Wednesday night, an hour after dark, modern technology had one of its rare and welcome failures. Around my neighborhood, across the street, and as far as the eye could see, the lights went out, and stayed that way.
If you know that remarkable silence that comes when a loud noise suddenly stops, you know the kind of dark this was. Walking outside, you noticed it for what it was, instead of just the absence of light between houses. There were stars and a half-moon, so that it was a dark with texture and variety to it.
And for the first time, it was actually night as night really is, bringing with it the lost things of night. There were new sounds of tree-frogs and night birds. There were fireflies in remarkable numbers, filling up all the woods behind the house. There was just the night itself, like a road connecting everything now that the houses were no longer isolated in their separate, private lights.
In front of my house there is a streetlight that burns all night, every night, a vigilant guard making sure that astronomy was impossible. Now it was out, and the once-invisible stars appeared. Whole new fields opened up in the sky. I dragged the telescope out to the sidewalk, trying to figure out the new constellations as I did.
Among the tens of thousands of cataloged objects in the sky, only three are called Great. There is the Great Nebula in Orion, the Great Andromeda Galaxy, and M13, the Great Globular Cluster in Hercules, a tight glowing ball of a million stars. Before the lights came back on, I turned the scope to Hercules.
Well, there's a reason this cluster is called Great. It practically leaps through the eyepiece at you. I'm used to straining my eyes for fugitive gray traces of galaxies. Sitting on a field of absolute black, M13 is a huge sphere of dusty light, thrice the diameter of Jupiter, and fringed by a hundred distinct points of individually distinguishable stars within it. It's large, obvious, and breathtaking as looking through the scope of a rifle and finding a tiger looking back at you.
This was, in fact, the stargazing equivalent of shooting ducks in a pond. I pointed the scope in the general direction of a second cluster nearby, and there it was, like a neon sign, clear enough to count the stars at its edges, thirty thousand light-years away. It was open season in the sky.
So I passed a happy hour, looking at the sky and at the things of night while others looked for candles. The constellations turned; the fireflies made their cool, silent arcs: everything surreal as only the real can be.
A paper appeared today by Richard Arenstorf, Professor Emeritus of Mathematics at Vanderbilt University, with a proposed proof of the Twin Primes conjecture. It looks like a legitimate assault. Arenstorf is a respected figure in mathematics, and has apparently been at work on this proof for a couple of decades. More to the point, it's a 38-page paper. Crackpots usually run out of steam after five or six.
The math involved is daunting, and may or may not hold up to scrutiny. Arenstorf seems to throw every mathematical trick in the book at the problem, which leaves plenty of room for error. To get an idea of the complexity of argument and depth of lemmas, here's his conclusion:
...the first integral exists by (76) and comparison with J* 23 in (84b), while the second integral exists using (91a), (105) & (105a) and arguing as for J* 22 above. The next to last integral J* 31 in (84c) exists similarly. Finally, convergence of the integrals J* 0 and J* 1 of (84a) follows with (90) & (90a) for k = 2 & 1 in the same way as for J* 22, using (87a), (96) and (99) additionally, to establish the crucial estimates corresponding to (109) & (111).This completes the proof of convergence of the integrals J* k = J* k (s), (k = 0, 1, 2) in (71a) and implies their continuity and holomorphy, as for J* 22 above. By the remarks surrounding (71a) we have thereby proven our assumption A* and the existence of the iterated integral H(s) in (53); which implies with (65) that lim(e → 0) J(s, e) = H(s) for s in 1/2 = s = 3/4, as we have seen. Since now we have 2piH(s) = J* 0 (s) + J* 1 (s) + J* 2 (s), H(s) also is holomorphic on 1/2 < s < 3/4 and continuous on 1/2 = s = 3/4; i.e. our proof of the Main Lemma is complete; and in conclusion, Theorems 1 & 2 are validated.
If Theorem 1 is indeed validated, it's a commanding accomplishment in number theory, and the conquest of one of the great problems of mathematics.
