A couple of commenters have asked what software package I use to create the graphs that appear in bit-player posts–illustrations like the one below, which is a slightly improved version of something I posted last week. Let’s call it Figure 1.

Prompted by these inquiries, I immodestly ask myself: Why do my graphs look so darn good? I immodestly answer: It’s not because of any packaged software! I don’t need a cake mix, or even a recipe. These are home-baked graphs, made from scratch out of locally grown organic pixels.

I have strong opinions about the aesthetics of scientific illustrations, and I could certainly spout off about the design elements of Figure 1, such as that putty-colored background, just dark enough to allow drop-out white grid lines, yet neutral enough to avoid competing with the data curves, which also have a distinctive color scheme on which I could discourse at length. Yes, I can talk the Tufte talk. But I think the commenters were really asking how I create the graphs rather than why they’re so elegant, and so I’m going to focus here on the practical programming problem.

Most of my experience in drawing pictures with a computer comes from the world of print publishing, where the final product is ink on paper rather than pixels on a screen. Compared with the online environment, print has some advantages, notably higher resolution (up to 1,000 dots per centimeter) and precise control over typography and color. But print also has obvious limitations: On a magazine page, there are no mouseovers or clickable buttons, and you can’t make a square knot twirl in 3D.

Thirty years ago, the big challenge for computer-generated illustrations was not how to draw the picture but how to get it out of the computer and onto the printing press. You couldn’t just export a PDF and place it in a Quark or InDesign document; none of those things existed. The only practical option was to print out the artwork, photograph it, and “strip” the negative into the page-size film that would be used to make the press plate. Because of this emphasis on printouts, most of the effort went into programming the printer rather than the computer.

The figure below is the first published computer-generated illustration I had a hand in creating. It appeared in Scientific American in 1983.

The array of 28² tiny bar graphs was produced with an Epson MX-80 dot-matrix printer, using escape codes to fire combinations of the eight pins in the printhead. Of course the MX-80 was a black-and-white device. The two-color illustration was created from two separate printouts. Also, the Epson letterforms were replaced with typeset characters.

The world of computer-generated illustrations changed dramatically with the arrival of PostScript, the “page description language” created by John Warnock and his colleagues at Adobe Systems (based in part on earlier work at Evans and Sutherland and Xerox PARC). PostScript was designed as a complete programming language rather than just a file format or a set of drawing commands. And something else set it apart as well: attention to details of graphic design. With most earlier software (such as programs based on the Apple Quickdraw library), trying to create publishable figures was an exercise in frustration. For example, the apparent weight of a line would vary depending on its orientation: lighter when vertical or horizontal, heavier when diagonal. PostScript allows very precise control over such niceties of presentation. To take another example, where lines meet the edge of a graph, you don’t want to have to choose between falling short and overshooting; PostScript provides the tools needed to make it look right.

(The version in the rightmost panel is created by allowing the colored lines to extend outside the background box, and then applying a clipping mask that cuts off all objects at the boundary of the box.)

Obsessing over minute details like these may seem comically fussy, but I believe that neatness counts in these matters. To some extent, illustration is an art of illusion. Graphs and diagrams work best when you can look through them rather than at them. The viewer should be seeing the underlying information or abstraction–the array of correlation coefficients, the function y = f(x), or whatever–rather than noticing the mechanics of how the drawing was constructed. A ragged edge is the kind of distraction that destroys the illusion.

Although PostScript was a giant step forward from the MX-80 command set, in the early years it was still just another printer language, not a computer language. The only way I could execute a PostScript program was to send it to a laser printer and wait to see what came out. Sometimes it was a long wait. I had no way of running a PostScript program on the computer itself. (Ghostscript came later.)

My first PostScript illustrations were created as hand-written PostScript programs; the same language was used both for doing the computations and for presenting the results. The faces at right were created in this way. (They were inspired by the work of Herman Chernoff and drawn to illustrate an American Scientist article by Robert Levine in 1990.) The dual role of the language caused me a moment of disorientation just now when I went looking for my records of this project. I found an EPS (encapsulated PostScript) file, which I knew was the finished illustration, but where was the source code? And then I remembered: It’s the same file! Open it up in Ghostscript or Adobe Illustrator and you see those silly faces smiling or scowling at you; open the same file in a text editor, and you see procedures for drawing elements of the faces:

   /draweyes
     { newpath
       dx dy eyewidth eyeheight 0 360 ellipse stroke
       ex ey eyewidth eyeheight 0 360 ellipse stroke
     } bind def
   /drawpupils
     { fx fy pupilsize pupilsize 0 360 ellipse fill
       gx gy pupilsize pupilsize 0 360 ellipse fill
     } bind def

Bill Casselman, the graphics editor of the Notices of the American Mathematical Society, still favors this direct-to-PostScript methodology. He has written an excellent guidebook, taking you from the basics of PostScript through an elaborate library for rendering three-dimensional objects.

