Someone shows you a coin with a head and a tail on it. You watch him flip it ten times and all ten times it comes up heads. What is the probability that it will come up heads on the eleventh flip?

A novice gambler would tell you, “Tails is more likely than heads, since things have to even out and tails is due to come up.”

A math student would tell you, “We can’t predict the future from the past. The odds are still even.”

A professional gambler would say, “There must be something wrong with the coin or the way it is being flipped. I wouldn’t bet with the guy flipping it, but I’d bet someone else that heads will come up again.”

Yes I know the math student would really say “individual trials are uncorrelated,” not “we can’t predict the future from the past.” The lesson still holds.

Happy Labor Day, everyone.

Yes, you read that subtitle correctly. Let’s be clear: this book is probably not for you. That’s because you, I have no doubt, already love calculus. You carry a table of integrals in your back pocket, and you practice substituting variables to while away the time in the DMV. This isn’t the book for people who already appreciate the austere beauty of a differential equation, or even for people who want to study up for their AP exam.

No, this is the book for people who hate math. It’s for people who look at you funny and turn away at parties when you mention that you enjoy science. It’s for your older relatives who think you’re crazy for appreciating all that technical stuff, or your nieces and nephews who haven’t yet been captivated by the beauty of mathematics. The Calculus Diaries is the book for people who need to be convinced that math isn’t an intimidating chore — that it can be fun.

Know anybody like that? Any gift-giving holidays coming up?

Now it’s true, I know the author. In fact, I appear as a character in the book (to a certain degree of comic effect). I’m the one who gets soaked when we ride Splash Mountain at Disneyland, but also the one who maximizes his winnings at craps by clever betting in Vegas. You get the idea: this isn’t a textbook, it’s a tour through the real world (and occasional fantasy worlds), pointing out that math is all around us, and that perceiving it is kind of cool.

When you understand math, how you think about the world changes. Every day, we all change position by accumulating velocity, or do informal optimization problems when making a decision. But most people don’t know about the wonderful insights that math can add to these processes. You know, because you are a mathphile. But you are outnumbered by the mathphobes. You have a secret that they don’t know, but now there’s a way to share it. What are you waiting for?

Doesn’t make the current fiasco seem so bad, does it? That little blob on the left looks a lot smaller than the blob right next to it, representing Saddam Hussein’s dump of oil into the Persian Gulf during the first Gulf War. In fact, when you think about it, it looks a lot smaller. Which is weird, when you look at the numbers and see that the current spill is 38 million gallons (as of May 27), while the Iraqi spill was 520 million gallons, a factor of about 14 times bigger. The blob representing Iraq’s spill seems a lot more than 14 times the size of the blob for the current spill. You don’t think — no, they couldn’t have done that. Could they?

Yes, they did. When measure the diameter of the circle representing the Iraqi spill, I get about 360 pixels (in the high-res version), while the smaller spill is about 26 pixels — a factor of about 14 larger. But that’s the diameter, not the area. The area of a circle, as many of us learned when we were little, is proportional to the square of its radius: A = π r². The radius is just half the diameter, so the area is proportional to the diameter squared, not to the diameter. In other words, that big blob is about (14)² = 196 times the area of the little one, when it should be only 14 times bigger.

I remember reading on some other blog about this same mistake being made in a completely different context, but I have no recollection of where. (Update: it was at Good Math, Bad Math, sensibly enough.) Probably won’t be the last time.

Sometimes you’ll be happily calculating along, and wind up with an equation you wouldn’t want to meet at night in a dark alley. But, a quick flip through Abramowitz & Stegun frequently turns up your nemesis, along with handy tricks for disarming it. Additional satisfaction comes when you write the paper, and get to throw off lines like “The solutions to equation 4 are confluent hypergeometric functions (of the first kind)”.

However, it will be hard top my amusement when I once discovered that the solutions to my problem were closely related to Anger functions.

I can only hope that Dr. Hate and Professor Loathing someday derive equally useful function forms.