Arenstorf's paper (pdf)
If you were to select one discipline to serve as an index of man's accomplishment, you could make no better choice than cartography. Maps of the world are maps of the mind; in what they show, and how much of it, they directly illustrate the measure of our discoveries.
Mapmakers have constantly grappled with the determination of shapes. In the early 15th century, an accurate map of the coast of Africa was no small accomplishment, as well as being a state secret. A curve one way meant a favorable bay while the other way meant an obstacle. Flat contested with curved for the official shape of the world for centuries. By the late 15th century the shape was decided, but size was still uncertain, Columbus having taken the world for about half the circumference it really is.
Two hundred years after that, the maps had gone up into the heavens and out among the planets, and elliptical versus circular was the new question of shape for the orbit of the earth and everything else celestial. And then in the 20th century the problem was once again flat versus curved. A flat universe held sway for a long time, but eventually curved held sway, with the idea of Einstein's closed but unbounded geometry winning out.
And now all that's left, it seems, in the last and largest map, is size. The latest numbers are in, and 150 billion light years or so seems to be the best new estimate for the width of the universe. Twice that would take you back to where you started, with plenty of New Worlds along the way. We can only see twenty or thirty billion light-years of that, though, since light from further off is still trudging its way to us.
But the question of size is complicated by the fact of expansion, and this is where all the interest lies to me. It's the first new cartographic problem in centuries, and the most profound. The orbiting Chandra telescope has confirmed, by an analysis of galactic clusters, what the Hubble scope's examination of supernovae suggested: that the universe is expanding not steadily as from the aftermath of one great explosive push, but at an ever-increasing rate, as if the explosion were still under way and going stronger all the time. This is exactly what should not be happening by the normal rules of mass, gravity, and momentum. But in looking at how far away different types of stars are, as opposed to how far they should be if they had been carried on a steady tide of expansion, an even more baffling picture emerges:
That's where we are now, in this second stage of acceleration. And if this hasn't startled you out of your seat yet, I'll make it plainer: at some point in the life of the universe as it was following the rules of physics as we know them, something utterly unknown came along, took hold, and speeded everything up.
This is called "dark energy", meaning "we are completely clueless".
To me this is stranger than time slowing down as you go faster, stranger than photons being in two places at the same time, stranger than electrons obeying Pauli exclusion. It is akin to the discovery of plate tectonics on Earth, the phenomenon that ensures that cartogrpahers a thousand years from now will still have jobs redrawing the continents.
So we stand now where Columbus did. We feel sure of a basic curvature. We are making our best guess at extent. And we are keeping our erasers handy.
Chandra and the map of the universe
Even if you haven't been to Las Vegas or Atlantic City, you've noticed that there is now a World Series of Poker and a World Series of Blackjack — not just in casinos, but riding the airwaves. People are playing cards on TV, playing well, and winning astronomical amounts of money doing it.
A quiet revolution has been going on in these venerable gambling games over the last decade. They've moved out of the closed world of casinos and into the mainstream. At the same time, the knowledge of playing them well has been transformed from something private and anecdotal to public and analytical. The revolution in cards is an information revolution.
In fact, these games are probably being played at the highest level of skill in history, a skill that's been dramatically improved in a very short period of time. The World Series of Poker, for instance, was most recently won not by a 20-year veteran of the game, but a novice with less than two years of experience. And knowledge has not only gone public; it has also been transformed into science.
If there is a world race for dominance in card-playing, the U.S. has won it (I almost wrote "hands-down"). This radical success, oddly enough, has been brought about by applying the same method that drives scientific research every day: make lots of money available to experimenters; let them compete as often as possible; reward any method that succeeds, no matter how outlandish, with more money. Lubricate the entire research machine with plenty of publication, because the sharing of knowledge is what elevates competition into research.