But here I part company from Casselman; I’d rather not do all my computing in PostScript. It’s not that I have anything against the language itself, but the development environment is not to my taste. I therefore adopted the modus operandi of writing a program in my language of choice (usually some flavor of Lisp) and having that program write a PostScript program as its output. After doing this on an ad hoc basis a few times, it became clear that I should abstract out all the graphics-generating routines into a separate module. The result was a program I named lips (for Lisp-to-PostScript).

Most of what lips does is trivial syntactic translation, converting the parenthesized prefix notation of Lisp to the bracketless postfix of PostScript. Thus when I write (lineto x y) in Lisp, it comes out x y lineto in PostScript. The lips routines also take care of chores such as opening and closing files and writing the header and trailer lines required of a well-formed PostScript program.

But the lips interface is low-level, confined to drawing individual dots, line segments, rectangles and the like. Assembling a complete graph out of these primitives is tedious. For example, the grid of white lines in Figure 1 would have to be drawn one line at a time, with each line specified by a sequence of commands such as

    (newpath)
    (moveto u v)
    (lineto x y)
    (stroke)

Before you can issue those commands, you have to calculate u, v, x and y. Clearly, a higher-level front end is needed; like everyone else, I call mine plot.

At the core of any plotting program is a simple operation: mapping points from an abstract user space to coordinates in a rectangular pane, the page space. In Figure 1, the y axis runs from 0 to 5000; values in this range have to be scaled to the dimensions of the graph, which is about 300 PostScript points, or 11 centimeters. Mathematically, the transformation is straightforward. Indeed, if I wished I could leave all the arithmetic to the PostScript interpreter, simply passing in the appropriate matrix elements for scaling and translation. This is an attractive option; it would allow plot to work entirely in user space. But a few niggling details get in the way. Consider the tick marks along the y axis in Figure 1. Their vertical positions are conveniently expressed in user coordinates: one tick every 500 units. But what about the length of the ticks–their horizontal extent? This dimension is purely concerned with the appearance of the graph and has nothing to do with the content; it ought to be expressed in unscaled units of points or pixels.

Here’s a possible solution: Let everything inside the rectangular frame of the graph–the area with the putty-colored background in Figure 1–go through the scaling engine, but define everything outside the frame, including the tick marks and the axis labels, directly in page coordinates. If you think this is the final answer, take a look at Figure 2:

In this nonsensical graph (constructed just for this occasion), data points are indicated by stars, crosses and diamonds. The positions of those glyphs ought to be defined in user space, but the drawing commands that create the shapes are properly defined in page coordinates. If we tried to draw the glyphs in user space, their size and shape would vary with position in the graph.

What’s the best way to deal with this messy situation? Is there some tidy solution that will reconcile the two coordinate systems and allow all dimensions to be treated uniformly? I don’t believe so; it’s just in the nature of graphs to mix up elements from these two disparate realms. We look through a window into a world of data or mathematical abstractions, but we also draw our own little doodles on the window itself.

Of course there are solutions; they’re just not as pretty as I would like. My own strategy for coping is to attach extra information to each geometric point, indicating whether or not the x and y coordinates are to go through the scaling transformation. This is less troublesome than it might seem; from the user’s point of view, it’s almost always invisible.

In writing the lips and plot programs, I walk a path that is already worn smooth by many earlier footsteps. I don’t know who wrote the first computer program for plotting data, but it probably came soon after the first program for producing data. Today we have hundreds of clever, comprehensive, well-designed and well-maintained programs for plotting and graphing. Gnuplot is very capable; Grace is one I’ve never used but I’ve heard good things about it; Mathematica, Sage, R, MATLAB, Octave and the like all have elaborate graphics facilities built in; the Python world, as usual, has an overabundance of options; there are a few libraries for my beloved Lisp; you can even do dataviz online.

All of which raises the question of why I bother to roll my own. I’ll never keep up–or even catch up–with the efforts of major software companies or the huge community of open-source developers. In my own program, if I want something new–treemaps? vector fields? the third dimension?–nobody is going to code it for me. And, conversely, anything useful I might come up with will never benefit anyone but me.

The trouble is, every time I try working with an external graphics package, I run into a terrible impedance mismatch that gives me a headache. Getting what I want out of other people’s code turns out to be more work than writing my own. No doubt this reveals a character flaw: Does not play well with others.

In any case, the time for change is coming. My way of working is woefully out of date and out of fashion. PostScript is a technology that even Adobe seems to regard as outmoded. And making ultraprecise PostScript graphs is quite silly when their destination is the web; before I can put them online, I have to convert them to low-res PNG images. Furthermore, a PostScript-based workflow loses out on all the interactive richness of the web. These are deathly still images. How can I expect to earn any web cred when my work is not even clickable, much less multitouch-enabled?