An eccentric benefactor holds two envelopes, and explains to you that they each contain money; one has two times as much cash as the other one. You are encouraged to open one, and you find $4,000 inside. Now your benefactor — who is a bit eccentric, remember — offers you a deal: you can either keep the $4,000, or you can trade for the other envelope. Which do you choose?

If you’re a tiny bit mathematically inclined, but don’t think too hard about it, it’s easy to jump to the conclusion that you should definitely switch. After all, there seems to be a 50% chance that the other envelope contains $2,000, and a 50% chance that it contains $8,000. So your expected value from switching is the average of what you will gain — ($2,000 + $8,000)/2 = $5,000 — minus the $4,000 you lose, for a net gain of $1,000. Pretty easy choice, right?

A moment’s reflection reveals a puzzle. The logic that convinces you to switch would have worked perfectly well no matter what had been in the first envelope you opened. But that original choice was complete arbitrary — you had an equal chance to choose either of the envelopes. So how could it always be right to switch after the choice was made, even though there is no Monty Hall figure who has given you new inside information?

Here’s where the non-normalizable measure comes in, as explained here and here. Think of it this way: imagine that we tweaked the setup by positing that one envelope had 100,000 times as much money as the other one. Then, upon opening the first one, you found $100,000 inside. Would you be tempted to switch?

I’m guessing you wouldn’t, for a simple reason: the two alternatives are that the other envelope contains $1 or $10,000,000,000, and they don’t seem equally likely. Eccentric or not, your benefactor is more likely to be risking one dollar as part of a crazy logic game than to be risking ten billion dollars. This seems like something of a extra-logical cop-out, but in fact it’s exactly the opposite; it takes the parameters of the problem very seriously.

The issue in this problem is that there couldn’t be a uniform distribution of probabilities for the amounts of money in the envelopes that stretches from zero to infinity. The total probability has to be normalized to one, which means that there can’t be an equal probability (no matter how small) for all possible initial values. Like it or not, you have to pick some initial probability distribution for how much money was in the envelopes — and if that distribution is finite (“normalizable”), you can extract yourself from the original puzzle.

We can make it more concrete. In the initial formulation of the problem, where one envelope has twice as much money as the other one, imagine that your assumed probability distribution is the following: it’s equally probable that the envelope with less money has any possible amount between $1 and $10,000. You see immediately that this changes the problem: namely, if you open the first envelope and find some amount between $10,001 and $20,000, you should absolutely not switch! Whereas, if you find $10,000 or less, there is a good argument for switching. But now it’s clear that you have indeed obtained new information by opening the first envelope; you can compare what was in that envelope to the assumed probability distribution. That particular probability distribution makes the point especially clear, but any well-defined choice will lead to a clear answer to the problem.

.

Let’s assume that the buses are supposed to arrive every 15 minutes. If the buses adhered to their schedule, and I showed up at a random time, I should generally have to wait roughly half the mean bus arrival time: 7.5 minutes. If the buses were totally random, then I would have to wait the average time between bus arrivals: 15 minutes (if you haven’t thought about this before, this statement should sound crazy; perhaps I’ll do a future post on it). So the question is: why did I always end up waiting roughly 30 minutes or more?

I always assumed that the Universe was conspiring against me. This is a common feeling in graduate school. However….

I just stumbled across a blog post of a friend of mine from graduate school, Alex Lobkovsky. In it, he discusses precisely this problem, and presents various reasons for the bunching of buses. I have no doubt that he was inspired from similar suffering. Perhaps at the very same bus stop.

At the end of the day, there’s a fairly straightforward solution. Imagine all of the buses are roughly on time. Now imagine that one bus (call it bus S) happens to fall behind. Because S is running behind, more time has elapsed since the previous bus has passed. This means that more waiting passengers have accumulated, at more bus stops. This in turn means that bus S has to stop more often, and has to pick up more people at each stop. Hence, bus S falls even farther behind. Which means even more people accumulate at each stop. Which means the bus falls even farther behind. And so on. In short: a slow bus gets slower and slower.

