My All 2020 Mathematics A to Z: Michael Atiyah

To start this year’s great glossary project Mr Wu, author of the blog, had a great suggestion: The Atiyah-Singer Index Theorem. It’s an important and spectacular piece of work. I’ll explain why I’m not doing that in a few sentences.

Mr Wu pointed out that a biography of Michael Atiyah, one of the authors of this theorem, might be worth doing. GoldenOj endorsed the biography idea, and the more I thought it over the more I liked it. I’m not able to do a true biography, something that goes to primary sources and finds a convincing story of a life. But I can sketch out a bit, exploring his work and why it’s of note.

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Michael Atiyah.

Theodore Frankel’s The Geometry of Physics: An Introduction is a wonderful book. It’s 686 pages, including the index. It all explores how our modern understanding of physics is our modern understanding of geometry. On page 465 it offers this:

The Atiyah-Singer index theorem must be considered a high point of geometrical analysis of the twentieth century, but is far too complicated to be considered in this book.

I know when I’m licked. Let me attempt to look at one of the people behind this theorem instead.

The Riemann Hypothesis is about where to find the roots of a particular infinite series. It’s been out there, waiting for a solution, for a century and a half. There are many interesting results which we would know to be true if the Riemann Hypothesis is true. In 2018, Michael Atiyah declared that he had a proof. And, more, an amazing proof, a short proof. Albeit one that depended on a great deal of background work and careful definitions. The mathematical community was skeptical. It still is. But it did not dismiss outright the idea that he had a solution. It was plausible that Atiyah might solve one of the greatest problems of mathematics in something that fits on a few PowerPoint slides.

So think of a person who commands such respect.

His proof of the Riemann Hypothesis, as best I understand, is not generally accepted. For example, it includes the fine structure constant. This comes from physics. It describes how strongly electrons and photons interact. The most compelling (to us) consequence of the Riemann Hypothesis is in how prime numbers are distributed among the integers. It’s hard to think how photons and prime numbers could relate. But, then, if humans had done all of mathematics without noticing geometry, we would know there is something interesting about π. Differential equations, if nothing else, would turn up this number. We happened to discover π in the real world first too. If it were not familiar for so long, would we think there should be any commonality between differential equations and circles?

I do not mean to say Atiyah is right and his critics wrong. I’m no judge of the matter at all. What is interesting is that one could imagine a link between a pure number-theory matter like the Riemann hypothesis and a physical matter like the fine structure constant. It’s not surprising that mathematicians should be interested in physics, or vice-versa. Atiyah’s work was particularly important. Much of his work, from the late 70s through the 80s, was in gauge theory. This subject lies under much of modern quantum mechanics. It’s born of the recognition of symmetries, group operations that you can do on a field, such as the electromagnetic field.

In a sequence of papers Atiyah, with other authors, sorted out particular cases of how magnetic monopoles and instantons behave. Magnetic monopoles may sound familiar, even though no one has ever seen one. These are magnetic points, an isolated north or a south pole without its opposite partner. We can understand well how they would act without worrying about whether they exist. Instantons are more esoteric; I don’t remember encountering the term before starting my reading for this essay. I believe I did, encountering the technique as a way to describe the transitions between one quantum state and another. Perhaps the name failed to stick. I can see where there are few examples you could give an undergraduate physics major. And it turns out that monopoles appear as solutions to some problems involving instantons.

This was, for Atiyah, later work. It arose, in part, from bringing the tools of index theory to nonlinear partial differential equations. This index theory is the thing that got us the Atiyah-Singer Index Theorem too complicated to explain in 686 pages. Index theory, here, studies questions like “what can we know about a differential equation without solving it?” Solving a differential equation would tell us almost everything we’d like to know, yes. But it’s also quite hard. Index theory can tell us useful things like: is there a solution? Is there more than one? How many? And it does this through topological invariants. A topological invariant is a trait like, for example, the number of holes that go through a solid object. These things are indifferent to operations like moving the object, or rotating it, or reflecting it. In the language of group theory, they are invariant under a symmetry.

It’s startling to think a question like “is there a solution to this differential equation” has connections to what we know about shapes. This shows some of the power of recasting problems as geometry questions. From the late 50s through the mid-70s, Atiyah was a key person working in a topic that is about shapes. We know it as K-theory. The “K” from the German Klasse, here. It’s about groups, in the abstract-algebra sense; the things in the groups are themselves classes of isomorphisms. Michael Atiyah and Friedrich Hirzebruch defined this sort of group for a topological space in 1959. And this gave definition to topological K-theory. This is again abstract stuff. Frankel’s book doesn’t even mention it. It explores what we can know about shapes from the tangents to the shapes.

And it leads into cobordism, also called bordism. This is about what you can know about shapes which could be represented as cross-sections of a higher-dimension shape. The iconic, and delightfully named, shape here is the pair of pants. In three dimensions this shape is a simple cartoon of what it’s named. On the one end, it’s a circle. On the other end, it’s two circles. In between, it’s a continuous surface. Imagine the cross-sections, how on separate layers the two circles are closer together. How their shapes distort from a real circle. In one cross-section they come together. They appear as two circles joined at a point. In another, they’re a two-looped figure. In another, a smoother circle. Knowing that Atiyah came from these questions may make his future work seem more motivated.

But how does one come to think of the mathematics of imaginary pants? Many ways. Atiyah’s path came from his first research specialty, which was algebraic geometry. This was his work through much of the 1950s. Algebraic geometry is about the kinds of geometric problems you get from studying algebra problems. Algebra here means the abstract stuff, although it does touch on the algebra from high school. You might, for example, do work on the roots of a polynomial, or a comfortable enough equation like x^2 + y^2 = 1 . Atiyah had started — as an undergraduate — working on projective geometries. This is what one curve looks like projected onto a different surface. This moved into elliptic curves and into particular kinds of transformations on surfaces. And algebraic geometry has proved important in number theory. You might remember that the Wiles-Taylor proof of Fermat’s Last Theorem depended on elliptic curves. Some work on the Riemann hypothesis is built on algebraic topology.

(I would like to trace things farther back. But the public record of Atiyah’s work doesn’t offer hints. I can find amusing notes like his father asserting he knew he’d be a mathematician. He was quite good at changing local currency into foreign currency, making a profit on the deal.)

