Compared to him we are all math-challenged

kw: book reviews, nonfiction, mathematics, mathematicians, memoirs, short biographies

The Weil Conjectures: On Math and the Pursuit of the Unknown, by Karen Olsson, is part biography (of André Weil and his sister Simone), part memoir, part rumination. The one thing it is not, is a description of the Weil Conjectures. Let's see why, beginning with this statement, not from the book, but from the Online Encyclopedia Britannica:

"Weil believed that many fundamental theorems in number theory and algebra had analogous formulations in algebraic geometry and topology. Collectively known as the Weil conjectures, they became the basis for both these disciplines."

The four (not two) disciplines mentioned are utter mysteries to everyone but a literal handful (really!, like four or five) of mathematicians who have a sufficient grasp of all of them for the quote above to make sense. Ms Olsson is not one of those few, but if she were, she might find describing these things to a popular audience (including me, an accomplished mathematician, but not in this league) a task similar to that shunted aside in 1965 by Dr. Robert Feynman; when asked after the Nobel Prize ceremony in which he was awarded the Prize, "Can you tell us briefly what it was that you did?", he replied, "Buddy, if I could tell you that in one minute, it wouldn't be worth a Nobel Prize."

In nearly every page the book switches gears, between a mini-biography of Simone Weil, one of André Weil, the author's memories of mathematical study (which she dropped after receiving her degree at Harvard), her more recent attempt to rekindle her early love of math, and vignettes about historical mathematicians whose work underlay that which André Weil labored upon.

The word "mathematics" literally means "learnings", emphasizing plurality. The power of the Weil Conjectures lies in making connections between and among areas that have been treated as separate, to show that they are all one. With that in mind, I find a multi-threaded book like this to be just the vehicle needed, to induce in someone like me at least a bit of the feeling that must have inhabited Weil as he considered his conjectures and the way the disciplines so treated reflected from one another to show mathematics as a conceptual whole. The satisfaction of finishing this book lies not in comprehending Weil's work, but in seeing that whole, though from afar.

He placed us in the sky

kw: book reviews, nonfiction, biographies, mathematicians, astronomers

Sunday coming, in just a few days, will mark the 549^th birthday of Nicolaus Copernicus, or very close to it. Born Niklas Koppernigk in German-speaking Varmia, then part of Prussia, now part of Poland, he Latinized his name as an adult. He is the classic example of one who, while seldom straying a day's journey from home, roamed the cosmos in his mind, saw better than others how it worked (at least the local billion miles or so), and wrote the book that dragged the rest of educated humanity, kicking and screaming, into the new view. His book On the Revolutions of the Heavenly Spheres was only removed from the Roman Catholic Index of Prohibited Books 177 years ago, after a residence of more than two centuries.

There are partial biographies of Copernicus aplenty, but none other as accessible yet so thorough as A More Prefect Heaven: How Copernicus Revolutionized the Cosmos by Dava Sobel. One great value of the book is placing Copernicus in his milieu as a canon, an unordained church official, nonetheless required to live celibate. His official duties occupied much of his time and energy. Yet he was able to compile a great mass of observations, particularly of the Moon. He had as a primary ambition the desire to revise the Moon's orbit; the accepted orbit inherited from Ptolemy required the Moon's apparent size to vary by a factor of four, which it clearly does not do. He succeeded in this, and these and other calculations convinced him that the Sun, not Earth, was at the center of all orbits except the Moon's.

Secondly, from close reading of many of his letters and other writings, Ms Sobel is able to open a window into the thought processes that enabled Copernicus to unveil a sun-centered Solar system. Now that he has led the way, we expect every child to come to the same realization, to understand that the apparently solid ground under their feet is in fact rotating about an axis that spins both it and them at hundreds of miles (or km) per hour. Not only so, the yearly motion about the Sun requires a planetary velocity of about 66,000 mph, or nearly 110,000 kph. Copernicus did not know this latter figure, having no accurate parallax for the Sun, but he guessed its magnitude within a factor of about ten.