Now let us consider the bus behind bus S; we’ll call it bus F. Bus F starts out roughly on schedule. But because bus S is running late, less time than average has elapsed between when bus S last passed and when bus F arrives. This means fewer people have accumulated, at fewer stops. Which means bus F makes fewer stops, and picks up fewer people. Which means that it starts to run faster than average. Which means even fewer people accumulate. Which means it runs even faster. And so on. In short: a fast bus gets faster and faster.

Putting this all together: if a random fluctuation creates a slow bus, then it will get slower and slower, and the bus behind it will get faster and faster, until the two buses meet up. At this point, the buses stick together, and are essentially incapable of separating. Thus, in general, buses will bunch up. This will usually happen in pairs, though on occasion triples and even quads may occur. This argument predicts that the arrival of buses will be random, with pairs of buses arriving more often than not, being separated by on average double the mean bus separation. And this is precisely what I discovered, the hard way, shivering at the corner of 55th St. and Hyde Park Boulevard. (N.B. I spent a year in Berlin. There, the buses are fermions, and always arrive exactly on time. It’s the stereotype, but it turns out to be true.)

After writing this post, I found that wikipedia has already figured it all out. Regardless, it’s nice to know that my suffering was due to statistics, and not because the Universe is out to get me.

Nevertheless, as one gets older, reads more, and hopefully gains a more sophisticated knowledge of the subject, one’s tastes tend to change somewhat. In my case, a gradual shift in my interests from pure to applied mathematics, and finally to theoretical physics opened up an increasing range of giants to understand and respect. Somehow though, I have always retained a soft spot for Russell; perhaps because of his atheism, perhaps because of his breadth, but more so I think, because when I think of him I can quite viscerally recall the way reading about him made me feel about mathematics.

Because of this, while I have never become an avid reader of graphic novels, I’m hoping to get hold of a copy of Apostolos Doxiadis’ Logicomix, which I learned about via The Guardian, and which the web site describes as

Covering a span of sixty years, the graphic novel Logicomix was inspired by the epic story of the quest for the Foundations of Mathematics.

This was a heroic intellectual adventure most of whose protagonists paid the price of knowledge with extreme personal suffering and even insanity. The book tells its tale in an engaging way, at the same time complex and accessible. It grounds the philosophical struggles on the undercurrent of personal emotional turmoil, as well as the momentous historical events and ideological battles which gave rise to them.

The role of narrator is given to the most eloquent and spirited of the story’s protagonists, the great logician, philosopher and pacifist Bertrand Russell. It is through his eyes that the plights of such great thinkers as Frege, Hilbert, Poincaré, Wittgenstein and Gödel come to life, and through his own passionate involvement in the quest that the various narrative strands come together.

The web site contains a few samples of what to expect, plus a nice summary of the cast of characters. To a graphic novel newbie like myself, it isn’t obvious what to expect from a telling of this kind of sweeping academic story in such a format. But the team involved looks promising, and I’m sufficiently fascinated by the subject matter that I’m really looking forward to getting a look at Logicomix.

But really, why can’t we visualize things in more than three dimensions of space? Could a Flatlander, living in a world with only two spatial dimensions, learn to visualize our three-dimensional world? Could we somehow, through practice or direct intervention in the brain, train ourselves to truly visualize more dimensions?

I can think of a couple of explanations why it’s so hard, with different ramifications. One would be simply that our imaginations aren’t good enough to project our consciousness into a constructed world so very different from our own. Could you, for example, really imagine what it’s like to live in two dimensions? Sure, you can visualize Flatland from the outside, but what about asking what it’s like to really be a Flatlander? The best I can do is to imagine a line, flickering with colors, surrounded by darkness on either side. But the darkness is still there, in my imagination.

The other possible explanation is that the process of visualization takes up a three-dimensional space in our actual brain, preventing us from “tuning a dimensionality knob” on our imaginations. The truth is certainly more complicated than that (and I’m not experts, so anyone who is should chime in); the visual cortex itself is effectively two-dimensional, but somehow our brain reconstructs a three-dimensional image of the space around us.

Maybe this could be a new tantric discipline: visualization in higher dimensions. Or maybe the Maharishi already offers a course?

