A peek inside the mind of a mathematician

This is not new, but it's one of the very best math videos on the inter-tubes. It's designed by Bill Thurston, who's known not only for his mathematical talent (winning him a Fields medal), but for his incredible ability to teach.

The video is about Thurston's technique for everting a sphere. But the real gems are the little visual techniques (complete with sound effects), so important for this kind of mathematics. Thurston's video not only makes them accessible, but provides a rare peek into the mind of one of the greatest mathematicians of our time.




See also my note on the history of sphere eversion, and on a more efficient method; you might also try this software from the Geometry Center, which provides an even more hands on experience of turning a sphere inside out. Enjoy!

Springer Publishes Crank 'Proofs'

crank, n.
A pejorative term for a person who holds a belief that a vast majority of their contemporaries consider false. A "cranky" belief is so wildly at variance with commonly accepted truth as to be ludicrous, and arguing with cranks is useless, because they will invariably dismiss all evidence or arguments which contradict their unconventional beliefs. (Wikipedia.org)

Two propositions, Fermat's Last Theorem and the Goldbach Conjecture, truly stand out among cranky topics, in drawing the vast majority of bogus "proofs" (although the Riemann Hypothesis may be another contender.) This is well-known among publishers like arxiv.org and philsci-archive, which get plenty of crank submissions. These topics are generally treated with distrust. So it's just stunning that Springer, one of the most trustworthy scientific publishers, may have just printed a book containing crank "proofs" of both these propositions!

The author, Nico F. Benschop of Crank.net fame, has seen basically the same argument refuted by number theorist Robin Chapman seven years ago. (Chapman can be found dismissing this book anew in a sci.math post last week.) You can take a look at the work here through Google Books; on page 133, you'll find the following alleged "proof" of the Goldback conjecture:

Yes, it's three paragraphs of jargon, and it's not clear if it's worth the effort to decipher it. Thanks to John Baez for pointing this out -- Springer really managed to miss a whopper!

Saving Mathematics

The despicable state of K-12 mathematics in the US has been summarized in one charming, witty, furious article by Paul Lockhart. That's Lockhart the mathematician-turned-gradeschool-teacher, not Lockhart the astronaut. He currently teaches at New York's prestigious Saint Ann's School. Here's the article (it's well worth the 30-minutes it takes to read):

PDF (or alternatively PDF.)
Much of Lockhart's effort goes into cataloging what we all woefully remember: how math, as taught in most K-12 schools, is ugly and tedious. This is an absurd state of affairs. The practice of mathematicians is both artistic and refreshing. Something's gone wrong. And Lockhart's got a some great ideas about how to fix it:

what about the real story? The one about mankind’s struggle with the problem of measuring curves; about Eudoxus and Archimedes and the method of exhaustion; about the transcendence of pi? Which is more interesting— measuring the rough dimensions of a circular piece of graph paper, using a formula that someone handed you without explanation (and made you memorize and practice over and over) or hearing the story of one of the most beautiful, fascinating problems, and one of the most brilliant and powerful ideas in human history? We’re killing people’s interest in circles for god’s sake! (p.9)

It's always interesting to hear how different math-lovers manage to overcome their arid early math education. In my case, it was a book. My friend Patrick suggested it to me in my first year of college; it was Richard Courant's classic "What Is Mathematics?" and it was like reading about a completely new subject.

There are many books of this kind. The goal is not to batter you with formalism, but to help you develop the imagination and the curiosity and the discovery that makes mathematics so beautifully enjoyable. I remember my first experience with Courant's book like a breath of fresh air. And I've always wondered: why on earth wasn't math like that in school?

I think it can be. But as Lockhart laments, it's much more demanding to teach in this way. On the other hand, he assures us, it's much more rewarding as well. Here's hoping Lockhart's ideas take off.

Visualize the Eversion of a Sphere

It is famously possible to continuously deform a sphere until it's inside out. You have to imagine the sphere is something like a bubble, which can 'pass through' itself. This transformation is called an eversion.