If I continue in my stubborn, do-it-yourself mode, I could replace the PostScript back end with one that generates SVG. This wouldn’t be a major undertaking. But is SVG the right answer? It’s been around for more than a decade and you still don’t see much of it in the wild. And there are horrid browser incompatibilities. I suspect that Javascript (and JQuery) has a brighter future. And if I can get over my abreaction to libraries, there are plenty of options. Advice anyone?

Today I learned of the death of Herb Grosch, proud provocateur and mischief-maker of the computing industry. Anybody who ever knew Herb, however slightly or briefly, has a story to tell, so here’s mine.

Grosch held senior positions at major household-name companies (IBM, General Electric) as well as in the U.S. government; he also spent time at MIT and Columbia and was editor of Computerworld. But through it all he cultivated the role of the outsider, or even the outcast; he saw himself as a lone wolf; at IBM they labeled him a “wild duck.” He boasted that he was the second scientist ever hired at IBM (after Wallace Eckert) and, more important, was the first employee with facial hair. Even when he was an insider he was an outsider. Grosch was elected president of the Association for Computing Machinery–but as a dissident candidate, opposing the slate annointed by the nominating committee.

The free-spirited maverick is a celebrated figure in American culture, but not always a well-rewarded one. Wild ducks don’t get tenure, or a pension.

My brief encounter with Herb came ten years ago, when I was working on an article (PDF) about the uses of ternary notation in computing. I read somewhere that Grosch had proposed a ternary architecture for the Whirlwind, Jay Forrester’s enormous early electronic computer. The idea of a base-three machine probably seems weirder today than it did in the early 1950s, but all the same the proposal was not adopted, and most histories of the Whirlwind project say little or nothing about the ternary option. I had no luck finding a copy of Grosch’s memo on the subject, so I tried getting in touch with him directly. With the help of friends I tracked him down, though not in the first place I looked. He was in Riga, Latvia.

Grosch couldn’t supply a copy of the memo; most of his papers, he said, were buried in a landfill in Switzerland. The best he could do was point me to a passage in his autobiography, summarizing a conversation with Forrester:

I said what might be genuinely gainful would be to store a ternary digit in each core, and calculate in base-three rather than binary fashion. There were materials–some kinds of permalloy, as I remember–that had north, south and neutral stable magnetic states. I told him I had taught my Poughkeepsie evening classes at IBM about a special kind of base-three arithmetic I called “signed ternary,” in which zero was in the middle of the number range. In this curious system there was no need for algebraic signs, no problem about the sign of zero, and you rounded perfectly by dropping digits.

Jay being a stiff type, I refrained from calling the ternary digits “tits,” a name which had been the source of much boyish amusement in the Poughkeepsie classes.

By the way, I’m still looking for a copy of that memo. Here’s the reference: Grosch, H. J. R. 1952. Signed ternary arithmetic. Memorandum M-1496, Digital Computer Laboratory, MIT.

Of course I had to ask Herb how and why he’d found his way to Latvia. It was a long story, involving a small NSF grant, archives of Datamation magazine, a collection of Soviet-era computer hardware in an attic at Latvia University, and a romance that Herb hoped would develop into his fifth marriage. But the grant ran out, the marriage plans faltered, and Herb was heading back to U.S. in dire need of employment. His hopes, he said, were “not much above fast-foodery: editing, or tech writing, or even clerking at Barnes and Noble.” He was 82 at the time.

In the end, he did not have to stoop quite so far as editing or tech writing, or Barnes and Noble; he became Adjunct Distinguished Professor of Computer Science at the University of Nevada Las Vegas. But that posting didn’t last long. A few years later he turned up in Toronto, teaching the history of computing, but by then I’d lost touch with him.

In a “Dear Everybody” message some years ago, Grosch wrote:

I begin this letter on the fourth of six recent Binary Days, 01.11.01…. The next Binary Day will be nine years from now, 01.01.10, and I will write again then if I survive. Shorter recipient list, I assume!

Herb did make it until 01.01.10. He died January 25th. For more on his life and works, see the ACM obituary and a web site at Columbia maintained by Frank da Cruz.

Fred Gruenberger may well have been the first blogger on computational topics. When he was writing, back in the 1970s, there was no RSS, and so he distributed his musings in a monthly newsletter called Popular Computing. A typical issue was 16 or 20 typewritten pages–stapled, folded, stamped and delivered by mail. It was always worth reading.

Gruenberger had been working and playing with computers since the 1940s. For a long stretch he was at the RAND Corporation, the famous think tank in Santa Monica. Later he taught at Cal State Northridge. In addition to Popular Computing he was involved in the startup of Datamation magazine and published at least a dozen books. I haven’t been able to learn much about his later years; he died in 1998.

A slogan that appeared in some issues of Popular Computing proclaimed: “The way to learn computing is to compute.” I took this advice to heart, although I was hampered by a total lack of hardware. Later on I acquired a programmable calculator, which helped on some of the problems and exercises.

