alternative history in which Hypatia figures it all out shortly before being killed! And the mathematician Sandro Graffi (who incidentally taught a course I took in college on the self-adjointness of quantum-mechanical Hamiltonians) has claimed:

Now an infrequently read work of Plutarch, several parts of the Natural History of Plinius, of the Natural Questions of Seneca, and of the Architecture of Vitruvius, also infrequently read, especially by scientists, clearly show that the cultural elite of the early imperial age (first century A.D.) were fully aware of and convinced of a heliocentric dynamical theory of planetary motions based on the attractions of the planets toward the Sun by a force proportional to the inverse square of the distance between planet and Sun. The inverse square dependence on the distance comes from the assumption that the attraction is propagated along rays emanating from the surfaces of the bodies.

I have no idea if the controversial last part of this claim is true. But it’s fun to imagine!

More importantly for what’s to come, we can express the semi-minor axis in terms of the semi-major axis and the eccentricity. Since

we have

so the semi-minor axis is

while

so the semi-major axis is

and thus they are related by

Remember this!

How total annual sunshine depends on eccentricity

We saw a nice formula for the total solar energy hitting the Earth in one year in terms of its angular momentum :

How can we relate the angular momentum to the shape of the Earth’s orbit? The Earth’s energy, kinetic plus potential, is constant throughout the year. The kinetic energy is

and the potential energy is

At the aphelion or perihelion the Earth isn’t moving in or out, just around, so by our earlier work

and the kinetic energy is

Equating the Earth’s energy at aphelion and perihelion, we thus get

and doing some algebra:

and solving for

But remember that the semi-major and semi-minor axis of the Earth’s orbit are given by

respectively! So, we have

This lets us rewrite our old formula for the energy in the form of sunshine that hits the Earth each year:

But we’ve also seen that

so we get the formula we’ve been seeking:

This tells us as a function of semi-major axis and eccentricity.

As we’ll see later, the semi-major axis is almost unchanged by small perturbations of the Earth’s orbit. The main thing that changes is the eccentricity . But if is small, is even smaller, so doesn’t change much when we change

We can make this more quantiative. Let’s work out how much the actual changes in the Earth’s orbit affect the amount of solar radiation it gets! As we’ll see, the semi-major axis is almost constant, so we can ignore that. Complicated calculations we can’t redo here show that the eccentricity varies between 0.005 and 0.058. We’ve seen the total energy the Earth gets each year from solar radiation is proportional to

When the eccentricity is at its lowest value, we get

When the eccentricity is at its highest value, we get

So, the solar power hitting the Earth each year changes by a factor of

In other words, it changes by merely 0.167%.

That’s very small And the effect on the Earth’s temperature would naively be even less!

Naively, we can treat the Earth as a greybody: an ideal object whose tendency to absorb or emit radiation is the same at all wavelengths and temperatures. Since the temperature of a greybody is proportional to the fourth root of the power it receives, a 0.167% change in solar energy received per year corresponds to a percentage change in temperature roughly one fourth as big. That’s a 0.042% change in temperature. If we imagine starting with an Earth like ours, with an average temperature of roughly 290 kelvin, that’s a change of just 0.12 kelvin!

The upshot seems to be this: in a naive model without any amplifying effects, changes in the eccentricity of the Earth’s orbit would cause temperature changes of just 0.12 °C!

This is much less than the roughly 5 °C change we see between glacial and interglacial periods. So, if changes in eccentricity are important in glacial cycles, we have some explaining to do. Possible explanations include season-dependent phenomena and climate feedback effects, like the ice albedo effect we’ve been discussing. Probably both are very important!

Adiabatic invariance

Why does the semi-major axis of the Earth’s orbit remain almost unchanged under small perturbations? The reason is that it’s an ‘adiabatic invariant’. This is basically just a fancy way of saying it remains almost unchanged. But the point is, there’s a whole theory of adiabatic invariants… which supposedly explains the near-constancy of the semi-major axis.

According to Wikipedia:

The Earth’s eccentricity varies primarily due to interactions with the gravitational fields of Jupiter and Saturn. As the eccentricity of the orbit evolves, the semi-major axis of the orbital ellipse remains unchanged. From the perspective of the perturbation theory used in celestial mechanics to compute the evolution of the orbit, the semi-major axis is an adiabatic invariant. According to Kepler’s third law the period of the orbit is determined by the semi-major axis. It follows that the Earth’s orbital period, the length of a sidereal year, also remains unchanged as the orbit evolves. As the semi-minor axis is decreased with the eccentricity increase, the seasonal changes increase. But the mean solar irradiation for the planet changes only slightly for small eccentricity, due to Kepler’s second law.

Unfortunately, even though I understand a bit about the general theory of adiabatic invariants, I have not gotten around to convincing myself that the semi-major axis is such a thing, for the perturbations experienced by the Earth.

Here’s something easier: checking that the semi-major axis of the Earth’s orbit determines the period of the Earth’s orbit, say . To do this, first relate the angular momentum to the period by integrating the rate at which orbital area is swept out by the planet:

over one orbit. Since the area of an ellipse is , this gives us:

On the other hand, we’ve seen

Equating these two expressions for shows that the period is:

So, the period depends only on the semi-major axis, not the eccentricity. Conversely, we could solve this equation to see that the semi-major axis depends only on the period, not the eccentricity.

I’m treating and as constants here. If the mass of the Sun decreases, as it eventually will when it becomes a red giant and puffs out lots of gas, the semi-major axes of the Earth’s orbit will change. It will actually increase! This is one reason people are still arguing about just when the Earth will get swallowed up by the Sun:

• David Appell, The Sun will eventually engulf the Earth—maybe, Scientific American, 8 September 2008.

And, to show just how subtle these things are, if the mass of the Sun slowly changes, while the semi-major axis of the Earth’s orbit will change, the eccentricity will remain almost unchanged. Why? Because for this kind of process, it’s the eccentricity that’s an adiabatic invariant!

Indeed, I got all excited when I started reading a homework problem in Landau and Lifschitz’s book Classical Mechanics, which describes adiabatic invariants for the gravitational 2-body problem. But I was bummed out when they concluded that the eccentricity was an adiabatic invariant for gradual changes in . They didn’t discuss any problems for which the semi-major axis was an adiabatic invariant.

I’ll have to get back to this later sometime, probably with the help of a good book on celestial mechanics. If you’re curious about the concept of adiabatic invariant, start here:

• Adiabatic invariant, Wikipedia.

and then try this:

• Marko Robnik, Theory of adiabatic invariants, February 2004.

And if you know how to show the Earth’s semi-major axis is an adiabatic invariant, please tell me how!