The fact is established by a theorem due to Stephen Smale (1959 - JSTOR),
which is not constructive -- Smale was not able to tell us exactly how we can carry out this deformation. However, recent computer-minimization techniques have allowed us to describe many (constructive) ways to turn a sphere inside out.

The process is difficult for most of us to visualize. Fortunately, three University of Illinois mathematicians have made a 7-minute computer animation that illustrates how it's done. (Warning: this video is trippy.)


Smale's theorem says that any two immersions of a sphere a real n-dimensional manifold are regularly homotopic. Thus, in particular, it follows that a sphere sitting in 3-space according to the immersion,


is regular homotopic to the same sphere turned inside out (that is, to inverse immersion). Smale is also well known for having proven higher-dimensional versions of the Poincaré conjecture in the 60's.

Riemann Hypothesis Gets Proved Again

Yet another proof of the Riemann Hypothesis (RH) has been proposed by BYU mathematician Xian-Jin Li. Dr. Li posted his proof on the arXiv a few days ago. (Update, 11:05am.)

Why I'm Not Holding My Breath. Although it's tempting to get excited over a proof of RH, here are a few reasons why I'm not holding my breath:

1. Pitkanen's proof of RH (2001). Withdrawn by author due to errors.


2. Castro & Hahecha's proofs of RH (2001, 2002, 2006). Approach has been rejected.


3. Shi's proof of RH (2003). Proof contains errors.


4. de Brange's proof of RH (2004). Remember all the media attention this got? A counterexample was later produced.


5. Chun-Xuan's proof of RH (2005). Proof contains errors.


6. Aizenberg's counterexample to RH (2007). Withdrawn by author due to errors.


7. Madrecki's proofs of RH (2007a, 2007b, and 2007c). Proofs contain errors.


8. Han's proof of RH (2008). Proof contains errors.

These once-hopeful provers of RH are only the tip of the iceberg. But that's not what's really fearful about proposing a proof of RH. Check out the way that Stanford mathematician Brian Conrad layed into this nut job, for being arrogant about a purported proof!

I suppose you can't escape when duty calls. (A note on Conrad's last sentence: he's evidently not a historian of mathematics.) At any rate, the expert verdict on Dr. Li's proof will be out within a year or so. I'm willing to be patient on this one.

What is the Riemann Hypothesis? The Riemann Hypothesis is that the non-trivial roots of the Zeta function on C, given by


all have real component Re(s) = 1/2. In his famous 1859 paper, Riemann wrote that,

Without doubt it would be desirable to have a rigorous proof of this proposition; however I have left this research aside for the time being after some quick unsuccessful attempts, because it appears to be unnecessary for the immediate goal of my study...

You can read more about RH at Mathworld, about failed proofs at Matthew Watkins's homepage, and about the $1 million prize out for its solution at ClayMath.org.

Edit: Update, 11:05am. Field's medalist Terence Tao has pointed out an error in the proof. It is possible that there are more.

Group Theory Of Your Spine

In more examples of the unreasonable effectiveness of mathematics, there now exists a robust application of group theory to the human spinal cord.

Modeling the Human Spine. A few years ago, Australian DOD scientist Dr. Vladimir Ivancevic unveiled a new model of the human spine. The model, called the Human Biodynamics Engine, was designed to study how various stresses (like those on a soldier carrying a load) can result in spinal injury. However, while older models were built up from the physics of compression, bending and shear, Ivancevic built his model using group theory. He published his result last week on the arXiv.

The 'Spinal Cord' Group. The central idea is to describe the degrees of freedom of each vertebrae using a group. The group that does the trick turns out to be the Special Euclidean Group SE(3). SE(3) is equivalent to the direct product group SO(3) ⊗ R³, which is just the group of rotations around a fixed point:


together with the group of spatial translations:


(both in three dimensions).