(via failblog)

• Exponential
• Arctangent
• Witch of Agnesi
• Stirling’s approximation
• Continuous but almost-nowhere differentiable

Consider this an open thread to recommend other stuff we should all be reading. Or your favorite functions.

If you are with me so far, suppose now that an advanced alien en route from Arcturus to the galactic center kidnaps you and drops you into some unknown, 2n-dimensional manifold. You suspect that it is Sⁿ x Sⁿ, but you are not sure.

Come on, the screenplay practically writes itself! I’m seeing Ewan McGregor, maybe Natalie Portman. Russell Crowe as the alien. SEEx could help with some of the mathy stuff. If any studio executives are reading this, call me, I’d be happy to bang out a treatment.

Seriously, the book is great fun, and as Scott says it’s surprisingly readable. Not really a popularization; neither equations nor high-level abstractions are shied away from. (After months of jousting with the “grammar checker” in Microsoft Word, I now deploy sentence fragments and the passive voice out of sheer spite.) But put into the hands of the right ambitious high-school student, it could be life-changing.

p.s. You haven’t really lived until you’ve seen Cliff Taubes do his little dance to illustrate the concept of “quantum fluctuations.”

Troy writes:

“I’m a high school math teacher who is trying to assemble an extra-credit reading list. I want to give my students (ages 16-18) the opportunity/motivation to learn about stimulating mathematical ideas that fall outside of the curriculum I’m bound to teach. I already do this somewhat with special lessons given throughout the year, but I would like my students to explore a particular concept in depth. I am looking for books that are well-written, engaging, and accessible to someone who doesn’t have a lot of college-level mathematical training. I already have a handful of books on my list, but I want my students to be able to choose from a variety of topics. Many thanks for all suggestions!”

There are some good suggestions in the comments, and some not so good ones. Surely our wise and mathematically sophisticated readers will be able to help. Add what you can there, and in the comments here if you like.

As far as I know, the only show that rises to this challenge is NUMB3RS, the CBS drama featuring Charlie Epps, a math professor at a suspiciously Caltech-esque university who teams up with his FBI-agent brother to solve crimes. The shows creators, Nicolas Falacci and Cheryl Heuton, had a goal from the beginning of creating an entertaining hour of television that would involve science in an intimate way. (I suppose math is almost as good.) As part of the effort, they’ve partnered with Wolfram Research to follow each episode with a web page delving into the various mathematical concepts that were discussed, including Mathematica notebooks to illustrate the various ideas:

The Math Behind NUMB3RS

Episode 11 this year was entitled “The Arrow of Time.” Here’s the opening:

You can see the full episode here; the math page is here. This stuff would make a great topic for a book.

Make your integrals dimensionless.

This probably seems silly to the theoretical physicists in the audience, who have a habit of changing variables and units to the point where everything is dimensionless and equals one. However, in astrophysics, you frequently are integrating over real physical quantities (numbers of photons, masses of stars, luminosities of galaxies, etc) that still have units attached. While students typically do an admirable job of setting up the necessary integrals, they frequently go off the rails when actually evaluating the integrals, as they valiantly try to propagate all those extra factors.

Here’s an example of what I mean. Suppose you want to calculate some sort of rate constant for photoionization, that when multiplied by the density of atoms, will give you the rate of photo-ionizations per volume. These sorts of rates are always density times velocity times cross section:

For a Planck spectrum of photons and a typical energy-dependent cross section above some threshold

which becomes

This integral looks like a rough customer. You can pull some factors out front, but you’re still left with that unpleasant business in the exponential. You’re also using an integrating variable that has units, making it a bit tougher to check the dimensions of your answer to make sure it’s sensible.

Instead, if you force the variable you’re integrating over to be dimensionless:

the integral reduces to something that you can start to wrap your brain around:

Now you have the business end of the integral out front, where you can check the units and the scaling of the answer to see if it makes sense. The integral is also something that is far simpler to evaluate (although in this case, it’s actually not a trivial integration, but at least you can recognize that early and plan on how to deal with it). If you’re in a situation where you have to integrate by parts, the dimensionless integral will save you a world of pain. Even if you make a mistake in evaluating the integral, you’re usually only off by a simple multiplicative factor like pi, or 2. All these things are good.