Parallax is the measurement of distance using trigonometry, by measuring the angular difference between measurements taken at different locations. Accurate measurements of stellar occultations and near misses by the Moon, taken by observers a few degrees apart in longitude (say, one in London and another in Paris, if both have a clear sky!) suffice to measure the Moon's parallax of about a degree (normalized to Earth's radius). That is, if one observer plots the Moon's position just at moonrise, at the same time a colleague plots its position while the moon is near the zenith—because the two observers are some 10,000 km apart—, their measurements will differ by about a degree. But the Sun is 400 times as far away as the moon, and naked-eye observations cannot discern a parallax of 1/400 degree, or nine arc-seconds.

As a church official, Copernicus knew there would be strong opposition to his model of the Universe. He knew as well as anyone the verses in the Bible that implied the Earth is "fixed forever" and that the Sun moved, except when Joshua commanded that it stand still long enough for a key battle to be finished. He published a short document that outlined his mathematical calculations using a new system with only 33 epicycles (Ptolemy had needed more than 100), without clearly stating that he believed the Earth was moving about the Sun. He delayed publishing anything further, until he was prodded by a Lutheran mathematician named Rheticus. Even then he dragged his feet and the book was only finished as he lay on his deathbed. He died at age seventy, having, perhaps, seen the last few pages of the book's galley proofs, but leaving it to others to proofread and complete. In April of 1543 the book was finished, and he died in May.

So carefully had the writing been done, and so hedged about with arguments intended to deflect scripture-based criticism, that it took the Holy Office of the Inquisition (established in 1542) 73 years to determine it belonged on the Index, and only as a book needing "correction", not wholly prohibited. The prohibition was dropped in 1835, after 219 years.

Concerning one fascinating bit of detective work, the author relates the work of Owen Gingerich, who studied about 600 copies of early editions of On the Revolutions (or De Revolutionibus, as it is frequently called). The book had three major editions and several minor ones, and was always printed with wide margins. Nearly every copy has copious notes, which indicates that anyone who thought enough of it to purchase a copy, read it carefully and checked calculations and thought about its implications. Interestingly, Johannes Kepler's copy has fewer annotations than most, but those that exist are very thorough and insightful. Nearly the only major scholar of the years prior to 1700 for whom a copy of the book is not known is Newton! Yet his work indicates he was familiar with it.

A word about production values. I love a book as well thought out as this, with such care taken to make the reading pleasant. The carefully-chosen quotes that begin each chapter, the archaic type face for chapter titles, and the decorated letter used in preference to a bare drop-capital, all indicate a fine historical sense.

One of my professors once remarked about such embellishments, nearly universal in well-produced 19^th Century books, that they showed the author had enjoyed the writing, and presaged an enjoyable reading experience. What a contrast to the dryness of most modern historical publications!

The other Leonardo

kw: book reviews, nonfiction, biographies, mathematicians

Three hundred years before Da Vinci, there was Pisano, Leonardo of Pisa, who lived from about 1175 to at least 1240. He became a legend in his own time, then was nearly forgotten except for a peculiar story about rabbits that became attached to his nickname. Through great struggles modern scholars have pieced together the depth and breadth of his influence, which led directly to the changeover throughout Europe from Roman Numerals and the cumbersome methods of calculation they required, to the Hindu-Arabic Digits, modern arithmetic, and algebraic methods that, once symbolic operations were devised to make them more easily understood, continued and advanced to the present day. Yet, because of a nickname based on a few instances in which he signed himself filis Bonacci, or "offspring of Bonacci", he is known today primarily as the originator of the Fibonacci Series, based on the multiplication of hypothetical rabbits.

The amazing biography of Leonardo of Pisa, The Man of Numbers: Fibonacci's Arithmetic Revolution, by Keith Devlin, brings together in a rather small book everything we know about Leonardo, which is, surprisingly, rather little. Nearly everything known with certainty is based on a handful of statements he made about himself: how he was brought by his father to an African trade center, where he learned the Hindu-Arabic system, his travels to learn more from its expert practitioners, and his determination upon returning to Pisa to publicize the methods throughout Italy and beyond.

Much more is inferred from his works themselves: Liber Abbaci (frequently and wrongly spelled Liber Abaci) and De Practica Geometrie in particular. The former book, first released in 1202 and revised in 1228, brings together all the streams of number and calculation that he had researched and learned and devised on his own, into a scholarly text that is in itself a great education in arithmetic and basic algebra. The latter book did much the same for geometric methods. But it was Liber Abbaci that he himself revised into a simpler text (now lost), which was endlessly copied, frequently with some of the geometrical methods of the other book appended, and effected a revolution in, first, mercantile calculations, and later, mathematics in general.