The problem reproduced above appeared in the December 1976 issue of Popular Computing (Vol. 4, No. 12). At the time, I made no attempt to work this one out, but evidently the problem seemed interesting enough to be worth filing away. When I came upon the old clipping recently, I gave it a closer look and realized I have no idea how to answer Gruenberger’s question, though the impediment now is not lack of hardware.

Gruenberger asks us to trace a planar path whose steps are indexed by the odd integers starting at 3. For each number N we turn right 90 degrees before taking a step if N is a prime congruent to 1 mod 6; we turn left 90 degrees before moving one unit if N is a prime congruent to −1 mod 6; otherwise we continue straight ahead in whatever direction we happen to be facing.

In his typewriter graphics, Gruenberger plotted the trajectory from N=3 through 97. Below I continue the path through N=199.

But something’s amiss here. Gruenberger wrote:

Eventually the path will cross itself, so that the cell containing 111 will also contain 147. Similarly, one cell will contain both 91 and 179.

Those two self-intersections are nowhere to be found in the diagram. When I first noticed this discrepancy, I assumed I must have made a mistake somewhere. (This eagerness to blame myself is not mere knee-jerk humility; I have years of experience to back it up.) Eventually, though, I concluded that it was Gruenberger who had made the wrong turn. I believe he mistakenly went left at 127, as shown in the brown trail below:

The brown continuation of the red path includes the two coincidences mentioned in Gruenberger’s problem statement. But the left turn at N=127 is incorrect, because 127 is a prime equal to (6×21)+1, and thus it should specify a right turn. The error is of no great consequence, but it does reveal something interesting: Gruenberger must have been plotting these paths by hand. Most likely he wrote a program to compute the series of residue classes, then traced out the trajectory on squared paper.

Setting aside this anomaly, Gruenberger was quite right that the path does intersect itself. Here’s the trail continued through N=1,001:

And if that’s not tangled enough, here’s what it looks like at N=10,001:

Gruenberger asks for “a list of the contents of those cells containing more than one number, arranged in the order of the smallest number in the cell.” It’s not hard to identify some cells that belong on such a list. The table below includes all multiply-occupied cells discovered when tracing the path up to N=1,001, sorted as Gruenberger requests:

                   x    y    values of N
                 -11   28    (137 337)
                 -15   27    (147 683)
                 -16   27    (149 349 685)
                 -18   26    (155 355)
                 -19   27    (159 691)
                 -19   28    (161 693)
                 -19   29    (163 695)
                 -17   31    (171 319)
                 -18   32    (175 315)
                 -19   32    (177 701)
                 -20   32    (179 703)
                 -22   31    (185 769)
                 -23   31    (187 771)
                 -24   31    (189 773)
                 -30   41    (245 269)
                 -30   42    (247 271)
                 -27   40    (281 733)
                 -26   40    (283 735)
                 -26   37    (289 725)
                 -23   35    (299 715)
                 -22   35    (301 761)
                 -21   35    (303 759)
                 -20   35    (305 757)
                 -17   27    (351 687)
                 -18   27    (353 689)
                 -17   24    (361 673)
                 -16   24    (363 675)
                 -15   24    (365 677)
                 -17   21    (379 667)
                 -17   22    (381 669)
                 -17   23    (383 671)
                 -20   22    (391 631)
                 -20   21    (393 633)
                 -20   20    (395 635)
                 -20   19    (397 637)
                 -22   19    (401 593)
                 -22   18    (403 591)
                 -22   17    (405 589)
                 -22   16    (407 587)
                 -27   15    (419 575)
                 -27   14    (421 573)
                 -28   14    (423 819)
                 -29   14    (425 549)
                 -32   14    (431 539)
                 -32   13    (433 537)
                 -26   10    (563 831)
                 -26   11    (565 829)
                 -27   13    (571 823)
                 -28   18    (607 811)
                 -22   32    (707 767)
                   4   -6    (923 971)
                   4   -7    (925 969)
                   4   -8    (927 967)
                   4   -9    (929 989)
                   5   -9    (931 991)

Is this list the answer to Gruenberger’s question? No, it’s not, because there’s no reason to stop at an arbitrary limit such as N=1,001. Indeed, the list above is not even a prefix of the complete answer. The smallest value of N appearing in the list is 137, but the trail will eventually revisit cells occupied by smaller values of N. For example, continuing the experiment to N=10,001 reveals a bunch of intersections quite close to the beginning of the path, including a site that’s visited five times:

                   x    y    values of N
                   1    0    (5 1621)
                   1    1    (7 1623)
                   2    1    (9 4725)
                   3    1    (11 1263)
                   3    2    (13 1265)
                   5    3    (19 1635)
                   6    3    (21 1637)
                   7    4    (25 7537)
                   7    5    (27 7319 7539)
                   7    6    (29 7505 7541)
                   6    6    (31 1643 7323 7503 7543)
                   6    7    (33 1645 7325)
                   6    8    (35 1647 7327)
                   6    9    (37 1649 7329)

One point still missing from this list is the origin–the site at x=0, y=0, N=3. Does the path ever revisit its starting point? If so, at what value (or values) of N does it come back home? Since I don’t know the answer to this question, I guess I’ll have to leave it as an exercise for the reader.