It’s possible to imagine this clear line in Atiyah’s career, and why his last works might have been on the Riemann hypothesis. That’s too pat an assertion. The more interesting thing is that Atiyah had several recognizable phases and did iconic work in each of them. There is a cliche that mathematicians do their best work before they are 40 years old. And, it happens, Atiyah did earn a Fields Medal, given to mathematicians for the work done before they are 40 years old. But I believe this cliche represents a misreading of biographies. I suspect that first-rate work is done when a well-prepared mind looks fresh at a new problem. A mathematician is likely to have these traits line up early in the career. Grad school demands the deep focus on a particular problem. Getting out of grad school lets one bring this deep knowledge to fresh questions.

It is easy, in a career, to keep studying problems one has already had great success in, for good reason and with good results. It tends not to keep producing revolutionary results. Atiyah was able — by chance or design I can’t tell — to several times venture into a new field. The new field was one that his earlier work prepared him for, yes. But it posed new questions about novel topics. And this creative, well-trained mind focusing on new questions produced great work. And this is one way to be credible when one announces a proof of the Riemann hypothesis.

Here is something I could not find a clear way to fit into this essay. Atiyah recorded some comments about his life for the Web of Stories site. These are biographical and do not get into his mathematics at all. Much of it is about his life as child of British and Lebanese parents and how that affected his schooling. One that stood out to me was about his peers at Manchester Grammar School, several of whom he rated as better students than he was. Being a good student is not tightly related to being a successful academic. Particularly as so much of a career depends on chance, on opportunities happening to be open when one is ready to take them. It would be remarkable if there wre three people of greater talent than Atiyah who happened to be in the same school at the same time. It’s not unthinkable, though, and we may wonder what we can do to give people the chance to do what they are good in. (I admit this assumes that one finds doing what one is good in particularly satisfying or fulfilling.) In looking at any remarkable talent it’s fair to ask how much of their exceptional nature is that they had a chance to excel.

My 2019 Mathematics A To Z: Zeno’s Paradoxes

Today’s A To Z term was nominated by Dina Yagodich, who runs a YouTube channel with a host of mathematics topics. Zeno’s Paradoxes exist in the intersection of mathematics and philosophy. Mathematics majors like to declare that they’re all easy. The Ancient Greeks didn’t understand infinite series or infinitesimals like we do. Now they’re no challenge at all. This reflects a belief that philosophers must be silly people who haven’t noticed that one can, say, exit a room.

This is your classic STEM-attitude of missing the point. We may suppose that Zeno of Elea occasionally exited rooms himself. That is a supposition, though. Zeno, like most philosophers who lived before Socrates, we know from other philosophers making fun of him a century after he died. Or at least trying to explain what they thought he was on about. Modern philosophers are expected to present others’ arguments as well and as strongly as possible. This even — especially — when describing an argument they want to say is the stupidest thing they ever heard. Or, to use the lingo, when they wish to refute it. Ancient philosophers had no such compulsion. They did not mind presenting someone else’s argument sketchily, if they supposed everyone already knew it. Or even badly, if they wanted to make the other philosopher sound ridiculous. Between that and the sparse nature of the record, we have to guess a bit about what Zeno precisely said and what he meant. This is all right. We have some idea of things that might reasonably have bothered Zeno.

And they have bothered philosophers for thousands of years. They are about change. The ones I mean to discuss here are particularly about motion. And there are things we do not understand about change. This essay will not answer what we don’t understand. But it will, I hope, show something about why that’s still an interesting thing to ponder.

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Zeno’s Paradoxes.

When we capture a moment by photographing it we add lies to what we see. We impose a frame on its contents, discarding what is off-frame. We rip an instant out of its context. And that before considering how we stage photographs, making people smile and stop tilting their heads. We forgive many of these lies. The things excluded from or the moments around the one photographed might not alter what the photograph represents. Making everyone smile can convey the emotional average of the event in a way that no individual moment represents. Arranging people to stand in frame can convey the participation in the way a candid photograph would not.

But there remains the lie that a photograph is “a moment”. It is no such thing. We notice this when the photograph is blurred. It records all the light passing through the lens while the shutter is open. A photograph records an eighth of a second. A thirtieth of a second. A thousandth of a second. But still, some time. There is always the ghost of motion in a picture. If we do not see it, it is because our photograph’s resolution is too coarse. If we could photograph something with infinite fidelity we would see, even in still life, the wobbling of the molecules that make up a thing.

A photograph of a blurry roller coaster passing through a vertical loop.
One of the many loops of Vortex, a roller coaster at Kings Island amusement park from 1987 to 2019. Taken by me the last day of the ride’s operation; this was one of the roller coaster’s runs after 7 pm, the close of the park the last day of the season.

Which implies something fascinating to me. Think of a reel of film. Here I mean old-school pre-digital film, the thing that’s a great strip of pictures, a new one shown 24 times per second. Each frame of film is a photograph, recording some split-second of time. How much time is actually in a film, then? How long, cumulatively, was a camera shutter open during a two-hour film? I use pre-digital, strip-of-film movies for convenience. Digital films offer the same questions, but with different technical points. And I do not want the writing burden of describing both analog and digital film technologies. So I will stick to the long sequence of analog photographs model.

Let me imagine a movie. One of an ordinary everyday event; an actuality, to use the terminology of 1898. A person overtaking a walking tortoise. Look at the strip of film. There are many frames which show the person behind the tortoise. There are many frames showing the person ahead of the tortoise. When are the person and the tortoise at the same spot?

We have to put in some definitions. Fine; do that. Say we mean when the leading edge of the person’s nose overtakes the leading edge of the tortoise’s, as viewed from our camera. Or, since there must be blur, when the center of the blur of the person’s nose overtakes the center of the blur of the tortoise’s nose.

Do we have the frame when that moment happened? I’m sure we have frames from the moments before, and frames from the moments after. But the exact moment? Are you positive? If we zoomed in, would it actually show the person is a millimeter behind the tortoise? That the person is a hundredth of a millimeter ahead? A thousandth of a hair’s width behind? Suppose that our camera is very good. It can take frames representing as small a time as we need. Does it ever capture that precise moment? To the point that we know, no, it’s not the case that the tortoise is one-trillionth the width of a hydrogen atom ahead of the person?