Makes sense, right? Those are the degrees of freedom of a piece of your spine. From there, it's easy to model the spinal cord as one big direct product group, made up of as many copies of SE(3) as there are vertebrae.



Of course, to make this idea useful, Ivancevic must argue that most spinal injuries are due to what he calls an 'SE(3)-jolt.' This is, effectively, a certain kind of local perturbation of the forces acting on the 'spinal cord' group. However, if the data continues to support Ivancevic's hypothesis, then he may have discovered a very useful new way to think about these injuries.

History: Applications of Groups. Mathematical groups began their history in the early 19th century, as a play-thing for pure mathematicians. Abel and Galois famously used them to study the roots of polynomials (especially the quintic).

Since then, applications of group theory have turned up in unexpected places. Group theory became inseparable from the study of both quantum theory and relativity in the mid-20th century, leading Wigner to famously call mathematics in general "unreasonably effective." (I recommend Octavio Bueno's very informative analysis of this history.)

In the last two decades, applications of group theory have surfaced in biology and in medicine as well. For example, groups have been applied to population genetics, epidemiology, neurobiology, and even anesthesiology. However, few of these applications have been very robust. Most deal only with combinatorial facts about the symmetric group S_n, the group of permutations of n objects.

n-Player Chess

Have you ever wanted to play a single chess game with two other people? What about with three or four other people? Is it possible to design a chess game that supports any finite number of players? Here's a precise description of this challenge, followed by my attempt at a solution.

The Challenge: Create an n-player chess board that (i) preserves all the traditional pieces and their degrees of freedom; (ii) is divided into cells with eight directions each (F B L R + diagonal); and (iii) minimally changes the strategy of traditional chess.

First example: here is an interesting 3-player board (link no longer exists). But it doesn't meet the challenge, because it's composed cells with 12 directions each. This means that the pieces also don't move in the traditional ways (For example, the queen can move in 12 directions instead of 8).

Second example: Meignorant found a 3-player board that has 4-sided cells. This board almost preserves the traditional motion of the pieces -- but not quite! In the middle six cells, there are 10 directions, which means that the pieces acquire additional degrees of freedom there.

Another drawback of both of these examples is that it's not obvious how n players can be added, instead of only 3. Here's a board design that makes this trivial.

A Solution. Start with a circle. Divide it into n equal portions. Then divide each portion into an 4x8 grid (4 cells along the radius, 8 along the circumference). For n=3 players, what you'll get is a board that looks like this:
One new board rule: the only way that a piece can pass from one of the n regions to another is by going through the center. In the board drawn above, this means that the three white lines separating the 3 regions are impassible barriers. However, straight lines pass through the central point just as they appear to.

The first two conditions are easily satisfied. All the pieces can keep their original degrees of freedom, because every cell (including each middle cell) has 8 directions (F B L R diagonal). Indeed, each cell even lies in one column and one row, each of length 8 -- just like in traditional chess.

But what about the strategy? The strategy changes about as little as I think is possible. For example, when n=2, the game just reduces to something very similar to traditional chess. (Can you figure out what's different? This difference can actually be avoided, but not without giving the board an unusual topology.)

On the other hand, things definitely start to change in higher-player chess games, although I don't know the extent of this until someone actually plays a game.

Note that from any given player's side of the board, only 1/(n-1) of each of the other players' sides of the board is accessible. (In the image above, 1/(3-1) = 1/2 of the other two regions are accessible from any given region). This might be insignificant when n is small. However, it means that for games with more than 10 players, there are always be regions of the board that you cannot access without first moving to another region.

This feature of many-player games may actually help to limit the chaos, by limiting how many people can attack a single player in a single round. It also makes the following question especially interesting.

A Final Question. For what values of n can a `Knight's Tour' be completed in n-player chess?