I suspect that the problem Gruenberger meant to pose (or thought he was posing) was to generate a list of self-intersection sites arranged in their natural order of occurrence–that is, the order in which the crossings are created when you construct the path starting from the origin. This natural-order list is not at all the same as a list “arranged in the order of the smallest number in the cell.” The natural-order list is easy to generate step by step. All you need to do is obey the left/right/straight rules, plot the resulting sequence of positions on the xy lattice, and leave behind a trail of breadcrumbs so you can check at each step to see if the site has been visited before. This task is a matter of straightforward computation–just the kind of assignment that Gruenberger favored. The natural-order list begins:

                   x    y    values of N
                 -30   41    (269 245)
                 -30   42    (271 247)
                 -18   32    (315 175)
                 -17   31    (319 171)
                 -11   28    (337 137)
                 -16   27    (349 149)
                 -18   26    (355 155)
                 -32   13    (537 433)
                 -32   14    (539 431)
                 -29   14    (549 425)
                 -27   14    (573 421)

Thus the prime path first crosses itself when N=269, a value that shares the same coordinates as N=245, namely x=−30, y=41. There are 56 such crossings up to N=1,001, and 112,988 self-intersections up to N=10⁶.

*     *     *

There is a wilder, conjectural answer to Gruenberger’s challenge–which I’m pretty sure he did not have in mind. It goes like this: Maybe the complete list of revisited values of N is simply the list of all N. In other words, maybe the Gruenberger prime path fills up the entire lattice of integers, crossing over itself everywhere many times.

In 1921 George Pólya published a celebrated proof that a random walk on the lattice of integers is recurrent in one or two dimensions, though not in higher dimensions. Recurrent means that the walk returns to each point along its length with probability 1, and indeed visits every point in its domain infinitely often. Is it possible that the prime path is also recurrent?

Pólya’s theorem is one of those mind-expanding results that seem impossible on first acquaintance, and then inevitable, and finally just so amazing that you want to go kiss a mathematician. I have to confess that I’ve never gotten all the way through Pólya’s original paper (it’s not long, but it’s in German). On the other hand, I can highly recommend a little book by Peter Doyle and Laurie Snell, Random Walks and Electric Networks, which gives several alternative proofs of the theorem; it was published in the MAA’s Carus Monograph series, and there’s a postprint available on the arXiv.

The key insight underlying Pólya’s result, as I understand it, is this: If you never revisit a former home, then you must be spending eternity somewhere else, and you can do that only if your universe has enough somewhere elses that you’ll never run out of new territories to visit. Suppose that, some eons after starting your journey, you find yourself at distance r from the origin. If you’re living in a one-dimensional universe, then there are just two places you could be at that moment, namely at +r or −r. It doesn’t matter how far you run; there are still just two points at any given distance from the origin. In two dimensions, a fugitive at distance r has a little more room to maneuver; the number of available points grows in proportion to r, forming a circle of radius r. But this is still not enough room to get lost in. Only in three dimensions or more is there a nonzero probability of escape. In three dimensions, the space available at radius r is proportional to r². In this three-dimensional world, the volume of empty space grows faster than a random walker’s expected distance from home.

What does all this have to do with Gruenberger’s prime path? Well, it’s no secret that the distribution of prime numbers looks convincingly random–if you look at it in just the right way. And in particular the distribution of primes in various residue classes, such as 6K+1 and 6K−1, seems to behave at least approximately like a random variable. All this suggests we might consider viewing the Gruenberger prime trail as if it were a random walk through the two-dimensional lattice of integers. Because the space is two-dimensional, it’s a good guess that the walk should be recurrent.

The original recurrence results of Pólya refer to a simple random walk, where at each step the walker chooses randomly among the available directions and then moves one unit in that direction. For example, in the two-dimensional lattice of integers there are four possible directions: north, south, east, west. The simple random walk is not the best model of the Gruenberger process, which is more like a nonreversing random walk–a path where on each step the walker can turn left or turn right or go straight ahead but can never make a 180-degree about-face. We can further refine the random-walk model of the Gruenberger process by biasing the choice made at each step to reflect the changing abundance of prime numbers. Primes grow scarcer as their magnitude increases; in the vicinity of a given value of N, the probability that a randomly chosen number is prime is approximately 1/log N. Since the Gruenberger path goes straight for all composite numbers and turns only when N is prime, the trail will have longer and longer straight segments, and rarer turns, as N increases. A random walk can mimic this behavior by choosing an action at each step according to this logic:

    if random(1.0) > 2/log(N)
       then go straight
       elseif randomboolean()
           then turn left
           else turn right

(The proportion of primes is given as 2/log(N) rather than 1/log(N) because the Gruenberger process is defined on odd numbers only, which immediately eliminates half of the composites.)