If we can’t show the frame where this overtaking happened, then how do we know it happened? To put it in terms a STEM major will respect, how can we credit a thing we have not observed with happening? … Yes, we can suppose it happened if we suppose continuity in space and time. Then it follows from the intermediate value theorem. But then we are begging the question. We impose the assumption that there is a moment of overtaking. This does not prove that the moment exists.

Fine, then. What if time is not continuous? If there is a smallest moment of time? … If there is, then, we can imagine a frame of film that photographs only that one moment. So let’s look at its footage.

One thing stands out. There’s finally no blur in the picture. There can’t be; there’s no time during which to move. We might not catch the moment that the person overtakes the tortoise. It could “happen” in-between moments. But at least we have a moment to observe at leisure.

So … what is the difference between a picture of the person overtaking the tortoise, and a picture of the person and the tortoise standing still? A movie of the two walking should be different from a movie of the two pretending to be department store mannequins. What, in this frame, is the difference? If there is no observable difference, how does the universe tell whether, next instant, these two should have moved or not?

A mathematical physicist may toss in an answer. Our photograph is only of positions. We should also track momentum. Momentum carries within it the information of how position changes over time. We can’t photograph momentum, not without getting blurs. But analytically? If we interpret a photograph as “really” tracking the positions of a bunch of particles? To the mathematical physicist, momentum is as good a variable as position, and it’s as measurable. We can imagine a hyperspace photograph that gives us an image of positions and momentums. So, STEM types show up the philosophers finally, right?

Hold on. Let’s allow that somehow we get changes in position from the momentum of something. Hold off worrying about how momentum gets into position. Where does a change in momentum come from? In the mathematical physics problems we can do, the change in momentum has a value that depends on position. In the mathematical physics problems we have to deal with, the change in momentum has a value that depends on position and momentum. But that value? Put it in words. That value is the change in momentum. It has the same relationship to acceleration that momentum has to velocity. For want of a real term, I’ll call it acceleration. We need more variables. An even more hyperspatial film camera.

… And does acceleration change? Where does that change come from? That is going to demand another variable, the change-in-acceleration. (The “jerk”, according to people who want to tell you that “jerk” is a commonly used term for the change-in-acceleration, and no one else.) And the change-in-change-in-acceleration. Change-in-change-in-change-in-acceleration. We have to invoke an infinite regression of new variables. We got here because we wanted to suppose it wasn’t possible to divide a span of time infinitely many times. This seems like a lot to build into the universe to distinguish a person walking past a tortoise from a person standing near a tortoise. And then we still must admit not knowing how one variable propagates into another. That a person is wide is not usually enough explanation of how they are growing taller.

Numerical integration can model this kind of system with time divided into discrete chunks. It teaches us some ways that this can make logical sense. It also shows us that our projections will (generally) be wrong. At least unless we do things like have an infinite number of steps of time factor into each projection of the next timestep. Or use the forecast of future timesteps to correct the current one. Maybe use both. These are … not impossible. But being “ … not impossible” is not to say satisfying. (We allow numerical integration to be wrong by quantifying just how wrong it is. We call this an “error”, and have techniques that we can use to keep the error within some tolerated margin.)

So where has the movement happened? The original scene had movement to it. The movie seems to represent that movement. But that movement doesn’t seem to be in any frame of the movie. Where did it come from?

We can have properties that appear in a mass which don’t appear in any component piece. No molecule of a substance has a color, but a big enough mass does. No atom of iron is ferromagnetic, but a chunk might be. No grain of sand is a heap, but enough of them are. The Ancient Greeks knew this; we call it the Sorites paradox, after Eubulides of Miletus. (“Sorites” means “heap”, as in heap of sand. But if you had to bluff through a conversation about ancient Greek philosophers you could probably get away with making up a quote you credit to Sorites.) Could movement be, in the term mathematical physicists use, an intensive property? But intensive properties are obvious to the outside observer of a thing. We are not outside observers to the universe. It’s not clear what it would mean for there to be an outside observer to the universe. Even if there were, what space and time are they observing in? And aren’t their space and their time and their observations vulnerable to the same questions? We’re in danger of insisting on an infinite regression of “universes” just so a person can walk past a tortoise in ours.

We can say where movement comes from when we watch a movie. It is a trick of perception. Our eyes take some time to understand a new image. Our brains insist on forming a continuous whole story even out of disjoint ideas. Our memory fools us into remembering a continuous line of action. That a movie moves is entirely an illusion.

You see the implication here. Surely Zeno was not trying to lead us to understand all motion, in the real world, as an illusion? … Zeno seems to have been trying to support the work of Parmenides of Elea. Parmenides is another pre-Socratic philosopher. So we have about four words that we’re fairly sure he authored, and we’re not positive what order to put them in. Parmenides was arguing about the nature of reality, and what it means for a thing to come into or pass out of existence. He seems to have been arguing something like that there was a true reality that’s necessary and timeless and changeless. And there’s an apparent reality, the thing our senses observe. And in our sensing, we add lies which make things like change seem to happen. (Do not use this to get through your PhD defense in philosophy. I’m not sure I’d use it to get through your Intro to Ancient Greek Philosophy quiz.) That what we perceive as movement is not what is “really” going on is, at least, imaginable. So it is worth asking questions about what we mean for something to move. What difference there is between our intuitive understanding of movement and what logic says should happen.

(I know someone wishes to throw down the word Quantum. Quantum mechanics is a powerful tool for describing how many things behave. It implies limits on what we can simultaneously know about the position and the time of a thing. But there is a difference between “what time is” and “what we can know about a thing’s coordinates in time”. Quantum mechanics speaks more to the latter. There are also people who would like to say Relativity. Relativity, special and general, implies we should look at space and time as a unified set. But this does not change our questions about continuity of time or space, or where to find movement in both.)

And this is why we are likely never to finish pondering Zeno’s Paradoxes. In this essay I’ve only discussed two of them: Achilles and the Tortoise, and The Arrow. There are two other particularly famous ones: the Dichotomy, and the Stadium. The Dichotomy is the one about how to get somewhere, you have to get halfway there. But to get halfway there, you have to get a quarter of the way there. And an eighth of the way there, and so on. The Stadium is the hardest of the four great paradoxes to explain. This is in part because the earliest writings we have about it don’t make clear what Zeno was trying to get at. I can think of something which seems consistent with what’s described, and contrary-to-intuition enough to be interesting. I’m satisfied to ponder that one. But other people may have different ideas of what the paradox should be.