The inspiration for the Web was a faster system for spreading scientific publications to the widest range of researchers. Card playing hitched a ride. Blackjack Confidential magazine used to be a mimeographed broadsheet mailed to a handful of top-level players. When its subscribers began to put their thoughts on the web, it was like particle physics secrets escaping from FermiLab. The audience expanded vastly, and the "confidential" part evaporated. The English-speaking world became one giant research lab for perfecting blackjack tournament play.
It's been a resounding success. The games have flourished, and the casinos that host them are multiplying.
I mention all this for its own sake and because the same system is now at work in a new field. Less than ten years ago, the X-Prize foundation was created with a simple mission: let inventors compete for ways to get into space, and give them a lot of money for winning. The prize stands at ten million dollars now: around the same amount that will be handed out in card playing tournaments this year.
That's all there is to it. There are smaller prizes for smaller successes, as in cards, and the game is open to all. The effect is that the world has become one giant spaceship laboratory. Or tournament, if you will. And indeed, it seems to be working; one vehicle has powered its way to an altitude of 38 miles, not up to Earth orbit yet, but 37 miles higher than you or I. If poker and blackjack are any indicators, they should make it all the way in just a few years.
The X-Prize, the leading ship, and some blackjack
There is great power in seeing things as they are, by eye, and not through the separation of a photograph. That's the appeal of the telescope over a subscription to Astronomy, and by extension, the appeal of the Hubble's visible-light images over the traces of radio telescopes. In the sky or in Hollywood, it's more exciting to see the stars in person.
That's what makes the new images from the Hubble so visceral, I think. These are pictures of something, very possibly a planet, in orbit about a sun. The medium is infrared light, but an accurate cousin to what visible light might show. Extrasolar planets have been detected and described before, but never imaged, and that makes these photographs seem alive with immediate newness. Big dot and small dot, but they hold the eye.
As if recognizing this fact, the Large Binocular Telescope is taking its final steps toward completion in Arizona. The LBT utilizes two enormous parabolic mirrors, each 25 feet across. Multiple-mirror arrays aren't new — most have a great many small mirrors, like the 36 that make up the telescope at the Keck Observatory — but the LBT has the unique potential to live up to its name and become the largest pair of binoculars on the planet. I suppose no scientist worthy of the name would uncouple the intended light receptors, CCD devices far more sensitive that the eye. But if one did, even for a moment, look through this huge apparatus instead of at its camera screens, he would have Hubble eyes, seeing the live light straight from the stars. No one on the web site will admit it, but everyone on the project must be thinking it.
What Hubble has seen, and the Large Binocular Telescope
Computer programs, as they evolve, accumulate ever-increasing quantities of unused code that has been superseded, like a traveler that has packed for a winter trip and lingered through to the summer. Much, and sometimes most of a program can called upon rarely or not at all. Human and animal DNA is analogous: there are immense stretches of nucleotides that seem to do nothing at all. By comparison, the amount of working DNA that actually produces proteins is rather small.
Software can get away with this extra baggage because computing storage and speed have expanded so rapidly that there is almost no penalty for excess. Nature, by contrast, is more parsimonious: what isn't used is eventually abandoned. That creates a puzzle in the interpretation of "junk" DNA: if it does nothing, why is it still around?
Findings from the University of California now suggest that this idle DNA must do something after all, and something vital indeed. A comparative study shows that hundreds of very long human sequences turn up in lower mammals with no change whatsoever in the tens of millions of years that we evolved away from them. For such amounts of DNA to be preserved against mutation (or to have ended the species in which it mutated in the slightest) suggests that this material is not junk at all, but more important than anything else in human and animal genomes. How it exercises that importance without coding for the production of proteins is anyone's guess.
It may, like some software or that fire extinguisher in the kitchen, become productive only under exotic conditions that happen so rarely that they haven't been observed or understood, but are absolutely essential at those times. Or it may be that animal evolution really is like the evolution of software: unlike, say, an unused pair of wings, extraneous DNA is very cheap to carry around and can somehow resist change in ways that protein-producing DNA can't. The answer will be very interesting, not only for biology, but for what it will tell us about the kinship between the course of computer software over the last fifty years and the course of human software over the last fifty million.