One way to compare the various kinds of random walks is to measure the root-mean-square displacement–the distance from the origin to the final position of the walker, averaged over many realizations of the random process. For a simple random walk, the RMS displacement for an N-step walk converges to \(\sqrt{N}\); for the nonreversing random walk the average displacement is \(\sqrt{2N}\). The biased random walk based on the distribution of primes also appears to yield an RMS distance proportional to the square root of the number of steps; numerically the curve looks something like \(\sqrt{10N}\). I’m not entirely sure that’s the true form of the curve, but the geometric details don’t really make much difference. If I understand correctly, all three of these random processes should be recurrent in the sense of Pólya.

Does the same reasoning apply to the Gruenberger prime path? There are two sides to this question.

The naysayer points out that Pólya’s theorem applies to random walks, but there’s nothing truly random about the sequence of primes. After all, we have a straightforward, deterministic algorithm for generating primes, as well as an efficient algorithm for testing whether any given integer is prime or composite. The essence of a random process is that every time you run it you get a different result, but there’s only one sequence of prime numbers, and so the Gruenberger prime path will come out exactly the same every time. According to this view of things, the kind of probabilistic reasoning that goes into the proof of Pólya’s recurrence theorem is out of bounds here. For randomness to make any sense, you need to average over some ensemble of independent instances. For example, you could average over the 50 salmon-pink paths in the graph below, which represent 50 independent realizations of a biased random walk; you can’t average over the prime path itself (green), because there’s only that one path.

The yeasayer retorts that a single path is all you need–if the path is infinitely long. Indeed, the salmon-colored trails above could be interpreted not as 50 distinct runs of a random process but as 50 segments of a single long path, which repeatedly loops around through the point at x=0, y=0, wanders off in various directions, and then comes back home yet again. In essence, everything that could possibly happen in an infinite set of random paths happens somewhere within a single infinite path; all possible variety is already present there.

I’m not sure how to settle this dispute between Dr. Yea and Professor Nay. When an argument hinges on the nature of randomness, the meaning of infinity and patterns in the distribution of the primes, I known I’m in over my head.

So I’ll leave that deep question unresolved and say a final word about a lesser curiosity. In the Gruenberger process, we’re using the congruence classes of prime numbers mod 6 as a kind of coin flip to decide which way to turn. Is it a fair coin flip? For small values of N, it certainly doesn’t look fair:

                               6x+1      6x−1
       primes <     100         11        12
       primes <    1000         80        86
       primes <   10000        611       616
       primes <  100000       4784      4806
       primes < 1000000      39231     39265

There’s a persistent excess of −1 primes, and the imbalance seems to be getting steadily larger. As a result, the prime path has a “winding number” that reaches 8.5 at N=10⁶; that is, the path makes eight and a half net counterclockwise revolutions. Does the windup continue with still larger N? I gather that the definitive answer is “Yes and No.” For more see the masterful paper by Andrew Granville and Greg Martin cited above.

[Correction 2010-02-19: reflected the accent on Pólya.]

The question again: Is there a four-coloring of the 17-by-17 grid in which none of the 18,496 rectangles have the same color at all four corners? As I said last time, Bill Gasarch would not have put a bounty on this problem if it had an easy solution. Over the past couple of weeks I’ve invested some 10¹⁴ CPU cycles in the search, and a few neural cycles too. I have nothing to show for the effort, except maybe a slightly clearer intuition about the nature of the problem.

If you generate a bunch of four-colored n-by-n grids at random, the average number of monochromatic rectangles per grid increases quite smoothly with n:

This gradual progression might lead you to suspect that the difficulty of finding or producing an n-by-n grid that is totally devoid of monochrome rectangles would also be a smooth function of n. The truth is quite different.

Finding a solution is easy for square grids of any size up to 15 by 15. The task suddenly becomes very hard at size 16 by 16. As for 17 by 17, it’s much harder still–and indeed is not yet known not to be impossible. (Details on the data behind this graph: For each size class from n=8 to n=17 I started with 1,000 randomly four-colored n-by-n grids. Then I applied a simple heuristic search (the first of the algorithms listed below) to each grid, running the program for 1,000 × n² steps. The graph records the number of times this procedure succeeded–i.e., produced a grid with no monochrome rectangles–at each grid size. Up to n=14, the search never failed; at n=16 and beyond, it never succeeded.)

This kind of sudden transition from easy to hard is a familiar feature in the realm of constraint satisfaction. Well-known intractable problems such as graph coloring and boolean satisfiability have the same structure. That doesn’t bode well for any of the simple-minded computational methods I’ve tried. Here’s a brief catalog of my failures. These are algorithms that I’m pretty sure are never going to pay off.