There are a handful of other paradoxes which don’t get so much love, although one of them is another version of the Sorites Paradox. Some of them the Stanford Encyclopedia of Philosophy dubs “paradoxes of plurality”. These ask how many things there could be. It’s hard to judge just what he was getting at with this. We know that one argument had three parts, and only two of them survive. Trying to fill in that gap is a challenge. We want to fill in the argument we would make, projecting from our modern idea of this plurality. It’s not Zeno’s idea, though, and we can’t know how close our projection is.

I don’t have the space to make a thematically coherent essay describing these all, though. The set of paradoxes have demanded thought, even just to come up with a reason to think they don’t demand thought, for thousands of years. We will, perhaps, have to keep trying again to fully understand what it is we don’t understand.

And with that — I find it hard to believe — I am done with the alphabet! All of the Fall 2019 A-to-Z essays should appear at this link. Additionally, the A-to-Z sequences of this and past years should be at this link. Tomorrow and Saturday I hope to bring up some mentions of specific past A-to-Z essays. Next week I hope to share my typical thoughts about what this experience has taught me, and some other writing about this writing.

Thank you, all who’ve been reading, and who’ve offered topics, comments on the material, or questions about things I was hoping readers wouldn’t notice I was shorting. I’ll probably do this again next year, after I’ve had some chance to rest.

My 2019 Mathematics A To Z: Hamiltonian

Today’s A To Z term is another I drew from Mr Wu, of the Singapore Math Tuition blog. It gives me more chances to discuss differential equations and mathematical physics, too.

The Hamiltonian we name for Sir William Rowan Hamilton, the 19th century Irish mathematical physicists who worked on everything. You might have encountered his name from hearing about quaternions. Or for coining the terms “scalar” and “tensor”. Or for work in graph theory. There’s more. He did work in Fourier analysis, which is what you get into when you feel at ease with Fourier series. And then wild stuff combining matrices and rings. He’s not quite one of those people where there’s a Hamilton’s Theorem for every field of mathematics you might be interested in. It’s close, though.

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When you first learn about physics you learn about forces and accelerations and stuff. When you major in physics you learn to avoid dealing with forces and accelerations and stuff. It’s not explicit. But you get trained to look, so far as possible, away from vectors. Look to scalars. Look to single numbers that somehow encode your problem.

A great example of this is the Lagrangian. It’s built on “generalized coordinates”, which are not necessarily, like, position and velocity and all. They include the things that describe your system. This can be positions. It’s often angles. The Lagrangian shines in problems where it matters that something rotates. Or if you need to work with polar coordinates or spherical coordinates or anything non-rectangular. The Lagrangian is, in your general coordinates, equal to the kinetic energy minus the potential energy. It’ll be a function. It’ll depend on your coordinates and on the derivative-with-respect-to-time of your coordinates. You can take partial derivatives of the Lagrangian. This tells how the coordinates, and the change-in-time of your coordinates should change over time.

The Hamiltonian is a similar way of working out mechanics problems. The Hamiltonian function isn’t anything so primitive as the kinetic energy minus the potential energy. No, the Hamiltonian is the kinetic energy plus the potential energy. Totally different in idea.

From that description you maybe guessed you can transfer from the Lagrangian to the Hamiltonian. Maybe vice-versa. Yes, you can, although we use the term “transform”. Specifically a “Legendre transform”. We can use any coordinates we like, just as with Lagrangian mechanics. And, as with the Lagrangian, we can find how coordinates change over time. The change of any coordinate depends on the partial derivative of the Hamiltonian with respect to a particular other coordinate. This other coordinate is its “conjugate”. (It may either be this derivative, or minus one times this derivative. By the time you’re doing work in the field you’ll know which.)

That conjugate coordinate is the important thing. It’s why we muck around with Hamiltonians when Lagrangians are so similar. In ordinary, common coordinate systems these conjugate coordinates form nice pairs. In Cartesian coordinates, the conjugate to a particle’s position is its momentum, and vice-versa. In polar coordinates, the conjugate to the angular velocity is the angular momentum. These are nice-sounding pairs. But that’s our good luck. These happen to match stuff we already think is important. In general coordinates one or more of a pair can be some fusion of variables we don’t have a word for and would never care about. Sometimes it gets weird. In the problem of vortices swirling around each other on an infinitely great plane? The horizontal position is conjugate to the vertical position. Velocity doesn’t enter into it. For vortices on the sphere the longitude is conjugate to the cosine of the latitude.

What’s valuable about these pairings is that they make a “symplectic manifold”. A manifold is a patch of space where stuff works like normal Euclidean geometry does. In this case, the space is in “phase space”. This is the collection of all the possible combinations of all the variables that could ever turn up. Every particular moment of a mechanical system matches some point in phase space. Its evolution over time traces out a path in that space. Call it a trajectory or an orbit as you like.

We get good things from looking at the geometry that this symplectic manifold implies. For example, if we know that one variable doesn’t appear in the Hamiltonian, then its conjugate’s value never changes. This is almost the kindest thing you can do for a mathematical physicist. But more. A famous theorem by Emmy Noether tells us that symmetries in the Hamiltonian match with conservation laws in the physics. Time-invariance, for example — time not appearing in the Hamiltonian — gives us the conservation of energy. If only distances between things, not absolute positions, matter, then we get conservation of linear momentum. Stuff like that. To find conservation laws in physics problems is the kindest thing you can do for a mathematical physicist.

The Hamiltonian was born out of planetary physics. These are problems easy to understand and, apart from the case of one star with one planet orbiting each other, impossible to solve exactly. That’s all right. The formalism applies to all kinds of problems. They’re very good at handling particles that interact with each other and maybe some potential energy. This is a lot of stuff.

More, the approach extends naturally to quantum mechanics. It takes some damage along the way. We can’t talk about “the” position or “the” momentum of anything quantum-mechanical. But what we get when we look at quantum mechanics looks very much like what Hamiltonians do. We can calculate things which are quantum quite well by using these tools. This though they came from questions like why Saturn’s rings haven’t fallen part and whether the Earth will stay around its present orbit.