Wallace’s big, famous book was of course Infinite Jest, but among his other words was a quirky history of the concept of infinity, Everything and More. Like everything he wrote, it was sprawling and inventive and chock full of discursive footnotes. One such footnote now seems especially poignant:

In modern medical terms, it’s fairly clear that G. F. L. P. Cantor suffered from manic-depressive illness at a time when nobody knew what this was, and that his polar cycles were aggravated by professional stresses and disappointments, of which Cantor had more than his share. Of course, this makes for less interesting flap copy than Genius Driven Mad By Attempts To Grapple With ∞. The truth, though, is that Cantor’s work and its context are so totally interesting and beautiful that there’s no need for breathless Prometheusizing of the poor guy’s life. The real irony is that the view of ∞ as some forbidden zone or road to insanity — which view was very old and powerful and haunted math for 2000+ years — is precisely what Cantor’s own work overturned. Saying that ∞ drove Cantor mad is sort of like mourning St. George’s loss to the dragon; it’s not only wrong but insulting.

Yes, it’s true. The decathlon combines ten different track and field events, so to come up with a final score we need some way to tally up all of the individual scores so that each event is of approximately equal importance. You know what that means: an equation. Let’s imagine that you finish the 100 meter dash in 9.9 seconds. Then your score in that event, call it x, is x = 9.9. This corresponds to a number of points, calculated according to the following formulas:

points = α(x₀-x)^β   for track events,

points = α(x-x₀)^β   for field events.

That’s right — power laws! With rather finely-tuned coefficients, although it’s unclear whether they occur naturally in any compactification of string theory. The values of the parameters α, x₀ and β are different for each of the ten events, as this helpful table lifted from Wikipedia shows:

Event	α	x0	β	Units
100 m	25.437	18	1.81	seconds
Long Jump	0.14354	220	1.4	centimeters
Shot Put	51.39	1.5	1.05	meters
High Jump	0.8465	75	1.42	centimeters
400 m	1.53775	82	1.81	seconds
110 m Hurdles	5.74352	28.5	1.92	seconds
Discus Throw	12.91	4	1.1	meters
Pole Vault	0.2797	100	1.35	centimeters
Javelin Throw	10.14	7	1.08	meters
1500 m	0.03768	480	1.85	seconds

The goal, of course, is to get the most points. Note that for track events, your goal is to get a low score x (running fast), so the formula involves (x₀-x); in field events you want a high score (throwing far), so the formula is reversed, (x-x₀). Don’t ask me how they came up with those exponents β.

You might think the mathematics consultants at the International Olympic Committee could tidy things up by just using an absolute value, |x-x₀|^β. But those athletes are no dummies. If you did that, you could start getting great scores by doing really badly! Running the 100 meter dash in 100 seconds would give you 74,000 points, which is kind of unfair. (The world record is 8847.)

However, there remains a lurking danger. What if I did run a 100-second 100 meter dash? Under the current system, my score would be an imaginary number! 61237.4 – 41616.9i, to be precise. I could then argue with perfect justification that the magnitude of my score, |61237.4 – 41616.9i |, is 74,000, and I should win. Even if we just took the real part, I come out ahead. And if those arguments didn’t fly, I could fall back on the perfectly true claim that the complex plane is not uniquely ordered, and I at least deserve a tie.

Don’t be surprised if you see this strategy deployed, if not now, then certainly in 2012.

D is for djinn, E for Euphrates,
F is for fohn, but not like when I call the ladies.

But I digress.

The first song on the album is “789″, about the nefarious dealings of the number 7.

1, 2, 3, 4 and more makes 7
Why is six afraid of 7?
Cause 7 ate 9

Recently the eldest kid piped up: “Seven eats all the numbers. There are no more numbers after 8.” I asked why. “Well, seven ate nine, so it’s 7-8-10, so then seven ate ten, so it’s 7-8-11, so then seven ate 11, and then it just keeps going.”