Artwerk is a brilliant invention I made last night: a web site where I have to write nothing at all.
Google, the great god of search engines, is soon to go public. Thus equipped with earthly form, and girded about with dollars, it is expected to drive all before it.
In the shadow of this second coming of Google, I want to remind us of the endurance of the meek.
We have left behind the age of mainframe processing just long enough to forget that it existed. This was an era when a single vast computing service performed tasks of swift and miraculous power, to which a host of subservient users submitted their requests. So powerful were these devices that they often returned an answer in less than a second.
Those same answers we now get in milliseconds; supercomputer problems of the 70s are dispatched each time play a DVD. The monumental preserves of data that were stored and managed by mainframes we now keep on our individual machines, duplicated a zillion times over. "A mainframe on every desk" was the slogan of personal computing, and it's a prediction that has come true today.
Byron Miller spearheads a project called MozDex: it's an open-source, free search engine. It heralds, I think, the second overthrow of the giants. The project still relies upon a large central store of data and indices, but the programming for it all is there for the taking, and the only thing that stands between Google and a Google on every desk is hundreds or thousands times more storage and data-transfer capacity. And this is scarcely an obstacle at all, as experience bears out.
Our increasing interconnection via the internet blurs the line between personal and central machines. But I don't think it changes the picture ahead. The direction of computing has always been the same direction as transportation, to abandon large public devices for versatile private ones by the prodigious application of technology. A search engine on every machine means an always and instantaneously available internet of data, but more importantly, one that is organized to suit each user's personal goals. This means that searching will be replaced by finding: a list of what we want rather than a list of what is there. The internet will return to being a means of connection and not a means of storage.
You can keep up with the course of Miller's project, which he describes to some extent in a recent article.
I've been browsing in Ribenboim's Little Book of Big Primes, a terrific condensation of mathematics on everything prime. Ribenboim is generally matter-of-fact and unfazed by the ever-more-staggering records and results he reports, but in the section titled "Are There Functions Defining Prime Numbers?" he unfolds a fact that draws him up short; even he calls it "astonishing":
There exists a polynomial, with integral coefficients, such that the set of prime numbers exactly coincides with the values of the polynomial as its variables range over the non-negative integers.
That is, there exists a finite Diophantine equation whose positive roots contain the primes, only the primes, and all the primes.
This seems slightly miraculous, or at least a minor violation of a law of Nature. It's possible because of some careful relaxing of restrictions on the solution set of the polynomial. While it is limited to integral coefficients, we ignore any negative values it produces, and confine its domain to the traditional values of number theory: zero and the natural numbers.
This is just enough to create a very flexible system for expressing sets of numbers. It has variables, like algebra, but like first-order logic it can express OR by multiplication:
(a)(b) = 0
Since this has solutions in integers only when a or b is zero. Similaly, it can express AND by addition:
a + b = 0
Since this is true only when both a and b are zero. Other representations are possible (ab = 1, say), but taken together there is much of the power of first-order math with universal quantifiers, or of a constraint-programming system.
Here, for example, is the construction of a simple polynomial whose values are exactly the set of composite numbers. A value x is composite only when it has (at least) two distinct factors:
x = (a)(b)
with the condition that neither of them is x; that is, we disallow (1)(x) and (x)(1). This can be done by simply writing:
x = (a+1)(b+1)
Or:
x - (a+1)(b+1) = 0
With our confinement to non-negative values, the polynomial we seek is just:
x - (a+1)(b+1)
And you can verify for yourself that x cannot take the values 2, 3, and 5 with positive choices for a and b.
In 1975, J. P. Jones demonstrated that the Fibonacci numbers are a Diophantine set by way of this polynomial:
2xy4 + x2y3 - 2x3y2 - y5 - x4y + 2y
The derivation is hardly clear from the expression, but it's probably built on the formula for the Fibonacci sequence that describes each value as the sum of two others (hence the two variables in the expression above).