It holds surprising power, too. Notice that the Hamiltonian is the kinetic energy of a system plus its potential energy. For a lot of physics problems that’s all the energy there is. That is, the value of the Hamiltonian for some set of coordinates is the total energy of the system at that time. And, if there’s no energy lost to friction or heat or whatever? Then that’s the total energy of the system for all time.

Here’s where this becomes almost practical. We often want to do a numerical simulation of a physics problem. Generically, we do this by looking up what all the values of all the coordinates are at some starting time t0. Then we calculate how fast these coordinates are changing with time. We pick a small change in time, Δ t. Then we say that at time t0 plus Δ t, the coordinates are whatever they started at plus Δ t times that rate of change. And then we repeat, figuring out how fast the coordinates are changing now, at this position and time.

The trouble is we always make some mistake, and once we’ve made a mistake, we’re going to keep on making mistakes. We can do some clever stuff to make the smallest error possible figuring out where to go, but it’ll still happen. Usually, we stick to calculations where the error won’t mess up our results.

But when we look at stuff like whether the Earth will stay around its present orbit? We can’t make each step good enough for that. Unless we get to thinking about the Hamiltonian, and our symplectic variables. The actual system traces out a path in phase space. Everyone on that path the Hamiltonian is a particular value, the energy of the system. So use the regular methods to project most of the variables to the new time, t0 + Δ t. But the rest? Pick the values that make the Hamiltonian work out right. Also momentum and angular momentum and other stuff we know get conserved. We’ll still make an error. But it’s a different kind of error. It’ll project to a point that’s maybe in the wrong place on the trajectory. But it’s on the trajectory.

(OK, it’s near the trajectory. Suppose the real energy is, oh, the square root of 5. The computer simulation will have an energy of 2.23607. This is close but not exactly the same. That’s all right. Each step will stay close to the real energy.)

So what we’ll get is a projection of the Earth’s orbit that maybe puts it in the wrong place in its orbit. Putting the planet on the opposite side of the sun from Venus when we ought to see Venus transiting the Sun. That’s all right, if what we’re interested in is whether Venus and Earth are still in the solar system.

There’s a special cost for this. If there weren’t we’d use it all the time. The cost is computational complexity. It’s pricey enough that you haven’t heard about these “symplectic integrators” before. That’s all right. These are the kinds of things open to us once we look long into the Hamiltonian.

This wraps up my big essay-writing for the week. I will pluck some older essays out of obscurity to re-share tomorrow and Saturday. All of Fall 2019 A To Z posts should be at this link. Next week should have the letter I on Tuesday and J on Thursday. All of my A To Z essays should be available at this link. And I am still interested in topics I might use for the letters K through N. Thank you.

My 2018 Mathematics A To Z: Manifold

Two commenters suggested the topic for today’s A to Z post. I suspect I’d have been interested in it if only one had. (Although Dina Yagoditch’s suggestion of the Menger Sponge is hard to resist.) But a double domination? The topic got suggested by Mr Wu, author of MathTuition88, and by John Golden, author of Math Hombre. My thanks to all for interesting things to think about.

Cartoon of a thinking coati (it's a raccoon-like animal from Latin America); beside him are spelled out on Scrabble titles, 'MATHEMATICS A TO Z', on a starry background. Various arithmetic symbols are constellations in the background.
Art by Thomas K Dye, creator of the web comics Newshounds, Something Happens, and Infinity Refugees. His current project is Projection Edge. And you can get Projection Edge six months ahead of public publication by subscribing to his Patreon. And he’s on Twitter as @Newshoundscomic.


So you know how in the first car you ever owned the alternator was always going bad? If you’re lucky, you reach a point where you start owning cars good enough that the alternator is not the thing always going bad. Once you’re there, congratulations. Now the thing that’s always going bad in your car will be the manifold. That one’s for my dad.

Manifolds are a way to do normal geometry on weird shapes. What’s normal geometry? It’s … you know, the way shapes work on your table, or in a room. The Euclidean geometry that we’re so used to that it’s hard to imagine it not working. Why worry about weird shapes? They’re interesting, for one. And they don’t have to be that weird to count as weird. A sphere, like the surface of the Earth, can be weird. And these weird shapes can be useful. Mathematical physics, for example, can represent the evolution of some complicated thing as a path drawn on a weird shape. Bringing what we know about geometry from years of study, and moving around rooms, to a problem that abstract makes our lives easier.

We use language that sounds like that of map-makers when discussing manifolds. We have maps. We gather together charts. The collection of charts describing a surface can be an atlas. All these words have common meanings. Mercifully, these common meanings don’t lead us too far from the mathematical meanings. We can even use the problem of mapping the surface of the Earth to understand manifolds.

If you love maps, the geography kind, you learn quickly that there’s no making a perfect two-dimensional map of the Earth’s surface. Some of these imperfections are obvious. You can distort shapes trying to make a flat map of the globe. You can distort sizes. But you can’t represent every point on the globe with a point on the paper. Not without doing something that really breaks continuity. Like, say, turning the North Pole into the whole line at the top of the map. Like in the Equirectangular projection. Or skipping some of the points, like in the Mercator projection. Or adding some cuts into a surface that doesn’t have them, like in the Goode homolosine projection. You may recognize this as the one used in classrooms back when the world had first begun.

But what if we don’t need the whole globedone in a single map? Turns out we can do that easy. We can make charts that cover a part of the surface. No one chart has to cover the whole of the Earth’s surface. It only has to cover some part of it. It covers the globe with a piece that looks like a common ordinary Euclidean space, where ordinary geometry holds. It’s the collection of charts that covers the whole surface. This collection of charts is an atlas. You have a manifold if it’s possible to make a coherent atlas. For this every point on the manifold has to be on at least one chart. It’s okay if a point is on several charts. It’s okay if some point is on all the charts. Like, suppose your original surface is a circle. You can represent this with an atlas of two charts. Each chart maps the circle, except for one point, onto a line segment. The two charts don’t both skip the same point. All but two points on this circle are on all the maps of this chart. That’s cool. What’s not okay is if some point can’t be coherently put onto some chart.