• Biased random walk. Start with a randomly colored grid. Repeatedly choose a site at random, then try changing its color; accept the move if it reduces the overall number of monochrome rectangles. This is the simplest of all the algorithms. None of the more-elaborate schemes is decisively better.
• Whack-a-mole. Find all the monochrome rectangles in the grid; choose one of them and alter the color of one corner, thereby eliminating the rectangle. In the simplest version of this algorithm, you choose the rectangle and the corner and the new color at random; in more sophisticated versions, you might evaluate the alternatives and take the one that offers the greatest benefit.
• Steepest descent. Examine all possible moves (for the 17-by-17 grid there are 867 of them) and choose one that minimizes the rectangle count.
• Lookahead steepest descent. Examine all possible moves, and then all possible sequels to each such move (for the 17-by-17 grid there are 751,689 two-move sequences); choose a sequence that minimizes the rectangle count. In principle this method could be extended to chains of three or more moves, but the cost soon gets out of hand. (The lookahead technique is the mirror image of backtrack search; it explores the tree of possible moves breadth-first instead of depth-first.)
• Color-balanced search. Allow only moves that maintain the overall balance of colors in the grid. For example, in a 16-by-16 four-colored grid, color balance implies 16 sites in each color. One way to maintain balance is to make moves that swap the colors of two sites. (There is no reason to think that a rectangle-free grid will have exact color balance; on the other hand, a solution for a large grid cannot depart too far from perfect balance. Thus a color-balanced search might be an effective trick for finding a neighborhood where solutions are more common.)
• Row-and-column-balanced search. Allow only moves that maintain the balance of colors within each row and each column of the grid. In a 16-by-16 grid, each row and column should have four sites in each of four colors. A simple way to maintain this detailed color balance is to search for “harlequin rectangles” with the color pattern \(\begin{array}{cc}a & b\\b & a\end{array}\) and permute them to \(\begin{array}{cc}b & a\\a & b\end{array}\).

Most of these techniques are greedy: At each stage the algorithm chooses an action that maximizes some measure of progress. On hard instances, a pure greedy strategy almost always fails; the search gets stuck in some local optimum. Thus it’s usually best to temper the greediness to some extent, occasionally choosing a move other than the one that yields the best immediate return. (The family of methods known as simulated annealing are more elaborate variations on this idea, based on insights from thermal physics.)

Here we see traces of three runs of the algorithm identified above as steepest descent, with differing values of a greediness parameter m. (The grid size is 15 &times 15.) At m=0 (no greediness at all), all moves are equally likely to be chosen, and the algorithm executes a random walk on the space of grid colorings. At m=1 (maximum greediness), the program always chooses the highest-ranked move, which works well until the system stumbles into a state where no move can reduce the rectangle count. A value of m=0.3 seems to be a good compromise. (I’ll say a little more below on the greediness parameter; indeed, I have a question about how best to define and implement it.)

After all this fussing with a dozen variations on local-search algorithms, I’m afraid the outlook for success is not promising. With a little patience and some tuning of parameters, any one of these algorithms can solve grids up to 15 &times 15. With a lot more patience and tuning, they’ll eventually yield answers for 16 &times 16. But none of the algorithms come even close to cracking the 17 &times 17 barrier. Solving that one is going to require a fundamentally new idea. Perhaps someone will find an analytic approach to constructing a solution, rather than blindly searching for one. Or perhaps someone will prove that no solution exists.

On the computational front, I suspect the best hope is a family of algorithms known in various contexts as belief propagation, survey propagation and the cavity method. I’ve been hoping that friends who are expert in these techniques might swoop in and solve the problem for me, but if not I may have to give it a try myself.

In the meantime, here’s the thing about greediness (an apt subject for this time of year?). We want to define a function greedy whose arguments are a vector of alternative moves ranked from best to worst and a number m such that 0 ≤ m ≤ 1. If the greediness parameter m is 0, the function returns a random element of the vector. If m = 1, the returned value is always the first (highest-ranked) move. Otherwise, we must somehow interpolate between these behaviors. One attractive notion is to return the first element of the vector with probability m, the second choice with probability m(1 − m), and so on. Thus for m = 1/2 the series of probabilities would begin 1/2, 1/4, 1/8…. For m = 1/3 the first few values are 1/3, 2/9, 4/27….