A modern person conversant with algebra would find Liber Abaci to be tough sledding. I contains hundreds of examples, worked out for the reader, but every one is what we call a "word problem", and the solution is also in narrative format that takes a great deal of getting used to. It was not until a few generations after Leonardo's death that symbolic notation was developed into things like the dreaded 3x+4y=12, where in Leonardo's time one had to say, "Twelve are found of the first quantity tripled and four times the second quantity…", except I have used "quantity" to represent special words the mathematicians used for their unknowns. (Note: this is only half a problem. There would be a second set of quantities so one could use cancellation to solve the system.)

While such methods are cumbersome compared to modern algebra, they made things possible that were impossible with Roman numerals. Why, it is really quite difficult just to multiply XXVI by LIII (the answer is MCCCLXXVIII). The notion of putting a line above the I, V, X, L, C, D and M to multiply each by 1,000 came along centuries after Roman numerals fell out of use for everything except cornerstones, clock faces and movie credits. Thus, calculations that produced quantities greater than a few thousand were not attempted.

It was quite a feat of detective work that demonstrates how Leonardo was behind the revolution. Nearly half the book traces the threads that bind the story together. A great part of the problem was that there was little notion of plagiarism. It was commonplace for a new book to be written that consisted largely of extended quotes from other works, entirely without attribution. It is very rare to find a prior author mentioned or cited in a medieval manuscript. So Leonardo of Pisa was cited once or twice in his own lifetime, or just after, and no more than two or three times later, except for that persistent rabbit puzzle.

In case someone doesn't know this: Suppose you have a mated pair of young and fertile rabbits that can bear young once per month, and rabbits of age one month can mate for the first time, a month later to bear their own little rabbits. Suppose further that each litter is exactly two, a male and a female. How many rabbits, total, will you have after one, two, and up to thirty months? You start with 1 pair. A month later you have 2 pair, the original pair plus a newborn pair. The next month you have 3 pair: the original pair, the young pair now old enough to mate, and a pair of newborns from the original pair. And so it goes: 1, 2, 3, 5, 8, 13, 21, 34, 55,… each month's quantity is the sum of the prior two quantities. If you assume immortal rabbits, this goes on without end. This is the Fibonacci series, and I encourage those who find it a new thing to look it up. I'll not go into it here.

Today we live in a world of numbers. By the time we're grown, even if some of us "hated math", we find many kinds of simple calculations second nature. We check grocery receipts to make sure we got correct change; we can keep track of our own bowling scores; we buy gasoline and do a quick division to see how many miles per gallon we are getting. We are used to the numerals 9 8 7 6 5 4 3 2 1, and would be practically crippled without them. It is not too much to say we owe more to this Leonardo, than to the better known one.

Pythagoras 1, myself a half

kw: book reviews, nonfiction, mathematicians, mathematics, history

Almost any article of advice to authors states that every equation found in a "popular" book will reduce the book's potential audience by half. Let's see, there seem to be multiple equations on almost every page of a 260-page book, so the potential audience, out of seven billions of humans, must be no more than one. In fact, if you start with all the atoms in the universe and divide by two 260 times, the result is very close to one!

I sometimes call myself a working mathematician. That doesn't mean I have any degrees in the subject, but I do use a lot of math in my work. Not just adding and subtracting, but calculus, trigonometry, and linear analysis. But I have long known my limitations. I hesitate to mention it, but it was the math needed for senior Physics that induced me to change my major to Geology one semester before graduating, delaying graduation by more than two years. Nonlinear analysis and non-Euclidean geometries defeated me.

Then there are proofs. I can understand the occasional proof, and I can even take delight in a particularly facile one, as yesterday's post shows. I did learn to derive certain algebraic relationships. Luckily, once my career hit its stride, tons of math resources showed up on the Internet, and I've been able to look up the "how to" of a great many derivations that are difficult for me.

I still enjoy reading books in the 510-519 section of the Dewey Decimal system, and math history hits that right in the middle. The book cataloged 516.22/K is Hidden Harmonies: The Lives and Times of the Pythagorean Theorem, by Robert Kaplan and Ellen Kaplan. It discusses in enormous detail the Theorem (often called just PT in the book), and shows dozens of ways it has been derived, or proved, or used in other proofs.