This sad fate can happen. Suppose instead of a circle you want to chart a figure-eight loop. That won’t work. The point where the figure crosses itself doesn’t look, locally, like a Euclidean space. It looks like an ‘x’. There’s no getting around that. There’s no atlas that can cover the whole of that surface. So that surface isn’t a manifold.

But many things are manifolds nevertheless. Toruses, the doughnut shapes, are. Möbius strips and Klein bottles are. Ellipsoids and hyperbolic surfaces are, or at least can be. Mathematical physics finds surfaces that describe all the ways the planets could move and still conserve the energy and momentum and angular momentum of the solar system. That cheesecloth surface stretched through 54 dimensions, is a manifold. There are many possible atlases, with many more charts. But each of those means we can, at least locally, for particular problems, understand them the same way we understand cutouts of triangles and pentagons and circles on construction paper.

So to get back to cars: no one has ever said “my car runs okay, but I regret how I replaced the brake covers the moment I suspected they were wearing out”. Every car problem is easier when it’s done as soon as your budget and schedule allow.

This and other Fall 2018 Mathematics A-To-Z posts can be read at this link. What will I choose for ‘N’, later this week? I really should have decided that by now.

The Arthur Christmas Season

I don’t know how you spend your December, but part of it really ought to be done watching the Aardman Animation film Arthur Christmas. It inspired me to ponder a mathematical-physics question that got into some heady territory and this is a good time to point people back to that.

The first piece is Could `Arthur Christmas’ Happen In Real Life? At one point in the movie Arthur and Grand-Santa are stranded on a Caribbean island while the reindeer and sleigh, without them, go flying off in a straight line. This raises the question of what is a straight line if you’re on the surface of something spherical like the Earth. Also, Grand-Santa is such a fantastic idea for the Santa canon it’s hard to believe that Rankin-Bass never did it.

Returning To Arthur Christmas was titled that because I’d left the subject for a couple weeks. You know how it gets. Here the discussion becomes more spoiler-y. And it has to address the question of what kind of straight line the reindeer might move in. There’s several possible answers and they’re all interesting.

Arthur Christmas And The Least Common Multiple supposes that reindeer move as way satellites do. By making some assumptions about the speed of the reindeer and the path they’re taking, I get to see how long Arthur and Grand-Santa would need to wait before the reindeer and sled are back if they’re lucky enough to be waiting on the equator.

Six Minutes Off makes the problem of Arthur and Grand-Santa waiting for the return of flying reindeer more realistic. This involves supposing that they’re not on the equator, which makes meeting up the reindeer a much nastier bit of timing. If they get unlucky it could make their rescue take over five thousand years, which would complicate the movie’s plot some.

And finally Arthur Christmas and the End of Time gets into one of those staggering thoughts. This would be recurrence, an idea that weaves into statistical mechanics and that seems to require that we accept how the conservation of energy and the fact of entropy are, together, a paradox. So we get into considerations of the long-term fate of the universe. Maybe.

What Second Derivatives Are And What They Can Do For You

Previous supplemental reading for Why Stuff Can Orbit:

This is another supplemental piece because it’s too much to include in the next bit of Why Stuff Can Orbit. I need some more stuff about how a mathematical physicist would look at something.

This is also a story about approximations. A lot of mathematics is really about approximations. I don’t mean numerical computing. We all know that when we compute we’re making approximations. We use 0.333333 instead of one-third and we use 3.141592 instead of π. But a lot of precise mathematics, what we call analysis, is also about approximations. We do this by a logical structure that works something like this: take something we want to prove. Now for every positive number ε we can find something — a point, a function, a curve — that’s no more than ε away from the thing we’re really interested in, and which is easier to work with. Then we prove whatever we want to with the easier-to-work-with thing. And since ε can be as tiny a positive number as we want, we can suppose ε is a tinier difference than we can hope to measure. And so the difference between the thing we’re interested in and the thing we’ve proved something interesting about is zero. (This is the part that feels like we’re pulling a scam. We’re not, but this is where it’s worth stopping and thinking about what we mean by “a difference between two things”. When you feel confident this isn’t a scam, continue.) So we proved whatever we proved about the thing we’re interested in. Take an analysis course and you will see this all the time.

When we get into mathematical physics we do a lot of approximating functions with polynomials. Why polynomials? Yes, because everything is polynomials. But also because polynomials make so much mathematical physics easy. Polynomials are easy to calculate, if you need numbers. Polynomials are easy to integrate and differentiate, if you need analysis. Here that’s the calculus that tells you about patterns of behavior. If you want to approximate a continuous function you can always do it with a polynomial. The polynomial might have to be infinitely long to approximate the entire function. That’s all right. You can chop it off after finitely many terms. This finite polynomial is still a good approximation. It’s just good for a smaller region than the infinitely long polynomial would have been.

Necessary qualifiers: pages 65 through 82 of any book on real analysis.

So. Let me get to functions. I’m going to use a function named ‘f’ because I’m not wasting my energy coming up with good names. (When we get back to the main Why Stuff Can Orbit sequence this is going to be ‘U’ for potential energy or ‘E’ for energy.) It’s got a domain that’s the real numbers, and a range that’s the real numbers. To express this in symbols I can write f: \Re \rightarrow \Re . If I have some number called ‘x’ that’s in the domain then I can tell you what number in the domain is matched by the function ‘f’ to ‘x’: it’s the number ‘f(x)’. You were expecting maybe 3.5? I don’t know that about ‘f’, not yet anyway. The one thing I do know about ‘f’, because I insist on it as a condition for appearing, is that it’s continuous. It hasn’t got any jumps, any gaps, any regions where it’s not defined. You could draw a curve representing it with a single, if wriggly, stroke of the pen.

I mean to build an approximation to the function ‘f’. It’s going to be a polynomial expansion, a set of things to multiply and add together that’s easy to find. To make this polynomial expansion this I need to choose some point to build the approximation around. Mathematicians call this the “point of expansion” because we froze up in panic when someone asked what we were going to name it, okay? But how are we going to make an approximation to a function if we don’t have some particular point we’re approximating around?

(One answer we find in grad school when we pick up some stuff from linear algebra we hadn’t been thinking about. We’ll skip it for now.)