A Knight's Tour is possible iff it's possible for a knight to complete a circuit in which it lands on every cell exactly once, and ends up back where it started. Many theorems have been proved about this for different board shapes. It has been proven, for example, that a Knight's Tour is possible on a traditional chess board.

However, solving the problem for n-player chess seems to be quite a bit more complex. But let me leave it as an open question for now.

Fundamental Rubik's Cube Problem Is Nearly Solved

Question: What is the maximum number of moves required to solve a Rubik's cube?

Nobody knows the answer yet. But it's either 21, 22, or 23 moves, and we're about to find out which one: we've got a determined programmer with a huge super-computer on the case.

Rapid Recent Progress. In 2006, Tom Rokicki showed that there are cube configurations requiring at least 20 moves to solve; this gave us a lower bound. Then last year, Kunkle and Cooperman proved (pdf) that any cube configuration can be solved in 26 moves; this gave us an upper bound.

But in recent months, rapid progress has been made. In March, Rokicki developed and posted a highly effecient solution-finding algorithm, with which he was able to reduce the upper bound to 25 moves. He now says he's already reduced that number to 23, and is currently working on 22.

Clearly this will all be over soon. In the worst case, Rokicki will have to keep going until he shows that every cube position can be solved in 21 moves -- then this would be the maximum, and the problem would be solved.

Solving a Rubik's Cube. There are roughly 40 quintillion possible positions on a Rubik's cube -- that's 40 with 19 zeros after it. So even a super-computer that can check a trillion cube positions a second (which doesn't exist) would require a length of time equal to the current age of the Universe in order to check every position.

The plan of attack is thus to vastly reduce the number of cube positions to be checked, and then apply a very efficient algorithm. Rokicki's algorithm, together with 8 GB of memory, allowed him to clip through 16 million positions per second, and gave him the 25 move upper bound. This algorithm is based on a similar algorithm developed by Herbert Kociemba, and used in his fascinating Cube Explorer software (which you can download for free).

However, to produce his most recent results, Rokicki had to head over to Sony Pictures Imageworks. Sony lent Rokicki their super-computers during the idle-time between productions, which allowed him to grind through cube positions much more quickly. Yes, the very same computers that brought you Spiderman 3 now bring you "God's Algorithm" for the Rubik's Cube. Unfortunately, you won't be able to impress your friends with arbitrary 23-move solutions using your laptop -- the computing power you would need is enormous.

History of Solutions. The Rubik's cube was developed in the 70's in Hungary, and released internationally in 1980. The first popularly distributed solution was published in David Singmaster's (1981) Notes on Rubik's Magic Cube.

Operating a Rubik's cube forms a mathematical group, the Cube Group. Knot theorist Morwen Thisthlethwaite, who was Singmaster's office-mate, noticed that subgroups of the Cube Group could be systematically described as nested one within the other. He was able to exploit this structural feature in order to produce an algorithm that solves the cube in a maximum of 52 moves.

Thistlethaite's approach is the kind of solution that one's inner-mathematician would really like to see: a solution that actually teaches us something about the structure of the Rubik's cube. Rokicki's brute-force combinatorics approach offers little in this regard, despite its practical effectiveness.

Toward the mathematical end, Douglas Hofstadter suggested in 1981 (reprinted here, chapter 14) that there are sophisticated group theoretic arguments pointing to an upper bound of 22 or 23 moves. Frey and Singmaster also conjectured in 1982 that the maximum number of moves required to solve the cube may be in the low 20's. These arguments are in intriguing agreement with Rokicki's results. However, the group theorists have made little progress on the problem since then, and so a super-computer generated solution may be all we can expect in the near future.

A 4-Line Proof of the Isoperimetric Theorem in 3D

Here's an example of what might be called "biological proof" of a mathematical claim.

Proposition (The Isoperimetric Inequality). The solid that minimizes the ratio of surface area to volume (SA/V) in Euclidean 3-space is the sphere.