In fact, proofs abound, and these were my downfall. I read through the first half of the book, found myself slogging, and eventually skimmed through the second half, digging out nuggets here and there. Luckily, I slowed down long enough to at least get the flavor of a derivation that has puzzled me. In standard books about Pythagorean Triples (explained next), I had seen these formulas (here goes 15/16 of my readership):

Given positive integers U and V which are mutually prime, one even and the other odd, find a, b, and c thus:

a = U²-V²
b = 2UV
c = U²+V²

Then a, b and c will all be integers and obey the relationship c²=a²+b². Such a set of three integers is a Pythagorean Triple, and the most familiar is 3, 4, 5, because 9+16=25. Try U and V as 2 and 1:

a = 4-1 = 3
b = 2x2x1 = 4
c = 4+1 = 5

Now if U and V are 12 and 9,

a = 144-81 = 63
b = 2x12x9 = 216
c = 144+81 = 225

OK, that's about as far as I ever got with Pythagorean Triples (PTs). I made them too easy to find, so I didn't dig into it further. Of course, lots of other people dug into it, some for their whole lives. One such, Waclaw Sierpinski, asked whether there is or is not an unending number of PTs for which a and b and c are each triangular numbers. A triangular number is one found by adding n consecutive numbers, starting at 1, such as 1+2+3+4+5 = 15. So far, this PT, 8778, 10296, and 13530 has been found. No other examples are known. Sierpinski is best known for his "gasket" consisting of a triangle in which the middle quarter (a triangle upside-down from the first) has been removed, then each remaining triangle has been similarly dissected, and so on infinitely. The result is a fractal with no area left at all. The result of 0.75ⁿ approaches zero as n approaches infinity.

OK, I am delaying the inevitable. This kind of playing around the edges is about the best I can do. I bow before a couple who are consummate, real mathematicians, the kind who can make more math, for the rest of us to either use if we are sufficiently competent, or stand aside while the gods of the math world do things with it that are beyond our grasp. I'll go back to my toy box now.

A coffee table math book

kw: book reviews, nonfiction, mathematicians, mathematics, photographs

Tic Tac Toe is a known game. That means the number of possible games is small enough that a complete analysis has been performed, and optimal strategy has been determined. If both players are experienced and make no mistakes, every game will end in a draw.

In the process of analyzing the game, the following graphic was produced, which summarizes every possible game, even the stupid ones, ignoring symmetry. There are about a quarter million of them.
This illustration is found on page 39 of The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics by Clifford A. Pickover. Before discussing the book further, let's look closer at the illustration. Click on the images for a larger version. The closeup below shows all games that begin with X in the upper-left corner.

This allows us to see a little further into the matrix, but there is quite a lot of detail here. I scanned at full resolution for the following little clip, which shows another level or two, but now we've reached the limitations of the printed page.

Although there are about a quarter million possible games, this can be reduced a lot by symmetry. There are about 27,000 actual games possible, but only just over 1,400 if players at least have the wit to block whenever possible. See the Mathematical Recreations discussion for details.

Back to the book. Dr. Pickover has traced mathematical skills back to the ability of ants to count: they find their way back to the nest by counting their steps. And we thought it took a mammalian brain to count! But we know little of animal abilities, so three essays suffice to dispose of the nonhuman animals.

I was a little taken aback by the fourth essay, on knots. Tying a knot to keep a bead from slipping hardly seems like a mathematical activity to me. I think the use of knots as a data-storage medium by the Incas some 5,000 years ago better fits the bill.

Though this is a fairly large book, fully half is full-page photographs or graphics; there are 250 short essays and 250 images. A lot of care went into selecting them. In a few cases, such as a pixieish image of Martin Gardner with his books, the mathematician in question is pictured, particularly for contributions that don't lend themselves to meaningful visualization. Or, in the case of Gardner, which of his thousands of Scientific American columns do we pick an image from?

It would be nice to have a smooth brass loopy ring like this to dip into soap solution and produce this minimal surface. This one (page 193) was computer-generated, and you can be sure the computer took lots longer to fine-tune the surface than a soap bubble film would. This leads me to think of the pronouncements of some (such as Max Segmark, discussed on page 516), that we live in a universe that is mathematics. The universe and the things in it just do what they do. Mathematically modeling what they do is frequently a memory-hogging, time consuming enterprise.