So, the Barenaked Ladies just inspired my seven-year old to discover the principle of mathematical induction, which is one of the first techniques you learn when you venture into the land of advanced mathematics. The idea is that if you can prove that something is true for some integer n, and that it is also true for n+1, then it has to be true for all integers greater than n. So, for a simple (and somewhat silly) example, if you can first prove that if n>0 then n+1>0, and then you also prove that 1>0, then all positive integers are greater than zero. I remember having a hard time wrapping my head around this idea when I first bumped into it in high school (though I got over it in college after enough algebra classes with Michael Artin). I just find it pretty nifty that you can get the idea from a kid’s song.

Some of the entries are straightforwardly hard-core mathematical, such as the one from J. Doyne Farmer or this from Shing-Tung Yau:

Mathematical truths have a uniquely austere beauty in their own right, but the visual presentation of such results in the form of equations can be striking even if the concepts being expressed aren’t immediately accessible. (Yau is talking about Ricci Flow, a crucial element in the recent proof of the Poincare Conjecture.) Meanwhile, many of the entries take the form of metaphorical pseudo-equations, using the symbols of mathematics to express a fundamentally non-quantitative opinion (Jonathan Haidt, Linda Stone). Some of the entries are dryly LaTeXed up (David Deutsch), some are hastily scribbled (Rudy Rucker), some tell fun little stories (George Dyson), and some are painstakingly elaborate constructions (Brian Eno). Several aren’t equations at all, but take the form of flowcharts or other representations of processes, such as this from Irene Pepperberg:

My favorites are the ones that look formidably mathematical, but upon closer inspection aren’t any more rigorous than your typical sonnet, like this one by Rem Koolhaas:

Or the ones that are completely minimalistic, a la James Watson or Lenny Susskind. Note that the more dramatic your result, the more minimal you are allowed to be.

The big challenge, of course, is to choose just one equation. There are a lot of good ones out there.

But there are a lot of options out there, and it would be a shame to overlook any of them. So we’d be remiss not to mention the unique opportunities offered by the Maharishi University of Management. Founded by the Maharishi Mahesh Yogi, spiritual advisor to the Beatles, and led by John Hagelin, highly-cited theoretical physicist and occasional Presidential candidate, the MUM offers a — did I already mention “unique”? — set of experiences to the enthusiastic student. And that’s not even counting the Yogic Flying!

Here, for example, are some of the course descriptions for the undergraduate major in mathematics.

Infinity: From the Empty Set to the Boundless Universe of All Sets — Exploring the Full Range of Mathematics and Seeing its Source in Your Self (MATH 148)

Functions and Graphs 1: Name and Form — Locating the Patterns of Orderliness that Connect a Function with its Graph and Describe Numerical Relationships (MATH 161)

Maharishi Vedic Mathematics: Mathematical Structure and the Transcendental Source of Natural Law (MATH 205)

Geometry: From Point to Infinity — Using Properties of Shape and Form to Handle Visual and Spatial Data (MATH 267)

Calculus 1: Derivatives as the Mathematics of Transcending, Used to Handle Changing Quantities (MATH 281)

Calculus 2: Integrals as the Mathematics of Unification, Used to Handle Wholeness (MATH 282)

Calculus 3: Unified Management of Change in All Possible Directions (MATH 283)

Linear Algebra 1: Linearity as the Simplest Form of a Quantitative Relationship (MATH 286)

Calculus 4: Locating Silence within Dynamism (MATH 304)

Complex Analysis: Transcending the Real Numbers to a Simpler and More Unified Numbering System (MATH 318)

Probability: Locating Orderly Patterns in Random Events to Predict Future Outcomes (MATH 351)

Real Analysis 1: Locating the Finest Impulses of Dynamism within the Continuum of Real Numbers (MATH 423)

Set Theory: Mathematics Unfolding the Path to the Unified Field — the Most Fundamental Field of Natural Law (MATH 434)

Foundations of Mathematics: The Unified Field as the Basis of All of Mathematics and All Laws of Nature (MATH 436)

Now, sure, any old university will be offering courses in real analysis and set theory. But will they learn about the unified field, and locate the finest impulses of dynamism? “Vector calculus” sounds kind if dry, but “Unified Management of Change in All Possible Directions”? Sign me up!