A Biological Argument. Consider the large class of animals capable of changing their ratio of surface area to volume. (And note that these animals live -- approximately -- in Euclidean 3-space.) What do these animals do when it's cold? They curl up into a ball! More precisely, they assume the closest approximation to a ball that they can manage. This is because any exposed surface area is a place where heat is lost, and curling up into a ball minimizes that surface area. So a spherical or "ball" shape keeps animals warmer. Now, here's how these ideas can be turned into a 4-line "biological proof" of the above proposition.

1. Animals overwhelmingly assume a spherical shape when they are cold.


2. If animals overwhelmingly assume some shape when they're cold, then it is because that shape is the warmest.


3. The warmest shape is the one that minimizes SA/V.


4. Therefore, the sphere is the shape that minimizes SA/V.

It's a pretty easy argument, and you can write it down in four lines. At the same time, there simply are no easy mathematical arguments for this proposition in 3-Dimensions (for a nice survey, see Osserman (1978).)

Now, here come the tricky questions: what's the status of an argument like the one that I've given here? What does it allow me to infer? And especially: how does it compare to an argument in which, say, the conclusion is produced by way of the calculus of variations?

And just to ratchet the trickiness up another notch: there are apparently many of these non-mathematical arguments for mathematical claims. For example, a lovely economic argument for a mathematical conclusion was recently described about by Kenny, which he calls part of the "unreasonable effectiveness of the sciences in mathematics."

But let me encourage a little caution. Arguments with correct conclusions are easy to come by. An example: Bibble, babble, blat, blit: therefore, the primes are infinite. We must not be mislead by the fact that this argument for the infinitude of the primes (which is drivel) ends with a correct conclusion.

The biological argument above is analogous, to an extent (though I hope not quite drivel). Suppose that premise (1) turned out to be wrong -- for example, our empirical data may have been poorly gathered, or we may have interpreted it incorrectly. Then this little "proof" would be just another bad argument for a correct conclusion.

This doesn't mean that non-mathematical arguments for mathematical claims can't lead to new knowledge. I think that they can, and in many cases they do (is this mathematical intuition?). But my suspicion is this: these non-mathematical tricks are instructive only insofar as they lead to good arguments.

Update, 2 July 08. I just found out that George Pólya (1954, 170) has already suggested this kind of argument could be made. But the difficulty of the exercise remains, as I suggest above, in how you work out the details.

Get Started Using Automatic Symbol Insertion in MS Word

So you write with mathematical symbols. You're not interested in learning LaTeX. But you're fed up with going to Insert > Symbol every three seconds in Microsoft Word. Here is an essential trick that will simplify your life.

Begin by writing down all the mathematical symbols that you plan to use. Unfortunately, this will involve going to Insert > Symbol a bunch of times -- but just this once. For example, you might write down:

δ ε π ƒ(x) x →
Now here's the trick.

1. Select the first symbol on your list (in my case, δ).

2. Go to Tools > AutoCorrect Options. In the "Replace" box, enter a code that you'll be able to remember represents your symbol. I usually prefix the name of the symbol with a front-slash; so in this example, I'll enter \delta into that box.

3. The δ symbol itself should have already automatically appeared in the "With" box on the right.

4. Make sure the "Formatted Text" dial is selected if you want to preserve the italics and other formatting you used.

5. Click ok.

That's it! Whenever you enter \delta into Microsoft word, it will automatically be replaced by δ. Now, just repeat this process for each of the symbols you wrote down, and you will have permanently entered these headache-relieving codes into Word's autocorrect database. So when I write down

|\x - \pi| < \delta \rarrow |\fx - L| < \epsilon
what appears is

|x - π| < δ → |ƒ(x) - L| < ε
The trick works for longer expressions too. For example, if I selected the expression above and go to Tools > Autocorrect Options, I can enter \LimitImp into the "Replace" box. Now, whenever I type

\LimitImp
what appears is
|x - π| < δ → |ƒ(x) - L| < ε
Happy calculating!