Like this bubble film: produce any un-knotted closed loop you like, and dip it in soap solution (add a little glycerine to strengthen the film). Presto! The surface that results will be a minimal surface, slightly modified by the Earth's gravitational attraction (So, to get a true minimal surface, do the experiment on the ISS, though "dipping" there has complications not found down here). Consider the "shortest path" problem. It is computationally difficult to find the exactly shortest, or quickest, path from here to, say, the Toronto Zoo. But if you tie a bunch of strings together with the exactly scaled knot-to-knot distances (or time-durations), for all the likely roads between here and there, you can solve the conundrum by simply picking up the knot that represents where you are, and the shortest route will be the one that goes straight to the destination without sagging. Then just write down all the routes. By the way, Google Maps and your GPS use a near-optimal route finder that is a lot quicker than one that could find the exactly optimal route.

This looks like another minimal surface, but in fact it represents an inside-out sphere (Page 255). Ordinary spheres have a constant, positive curvature, by definition. This item has a constant negative curvature. The kicker is that it has to go to infinity in two directions, but does so in a way that it has a finite volume.

A lot of math is like that, dealing with infinite things that aren't infinite when looked at another way. This may be why many people shy away from it. As a working mathematician, I've had to do some problems that work out like this: Add up everything Outside some curve of interest, then subtract the Universe, to get the sum of what is Inside. The ability of certain mathematical techniques to either add or subtract the Universe is both fascinating and appalling!

Another thing that puts a lot of people off of math is that it takes something like five years of advanced education to understand why the expression e^iπ+1=0 is in any way significant. Just consider this: π, pi, is the ratio of a circle's circumference to its diameter, while e, Euler's constant, is the basis of natural logarithms, which express things like our eye's adaptability: No matter how bright the illumination is, if it is cut in half, our response is the same. In a room dimly lit by a 10-candlepower light, reducing its brightness to 5 candlepower is quite noticeable. Go outside, where the illumination may be 10,000 candlepower. Shade half of it so there is now 5,000 candlepower, and the perceived reduction in light feels the same. This is a logarithmic response. So what do circles have to do with logarithms? Nobody really knows, but I do understand the math that makes that equation work.

The book, as any good history, is arranged in time order. Though the history of mathematics goes back at least 5,000 years, most of the discoveries have come in more recent times. As in any discipline, as new tools are discovered, it is possible to discover even more. The halfway essay, #125 (page 266) discusses a discovery from 1875, a mere 135 years ago, the Reuleaux Triangle, a curved, three-pointed shape that can be rotated inside a square such that it goes exactly into each corner in turn. It is the basis of drill bits that can cut a square hole. More recently, its ability to rotate in a potato shaped cavity is the basis of the Wankel engine (which the author does not discuss. I've owned a Mazda car with such an engine. Lovely machine).

Just for fun, I did another half-jump, to the two essays that straddle 137.5. They are dated 1946 and 1947. So one quarter of the mathematical discoveries discussed date since World War II and the development of electronic computing machinery. The computer is such a great tool for mathematicians, allowing us to see things we could only think about before. Not all are equally good at mental visualization. But the work of John Von Neumann, discussed in the 1946 essay, and the Gray Code (it is used for making communications less prone to error) of the 1947 essay are two of many breakthroughs that led to the computer-intensive world we have today. The lovely illustrations above are just two examples of things we can see that before we had to imagine.

Not only is the computer useful to show us things it is hard to draw, it excels in running simple, recursive processes that can produce quite complex patterns. Patterns like the ones on this Cone snail occur in cellular atomata, which have been studied since the 1940s, before there were computers to help out. If you know the algorithm to use, you can draw patterns like this on graph paper, just as Conway's "game of Life" can be done by hand on graph paper (yes, I'm just obsessive enough that, before the PC was affordable, I used to play Life on graph paper).