There is no such simple formula for the primes. If it's possible to build a finite polynomial for them, it must rely on a logical combination of mathematical facts about primes. At least two such contenders are possible; one is:
x is prime if and only if x>1 and for all y and z, y ≤ x, z ≤ x, either yz < x, or yz > x, or y=1, or z=1
Creating a polynomial from this is not a straightforward matter, unfortunately. In 1971 a method was proposed that could lead to a polynomial having degree 37 with 24 unknowns. This theoretical polynomial was soon after improved to degree 21, with 21 unknowns.
Finally, in 1976, Jones, Sato, Wada and Wiens actually created such a beast. It's of degree 25, in 26 unknowns.
(k+2){1 - [wz+h+j-q]2 - [(gk+2g+k+1)(h+j)+h-z]2 - [2n+p+q+z-e]2 - [16(k+1)3(k+2)(n+1)2+1-f2]2 - [e3(e+2)(a+1)2+1-o2]2 - [(a2-1)y2+1-x2]2 - [16r2y4(a2-1)+1-u2]2 - [((a+u2(u2-a))2 -1)(n+4dy)2 + 1 - (x+cu)2]2 - [n+l+v-y]2 - [(a2-1)l2+1-m2]2 - [ai+k+1-l-i]2 - [p+l(a-n-1)+b(2an+2a-n2-2n-2)-m]2 - [q+y(a-p-1)+s(2ap+2a-p2-2p-2)-x]2 - [z+pl(a-p)+t(2ap-p2-1)-pm]2}
An astonishing result indeed: all the primes in a paragraph.
More on this Diophantine representation
Indulge me in a bit of British royal genealogy for a moment, a brief stroll through a hall of 19th-century portraits that takes us from the throne of England by way of three generations to a surprise of geography and a mystery of personalities.
In 1840 Queen Victoria was married to Prince Francis Albert Augustus Charles Emmanuel, known thereafter as the Prince Consort. Both were 21, and in the course of the next 17 years produced no fewer than nine sons and daughters.
The last to be born, in 1857, was Beatrice — Her Royal Highness Beatrice Mary Victoria Feodora, Princess of the United Kingdom of Great Britain and Ireland, Princess of Saxe-Coburg and Gotha, Duchess of Saxony. In 1885, England's connection to Germany being strong as of old, she married His Serene Highness Heinrich Moritz, Prince von Battenberg.
Their son, Alexander Albert von Battenberg, married Lady Irene Frances Adza Denison in 1917. It was a new century, pulling itself up from the wreck of World War I, so Prince Alexander may be forgiven for not marrying a princess as his forebears did. Lady Irene's blood was quite blue, however: she was the daughter of the second Earl of Londesborough, and (jumping ahead a century) an ancestor of Lady Diana Spencer.
The Prince and the Lady had a daughter, Iris, on whom our attention falls. She was born January 13th, 1920 in Kensington Palace. At 21 — now Lady Iris — she married into the military, to Captain Hamilton Joseph Keyes-O'Malley, but the modern age, it seems, took its toll; they were divorced five years later. We might assume she lived a quiet and comfortable life thereafter. She did not. It was not a kind time for princesses:
After the divorce, she went to America in search of work and had a string of short-lived jobs, including selling brassieres and posing for a bubble-gum advertisement. In 1947 she was arrested for passing a worthless check in a Washington D.C. store. Lady Iris was cleared of the charge, but a check by immigration officials revealed that her visitor's permit had expired and was also working in the United States illegally. After a visit to Canada, she was permitted to return to the U.S. on a permanent visa. (Worldroots.com)
Ten years passed after that, with no word of Lady Iris that I can find. Then she reappears, and the noble line from Queen Victoria, on hard times certainly, but undimmed, makes a sudden, strange turn.
In May 1957, Lady Iris Victoria Beatrice Grace Mountbatten of Great Britain and New York married Michael Neely Bryan of Byhalia, Mississippi.
It is as if, walking down an oak-panelled hall lined with portraits of the aristocracy, one comes across B. B. King on the wall.