I need a name for the point of expansion. I’ll use ‘a’. Many mathematicians do. Another popular name for it is ‘x0‘. Or if you’re using some other variable name for stuff in the domain then whatever that variable is with subscript zero.

So my first approximation to the original function ‘f’ is … oh, shoot, I should have some new name for this. All right. I’m going to use ‘F0‘ as the name. This is because it’s one of a set of approximations, each of them a little better than the old. ‘F1‘ will be better than ‘F0‘, but ‘F2‘ will be even better, and ‘F2038‘ will be way better yet. I’ll also say something about what I mean by “better”, although you’ve got some sense of that already.

I start off by calling the first approximation ‘F0‘ by the way because you’re going to think it’s too stupid to dignify with a number as big as ‘1’. Well, I have other reasons, but they’ll be easier to see in a bit. ‘F0‘, like all its sibling ‘Fn‘ functions, has a domain of the real numbers and a range of the real numbers. The rule defining how to go from a number ‘x’ in the domain to some real number in the range?

F^0(x) = f(a)

That is, this first approximation is simply whatever the original function’s value is at the point of expansion. Notice that’s an ‘x’ on the left side of the equals sign and an ‘a’ on the right. This seems to challenge the idea of what an “approximation” even is. But it’s legit. Supposing something to be constant is often a decent working assumption. If you failed to check what the weather for today will be like, supposing that it’ll be about like yesterday will usually serve you well enough. If you aren’t sure where your pet is, you look first wherever you last saw the animal. (Or, yes, where your pet most loves to be. A particular spot, though.)

We can make this rigorous. A mathematician thinks this is rigorous: you pick any margin of error you like. Then I can find a region near enough to the point of expansion. The value for ‘f’ for every point inside that region is ‘f(a)’ plus or minus your margin of error. It might be a small region, yes. Doesn’t matter. It exists, no matter how tiny your margin of error was.

But yeah, that expansion still seems too cheap to work. My next approximation, ‘F1‘, will be a little better. I mean that we can expect it will be closer than ‘F0‘ was to the original ‘f’. Or it’ll be as close for a bigger region around the point of expansion ‘a’. What it’ll represent is a line. Yeah, ‘F0‘ was a line too. But ‘F0‘ is a horizontal line. ‘F1‘ might be a line at some completely other angle. If that works better. The second approximation will look like this:

F^1(x) = f(a) + m\cdot\left(x - a\right)

Here ‘m’ serves its traditional yet poorly-explained role as the slope of a line. What the slope of that line should be we learn from the derivative of the original ‘f’. The derivative of a function is itself a new function, with the same domain and the same range. There’s a couple ways to denote this. Each way has its strengths and weaknesses about clarifying what we’re doing versus how much we’re writing down. And trying to write down almost anything can inspire confusion in analysis later on. There’s a part of analysis when you have to shift from thinking of particular problems to how problems work then.

So I will define a new function, spoken of as f-prime, this way:

f'(x) = \frac{df}{dx}\left(x\right)

If you look closely you realize there’s two different meanings of ‘x’ here. One is the ‘x’ that appears in parentheses. It’s the value in the domain of f and of f’ where we want to evaluate the function. The other ‘x’ is the one in the lower side of the derivative, in that \frac{df}{dx} . That’s my sloppiness, but it’s not uniquely mine. Mathematicians keep this straight by using the symbols \frac{df}{dx} so much they don’t even see the ‘x’ down there anymore so have no idea there’s anything to find confusing. Students keep this straight by guessing helplessly about what their instructors want and clinging to anything that doesn’t get marked down. Sorry. But what this means is to “take the derivative of the function ‘f’ with respect to its variable, and then, evaluate what that expression is for the value of ‘x’ that’s in parentheses on the left-hand side”. We can do some things that avoid the confusion in symbols there. They all require adding some more variables and some more notation in, and it looks like overkill for a measly definition like this.

Anyway. We really just want the deriviate evaluated at one point, the point of expansion. That is:

m = f'(a) = \frac{df}{dx}\left(a\right)

which by the way avoids that overloaded meaning of ‘x’ there. Put this together and we have what we call the tangent line approximation to the original ‘f’ at the point of expansion:

F^1(x) = f(a) + f'(a)\cdot\left(x - a\right)

This is also called the tangent line, because it’s a line that’s tangent to the original function. A plot of ‘F1‘ and the original function ‘f’ are guaranteed to touch one another only at the point of expansion. They might happen to touch again, but that’s luck. The tangent line will be close to the original function near the point of expansion. It might happen to be close again later on, but that’s luck, not design. Most stuff you might want to do with the original function you can do with the tangent line, but the tangent line will be easier to work with. It exactly matches the original function at the point of expansion, and its first derivative exactly matches the original function’s first derivative at the point of expansion.

We can do better. We can find a parabola, a second-order polynomial that approximates the original function. This will be a function ‘F2(x)’ that looks something like:

F^2(x) = f(a) + f'(a)\cdot\left(x - a\right) + \frac12 m_2 \left(x - a\right)^2

What we’re doing is adding a parabola to the approximation. This is that curve that looks kind of like a loosely-drawn U. The ‘m2‘ there measures how spread out the U is. It’s not quite the slope, but it’s kind of like that, which is why I’m using the letter ‘m’ for it. Its value we get from the second derivative of the original ‘f’:

m_2 = f''(a) = \frac{d^2f}{dx^2}\left(a\right)

We find the second derivative of a function ‘f’ by evaluating the first derivative, and then, taking the derivative of that. We can denote it with two ‘ marks after the ‘f’ as long as we aren’t stuck wrapping the function name in ‘ marks to set it out. And so we can describe the function this way:

F^2(x) = f(a) + f'(a)\cdot\left(x - a\right) + \frac12 f''(a) \left(x - a\right)^2

This will be a better approximation to the original function near the point of expansion. Or it’ll make larger the region where the approximation is good.

If the first derivative of a function at a point is zero that means the tangent line is horizontal. In physics stuff this is an equilibrium. The second derivative can tell us whether the equilibrium is stable or not. If the second derivative at the equilibrium is positive it’s a stable equilibrium. The function looks like a bowl open at the top. If the second derivative at the equilibrium is negative then it’s an unstable equilibrium.