Nobody ever said the Maharishi wasn’t a good salesman.

The N-body problem is one of the most famous, and easily stated, problems in mathematical physics: find exact solutions to point masses moving under their mutual Newtonian gravitational forces (i.e. the inverse-square law). For N=2 the complete set of solutions is straightforward and has been known for a long time — each body moves in a conic section (circle, ellipse, parabola or hyperbola) around the center of mass. In fact, Kepler found the solution even before Newton came up with the problem!

But let N=3 and chaos breaks loose, quite literally. For a long time people recognized that the motion of three gravitating bodies would be a difficult problem, but there were hopes to at least characterize the kinds of solutions that might exist (even if we couldn’t write down the solutions explicitly). It became a celebrated goal for mathematical physicists, and the very amusing story behind how it was resolved is related in Peter Galison’s book Einstein’s Clocks and Poincare’s Maps. In 1885, a mathematical competition was announced in honor of the 60th birthday of King Oscar II of Sweden, and the three-body problem was one of the questions. (Feel free to muse about the likelihood of the birthday of any contemporary world leader being celebrated by mathematical competitions.) Henri Poincare was a favorite to win the prize, and he submitted an essay that demonstrated the stability of planetary motions in the three-body problem (actually the “restricted” problem, in which one test body moves in the gravitational field generated by two others). In other words, without knowing the exact solutions, we could at least be confident that the orbits wouldn’t go crazy; more technically, solutions starting with very similar initial conditions would give very similar orbits. Poincare’s work was hailed as brilliant, and he was awarded the prize.

But as his essay was being prepared for publication in Acta Mathematica, a couple of tiny problems were pointed out by Edvard Phragmen, a Swedish mathematician who was an assistant editor at the journal. Gosta Mittag-Leffler, chief editor, forwarded Phragmen’s questions to Poincare, asking him to fix up these nagging issues before the prize essay appeared in print. Poincare went to work, but discovered to his consternation that one of the tiny problems was in fact a profoundly devastating possibility that he hadn’t really taken seriously. What he ended up proving was the opposite of his original claim — three-body orbits were not stable at all. Not only were orbits not periodic, they didn’t even approach some sort of asymptotic fixed points. Now that we have computers to run simulations, this kind of behavior is less surprising (example here from Steve McMillan — note how the final “binary” is not made of the same “stars” as the original one), but at the time it came as an utter shock. In his attempt to prove the stability of planetary orbits, Poincare ended up inventing chaos theory.

But the story doesn’t quite end there. Mittag-Leffler, convinced that Poincare would be able to tie up the loose threads in his prize essay, went ahead and printed it. By the time he heard from Poincare that no such tying-up would be forthcoming, the journal had already been mailed to mathematicians throughout Europe. Mittag-Leffler swung into action, telegraphing Berlin and Paris in an attempt to have all copies of the journal destroyed. He basically succeeded, but not without creating a minor scandal in elite mathematical circles across the Continent. (The Wikipedia entry on Poincare tells a much less interesting, and less accurate, version of the story.)

However, just because the general solution to the three-body (and more-body) problem is chaotic, doesn’t mean we can’t find special exact solutions in highly-symmetric conditions, and that’s just what Cris Moore and Michael Nauenberg have recently been doing. The image at the top really is an exact solution to twenty-one equal-mass objects moving in a figure-eight under their mutual gravitational attraction. They’re moving in a plane, of course, but that’s not strictly necessary; here’s a close relative of the figure-8, perturbed outside the plane.

From there you can just go nuts; here’s an example with twelve objects orbiting with cubic symmetry — four distinct periodic paths with three particles each.

Knowledge of this exact solution, plus $3.50, will get you a grande latte at Starbucks. Mathematicians have all the fun.