Today, the computer is used throughout mathematics, and there is a tendency to use it as a crutch. We need to remember that, to a mathematician, the "real" numbers used by a computer are actually rational numbers of a particular type. The computer can represent, in the "double precision" conventions in use now, either 2⁶⁴ (1.8×10¹⁹) or 2⁹⁶ (7.9×10²⁸) values. That is "all" that can be expressed. That is a lot, true, but between any two of them there are infinitely many values that cannot be expressed. That means a calculation such as the tangent of ten million radians cannot be calculated with as much precision as we might like. You have to divide the ten million by a value of π that is accurate to either 15 or 24 decimal digits, and subtract the part larger than 1. That is a 7-digit value, leaving only 8 or 14 digits after the decimal with which to do further calculations. Things like this lead to our continual need to improve analytical techniques that do not depend on digital methods. Most of the last quarter of the book presents problems that were solved primarily using digital methods.

So it is good to see John Horton Conway, as late as 1986, coming up with the "audioactive sequence", which makes sense only when spoken aloud. It begins 1, 11, 21, 1211, 111221; it is pronounced "one", "one one", "two ones", "one two, one ones" and so forth. It has a sister sequence in which you don't count the numbers in sequence, but simply tally them up: 1, 11, 21, 1211, 3112, 211213, 312213; and a 4 has to show up before the items get longer than six digits. While one can do a lot of things with this sequence using a computer, it is quite a bit of fun to fool with "by ear". You get other sequences by starting with something other than 1. Try it!

There's lots of room for mathematics to grow. A book like this brings a variety of the familiar and the obscure to a more level playing field, and just may convince a few folks that they aren't so bad at math after all.

The most mathematical mind

kw: book reviews, nonfiction, biographies, mathematicians

For one to live mathematics, there are certain consequences. Human life is, as Spock frequently said, "Quite Illogical". If ever there was a true Vulcan among us, his name was Grigory Perelman. Further, he is one of the luckiest persons ever; without a series of powerful protectors, he'd never have attained a situation in which he could hear about the Poincaré Conjecture, let alone become the one to prove it.

In the declining days of the U.S.S.R., mathematics became a favored enclave for some very eccentric Russians. Favored because the Party powers could not understand it, and its practitioners were clearly apolitical. Even so, some celebrated mathematicians were disgraced when any whiff of political interest was imputed to them. But many were protected by their very inscrutability, and by the sudden realization, over sixty years ago, that their labors were crucial to the arms race with the West.

The peculiar development of mathematics in Russia is outlined in loving detail by one if its beneficiaries, Masha Gessen, in her biography of another, Grigory (Grisha) Perelman, Perfect Rigor: A Genius + The Mathematical Breakthrough of the Century. This is not just an unauthorized biography; it is one that its subject would almost certainly object to. Perelman lives in his head so deeply that, if something doesn't interest him (and there is very little that does interest him), then it ought not be done, not by him, not by anybody. He has so far lived apart from the world, and now lives in opposition to it. Everything the author learned about him she learned from his acquaintances.

Perelman won several awards and prizes for his breakthroughs in mathematics. The first, many years ago, he declined to receive because he hadn't completed a solution, merely showed how a solution could be achieved. This was a portent. There was one which he did receive, and which led to extended visits to the West, primarily France and America. It was to be the last.

On 12 November 2002, Perelman posted a paper on arXiv and sent a covering email message to a handful of mathematicians whom he deemed competent to judge his work. This paper, accompanied at intervals by two others, completed the proof of the Poincaré Conjecture, a problam posed a century earlier, about whether a sphere of any dimensionality is truly the simplest topological object. He had spent about eight years producing his proof. It took his colleagues a further four to completely confirm it.

In the end, he was offered the Fields Medal and the $1Million Clay Millennium Prize, both of which he turned down. First, considering them politically tainted (he hated politics with the visceral disgust most people reserve for vipers), he refused them with strained politeness. Then, as various persons, including his closest colleagues (it is hard to say if he considered any to be friends) implored him, and tried to arrange a compromise, he reacted with increasing stridency, and finally aggressive vulgarity, and cut himself off from all former associations. His rigid rules include that nobody ought to force friendship, or favor, on another. He is now 44. One wonders what will become of him.

Considering the unique status of mathematics in mid-cold-war Russia, it is likely that, had his talents lain elsewhere—physics or music or writing, for example—he'd have gone nowhere and probably ended up dying in a Gulag. Only someone deeply hidden in the protective bubble of Russian mathematics could survive while at the same time living by rigid internal rules than none other could comprehend. He was more stiffly logical than any Pharisee (as Pharisees are portrayed in the Christian Gospels). Having the added handicap of being a Jew in officially anti-Semitic Russia, it is amazing how favored a life he lived, prior to recent days at least.