This remarkable fact is recorded without comment in the records of British royal peerage, but it ushers in a mystery. What traced an arc between northwest Mississippi and Kensington Palace, centered in New York? The place and the time, certainly, had something to do with it; it was Spring, it was rural upstate New York. Still, we are left wondering.
It seems that Michael Neely Bryan was a jazz guitarist; how he got that way, and out of Byhalia, I have no idea. Something got him out, and something worked its spell on a descendent of the British royal line — man, guitar, or both. Byhalia, 1956 population unknown, but no more than 700 a few years ago, is not so far from Memphis. Memphis blues and jazz might pick a young guitarist up and propel him further north and east, if he were good. But even the most successful musician would find it hard to attract a destitute princess, and impossible to lure a wealthy one. It was not a match made by the pocketbook.
Passion, then, and perhaps music. The marriage was not in New York City, but in Pound Ridge, fifty miles upstate. Starting in the '40s, it underwent something of a Renaissance. It gained a reputation as a home for artists, writers, and musicians; the population was probably over 1,000 by the 1950s. Benny Goodman was one of the first to purchase a historic home there. It is exactly the kind of place that would draw a Mississippi musician: close to the City, but comfortably rural.
If the union was passionate, it was accordingly swift. Divorce took less than a year, immediately after the birth of their son. The record is quiet on Bryan after that. One hopes he stayed on in Pound Ridge, playing from time to time in New York, nodding to Benny Goodman as they crossed on the train home. At some point fortune or hope took him to California; he died in Glendale in 1972. Lady Iris wandered, too, to Toronto and another brief marriage. She never left, passing away in 1982. Their son still lives in Toronto. He may or may not be a jazz guitarist. We can only wonder if he ever made the long trip to Byhalia to visit his grandparents, or whether Mississippi figures in his mind at all. He may have forgotten Byhalia, and it him. But he remains, with Michael Neely Bryan, part of the roll of British peerage, where strange circumstances may yet make him King.
Bach, books, and beer were some of the ingredients of an all-B party I had once. The Bach and beer, at least, were brought forth again when Matt Haimovitz came to Soulshine Pizza a couple of months ago. Jeremy Eichler has written it all up in the New York Times, and while he picks out the same high points I did, he writes about it better. Check out the article for his version of a great night.
NYT - Inkwerk/Inkwerk gets you in
Thursday I went to hear a solo piano performance in the local Episcopal church. It was a great performance for a number of reasons, the setting among them. Live music has an immediate and three-dimensional texture, and the small church is acoustically perfect. It made for intense, almost tactile listening.
But there were two other reasons that made this an unusual performance. It was the first recital I've heard in a room with windows. And it was one of the very few evenings of New Age music I've ever attended. If both of these hadn't worked together so well, it would not have been nearly as memorable.
Kater's music is pretty typical new-age piano, though not at all bad. It stands out for its richness of sound: he plays big with left and right hands, and the sound is very full and active, not minimalist and spare. Think foie gras rather than crisp salad. There's a lot of repetition of fairly short motifs, all very listenable, but not especially attention-getting.
But the moment he began playing, I suddenly noticed the windows. Concert halls don't have windows. They avoid such distractions because full attention always goes to the music. But this, I realized, wasn't concert music. It was made more for hearing than listening, and it didn't draw you into it. Instead, it directed you away from it and toward everything else. It was background music that enriched the foreground.
Inside, the music played, and outside the light became more graceful, the trees greener, the scene more dramatic, as if heightened listening was also enlivening sight. It's an effect I've never noticed before, because every other musical setting is designed to deprive the other senses. But I now think that one of the native qualities of this sort of music is that it adds to the senses instead of dominating them. That's something I've never credited new-age music with, and certainly key to finding the greatest value in it.
So break out your George Winston, or order up some Peter Kater, and then open your windows and let the light in. If you give this music the scenery it deserves, I think you'll find more in it that you might suspect.