We can make better approximations yet, by using even more derivatives of the original function ‘f’ at the point of expansion:

F^3(x) = f(a) + f'(a)\cdot\left(x - a\right) + \frac12 f''(a) \left(x - a\right)^2 + \frac{1}{3\cdot 2} f'''(a) \left(x - a\right)^3

There’s better approximations yet. You can probably guess what the next, fourth-degree, polynomial would be. Or you can after I tell you the fraction in front of the new term will be \frac{1}{4\cdot 3\cdot 2} . The only big difference is that after about the third derivative we give up on adding ‘ marks after the function name ‘f’. It’s just too many little dots. We start writing, like, ‘f(iv)‘ instead. Or if the Roman numerals are too much then ‘f(2038)‘ instead. Or if we don’t want to pin things down to a specific value ‘f(j)‘ with the understanding that ‘j’ is some whole number.

We don’t need all of them. In physics problems we get equilibriums from the first derivative. We get stability from the second derivative. And we get springs in the second derivative too. And that’s what I hope to pick up on in the next installment of the main series.

The Arthur Christmas Problem

Since it’s the season for it I’d like to point new or new-wish readers to a couple of posts I did in 2012-13, based on the Aardman Animation film Arthur Christmas, which was just so very charming in every way. It also puts forth some good mathematical and mathematical-physics questions.

Opening the scene is “Could `Arthur Christmas’ Happen In Real Life?” which begins with a scene in the movie: Arthur and Grand-Santa are stranded on a Caribbean island while the reindeer and sleigh, without them, go flying off in a straight line. This raises the question of what is a straight line if you’re on the surface of something spherical like the Earth.

“Returning To Arthur Christmas” was titled that because I’d left the subject for a couple weeks, as is my wont, and it gets a little bit more spoiler-y since the film seems to come down on the side of the reindeer moving on a path called a Great Circle. This forces us to ask another question: if the reindeer are moving around the Earth, are they moving with the Earth’s rotation, like an airplane does, or freely of it, like a satellite does?

“Arthur Christmas And The Least Common Multiple” starts by supposing that the reindeer are moving the way satellites do, independent of the Earth’s rotation, and on making some assumptions about the speed of the reindeer and the path they’re taking, works out how long Arthur and Grand-Santa would need to wait before the reindeer and sled are back if they’re lucky enough to be waiting on the equator.

“Six Minutes Off” shifts matters a little, by supposing that they’re not on the equator, which makes meeting up the reindeer a much nastier bit of timing. If they’re willing to wait long enough the reindeer will come as close as they want to their position, but the wait can be impractically long, for example, eight years, or over five thousand years, which would really slow down the movie.

And finally “Arthur Christmas and the End of Time” wraps up matters with a bit of heady speculation about recurrence: the way that a physical system can, if the proper conditions are met, come back either to its starting point or to a condition arbitrarily close to its starting point, if you wait long enough. This offers some dazzling ideas about the really, really long-term fate of the universe, which is always a heady thought. I hope you enjoy.

October 2013’s Statistics

It’s been a month since I last looked over precisely how not-staggeringly-popular I am, so it’s time again.
For October 2013 I had 440 views, down from September’s 2013. These came from 220 distinct viewers, down again from the 237 that September gave me. This does mean there was a slender improvement in views per visitor, from 1.97 up to 2.00. Neither of these are records, although given that I had a poor updating record again this month that’s all tolerable.

The most popular articles from the past month are … well, mostly the comics, and the trapezoids come back again. I’ve clearly got to start categorizing the other kinds of polygons. Or else plunge directly into dynamical systems as that’s the other thing people liked. October 2013’s top hits were:

  1. Reading the Comics, October 8, 2013
  2. How Many Trapezoids I Can Draw
  3. Reading the Comics, September 11, 2013
  4. From ElKement: On The Relation Of Jurassic Park and Alien Jelly Flowing Through Hyperspace
  5. Reading the Comics, September 21, 2013
  6. The Mathematics of a Pricing Game

The country sending me the most readers again was the United States (226 of them), with the United Kingdom coming up second (37). Austria popped into third for, I think, the first time (25 views), followed by Denmark (21) and at long last Canada (18). I hope they still like me in Canada.

Sending just the lone reader each were a bunch of countries: Bermuda, Chile, Colombia, Costa Rica, Finland, Guatemala, Hong Kong, Laos, Lebanon, Malta, Mexico, the Netherlands, Oman, Romania, Saudi Arabia, Slovenia, Sweden, Turkey, and Ukraine. Finland and the Netherlands are repeats from last month, and the Netherlands is going on at least three months like this.

Reblog: Making Your Balls Bounce

Neil Brown’s “The Sinepost” blog here talks about an application of mathematics I’ve long found interesting but never really studied, that of how to simulate physics for game purposes. This particular entry is about the collision of balls, as in for a billiard ball simulation.

It’s an interesting read and I do want to be sure I don’t lose it.

The Sinepost

In this post, we will finally complete our pool game. We’ve already seen how to detect collisions between balls: we just need to check if two circles are overlapping. We’ve also seen how to resolve a collision when bouncing a ball off a wall (i.e. one moving object and one stationary). The final piece of the puzzle is just to put it all together in the case of two moving balls.

Bouncy Balls

The principle behind collision resolution for pool balls is as follows. You have a situation where two balls are colliding, and you know their velocities (step 1 in the diagram below). You separate out each ball’s velocity (the solid blue and green arrows in step 1, below) into two perpendicular components: the component heading towards the other ball (the dotted blue and green arrows in step 2) and the component that is perpendicular to the other…

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Tipping The Toy

My brother phoned to remind me how much more generally nervous I should be about things, as well as to ask my opinion in an utterly pointless dispute he was having with his significant other. The dispute was over no stakes whatsoever and had no consequences of any practical value so I can see why it’d call for an outside expert. It’s more one of physics, but I did major in physics long ago, and it’s easier to treat mathematically anyway, and it was interesting enough that I spent the rest of the night working it out and I’m still not positive I’m unambiguously right. I could probably find out for certain with some simple experiments, but that would be precariously near trying, and so is right out. Let me set up the problem, though, since it’s interesting and should offer room for people to argue I’m completely wrong.

Continue reading “Tipping The Toy”