Following Scott Aaronson’s advice, I instructed the good folks at Amazon to send me a copy of The Princeton Companion to Mathematics. (In exchange for money, of course.) It’s sprinkled with gems like this, in the article on “Differential Topology” by my former professor Clifford Taubes:

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.”

Via Slashdot, I came across the following question

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.

If you’re like me, all too often while relaxing and watching a good procedural drama on TV you find yourself wondering, “How did they solve that differential equation so quickly?” That’s why we need more hit prime-time TV shows with web pages that explain the mathematical content underpinning each episode.

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.

Having recently slogged through grading an enormous pile of graduate-level problem sets, I am compelled to share one of the most useful tricks I learned in graduate school.

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.

David Foster Wallace died last night. He was found at home by his wife — apparently he hanged himself. It’s a terrible tragedy for American literature.

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.

Chad laments that we don’t hear that much about the decathlon any more, because Americans aren’t really competitive. I also think it’s a shame, because any sport in which your score can be a complex number deserves more attention.

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.

The Barenaked Ladies’ “Snacktime” is on very heavy rotation in my house these days. It’s officially an album for children (which explains the heavy rotation, because if kids like something once, they like it for approximately the next billion times). However, a lot of it is laugh-out-loud funny for adults. For example, from the alternate alphabet song:

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.

Edge.org has collaborated with the Serpentine Gallery in London on a fun kind of artistic event: a collections of formulas, equations, and algorithms scribbled (or typeset) on pieces of paper and hung from the gallery walls like honest-to-goodness pieces of art. I was one of the people asked to contribute, along with another blogger or two. You can check out the entries online.

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.

It’s that time of year when eager young students are deciding where to embark on, or to continue, their higher educations. You can see our advice-giving posts on choosing an undergraduate school and choosing a graduate school.

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.

This will be familiar to anyone who reads John Baez’s This Week’s Finds in Mathematical Physics, but I can’t help but show these lovely exact solutions to the gravitational N-body problem. This one is beautiful in its simplicity: twenty-one point masses moving around in a figure-8.

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. (more…)