In a late chapter, Ms Gessen explores the notion that Perelman may be Aspergian, one "afflicted" with Asperger's Syndrome, also called "high-functioning Autism". He certainly fits the description, to which I can only say, it is unlikely that anyone could have solved the Conjecture who is not Aspergian. One Aspergian, John E. Robison, offers a clue: Learning about and conforming to social niceties takes a certain amount of brain power, probably quite a lot. If one never bothers to do so, whatever the reason, there is more brain power available for intellectual work. This squares with the experience of Aspergians I know. The high incidence of Aspergian tendencies among the most celebrated mathematicians attests to the same notion.

However Perelman came by his formidable mathematical intelligence, his last communication indicates he has decided to develop a different skill set. He simply said he was pursuing new interests. If he can hold out for a few years, and most people forget what he looks like so he doesn't get accosted in the street, he may again emerge, but in what form, only he knows. I wish him well.

A genius like no other

kw: book reviews, nonfiction, biographies, mathematicians

Here it is, the Dirac Equation for the electron, in the form it appears on his tombstone. The number of people who actually understand this equation is about equal to the number of symbols it contains, and may be fewer. The only thing I understand about it is that it is laid the foundation for the "Standard Model" of particle physics.

I've just finished reading the definitive biography The Strangest Man: The Hidden Life of Paul Dirac, Mystic of the Atom by Graham Farmelo. It is a largeish book, 438 pages plus 54 pages of endnotes and a great many bibliographic entries. The subject is worthy of the attention. As the author states in his final chapter, "Great thinkers are always posthumously productive." The Standard Model, string theory and antimatter are three of Dirac's legacies that continue as fruitful realms of endeavor, keeping a large number of scientists employed, many of them running the most expensive machines in the world.

The author points out several times that Dirac was always guided, rightly or wrongly, by mathematical beauty. A theory had to be beautiful for him to deem it worthy of being, possibly, true. This led to his disdain for renormalization, the technique that allows physicists to do electrodynamic calculations without being stumped by the theoretically infinite self-energy of the electron and other fermions. He thought it Ugly with a capital U. This quest for beauty led to the work that gained him a Nobel Prize at a nearly unheard-of young age: he received the Prize at age 31, for work he'd done just five years previously (the only Nobelist I've met, Charles Pedersen, was 83 when he received his Prize, for work done a few years prior to his 1969 retirement).

A characteristic of Paul Dirac's nature was a very direct way of speaking. While this led to nearly uniform abruptness of personal discourse—and lots of seeming rudeness—he became a highly sought-after public speaker, for he could make the concepts of mathematical physics and quantum mechanics clear to nearly any audience, giving them a feeling they got the gist of it, even though the math was far beyond their grasp. This is a rare gift. In the penultimate chapter, the author speculates that Dirac was "high functioning autistic", perhaps with Asperger's Syndrome. I know a couple of such people, and it is possible but by no means certain.

It is also possible that he, an exquisitely sensitive person, was stomped into deep hiding within a defensive shell by an absolute boor of a father. I know something of a life lived in secret, of building a shell to protect myself, being the only bright kid in a series of schools full of dolts. Fortunately, I had parents who were much warmer and involved than what is reported for Dirac. Then again, I am no genius.

Dirac was lucky to find a soul mate in Eugene Wigner's sister Manci. Paul and Manci were so different from one another that they could never compete, yet they were equally sensitive, and she became the unique one to draw him out of his shell…at times.

Dirac lived 4/5 of the Twentieth Century, from 1902 to 1984, from the ragged end of the horse-and-buggy era to ten years after we quit going to the moon, but were using town-sized machines to probe particles that endure only long enough to move a tenth of the diameter of a proton. In the quarter century since his death, how much farther have we gone? There is some chance that the latest big machine, the LHC, will begin to twang the "strings" that may underlie the structure of everything. Such a string is a million times smaller than the proton. My mind boggles.

Of course, my mind hit its boggle point about forty years ago when I hit my mathematical limit as a Junior Physics major, and switched to Geology. I have read Dick Feyman's Lectures, but I can't get very far with Dirac's The Principles of Quantum Mechanics. We mere mortals are glad for the demigods who can make math work because of its beauty.