This scheme works just fine for a vector of infinite length, but there’s a problem with shorter vectors. Consider what happens with the procedure call greedy(v=[1, 2, 3, 4], m=0.5). We have the following table of probabilities:

     1 --> 1/2
     2 --> 1/4
     3 --> 1/8
     4 --> 1/16

But on adding up those values, we come up 1/16th short of 1. What happens to the missing probability? I took an easy way out, distributing the “extra” probability equally over all the elements of the vector. The code looks something like this:

function greedy(v, m)
  for i=0 to length(v)
     if (i==length(v))
        return v[random(length(v))]
     elseif (random(1.0) < m)
        return v[i]

This procedure seems to give sensible results, but I wonder if there might be a better or more natural definition of greedy probabilities. Also, the running time for my code is logarithmic in the length of the vector (assuming m < 1). Is there a constant-time algorithm that gives the same results? (We don’t know the length of the vector in advance, so merely precomputing the table of probabilities is not an option.)

William Gasarch is not the Clay Mathematics Institute. He isn’t paying a million bucks for proofs of famous conjectures. But Gasarch is putting up 17² of his own dollars for the solution to an intriguing little stumper. And the prize problem appears to be somewhat easier than the Riemann hypothesis or the P=NP question. (Unless it’s impossible!)

Gasarch sets forth his prize challenge in a blog post, with further background in a paper and in the slides from a talk. All of those works are well worth reading, but for those who don’t want to chase down the references, here’s the gist. Our mission, should we choose to accept it, is to color the nodes of an n-by-m grid, using only a specified number of colors, and observing a particular constraint: Nowhere in the grid may the four corners of a rectangle all have the same color. (Only rectangles with sides parallel to the x and y axes are considered.) For example, here is a four-colored 15-by-15 grid that satisfies the no-monochromatic-rectangles constraint:

In this array of dots there are \( {15 \choose 2}^2=11025\) distinct rectangles. If you care to check through all of them one by one, you’ll find that in no case do all four corners have the same color. In contrast, here is a 16-by-16 grid that is almost but not quite rectangle-free:

Gasarch offers his $289 bounty for any four-colored 17-by-17 grid with no monochromatic rectangles. Why is that grid of particular interest? It’s a border case. Among square grids, all those up through 16-by-16 have been shown to have rectangle-free four-colorings. For the 18-by-18 grid and all larger squares, rectangle-free four-coloring has been proved impossible. For squares larger than 18-by-18, four-coloring has been proved impossible. The status of the 17-by-17 and 18-by-18 grids remains unsettled, but Gasarch believes that both are four-colorable.

Gasarch has much more to say about the mathematics behind this problem. Here I would like to muse on some computational aspects of searching for a 17-by-17 four-coloring.

To state the obvious first, this is not a problem we can expect to solve by exhaustive enumeration. There are 4²⁸⁹ possible colorings of the grid. Casting out symmetries brings that down only to 2 × 4²⁸⁷. There’s not world enough or time for checking them all.

Testing grids at random is also hopeless in the 17-by-17 case. This nonstrategy actually works quite well for small grids. For example, you can readily find a four-coloring of an 8-by-8 grid just by generating a few hundred thousand random colorings. But the method fails for larger grids because the proportion of all colorings that are rectangle-free falls steeply with grid size. (Consider the 2-by-2 grid: There are 256 four-colorings, and all but four of them are rectangle-free.)

To make any progress toward the 17-by-17 case, we’ll have to do at least a little thinking, rather than expecting the computer to do all the work. Here’s one idea that’s very easy to implement: Find a monochrome rectangle somewhere within the grid, change the color of one of its corners, and repeat until you can’t find any more rectangles. This algorithm works reasonably well for grids up to about 12-by-12, but then it runs out of steam. On larger grids, changing the color of a node to eliminate one rectangle is likely to create another rectangle elsewhere (or several more of them). As a result, the system merely takes a random walk, with trendless fluctuations in the number of rectangles at any given moment. You discover a rectangle-free coloring only if the walk happens to stumble on the zero point.

I found the 15-by-15 four-coloring shown above with an algorithm that’s a little more effective even thought it’s no more sophisticated than the corner-twiddling method. The program repeatedly chooses a node at random and tries assigning it all four possible colors, tallying up the number of rectangles for each color choice. Some color or set of colors must minimize the rectangle count; from among these optimal colors the program chooses one at random and sets the node to that color before repeating the loop. This is a “greedy” method: At each step the number of rectangles can decrease or remain constant but can never increase. Greedy methods are notorious for getting stuck in local optima that are inferior to the global optimum. Maybe that’s what happens to my program when I try it on 16-by-16 and 17-by-17 grids. Or maybe the search space is just too large. In any case, when I woke up this morning and checked the results of an overnight run, I did not find a rectangle-free four-colored 17-by-17 grid awaiting me.

Of course I really didn’t have to do any algorithm analysis at all to know that I wasn’t going to win $289 and eternal fame with a day or two of idle hacking. If the problem were that easy, Gasarch and his students would have solved it for themselves long ago.

In spite of these various failures and frustrations, the grid-coloring problem still looks tantalizingly solvable. If a four-coloring of the Gasarch grid exists, it seems like we should be able to find it by some practical computation.

There are certainly lots of approaches more powerful than the blind dart game I’ve been playing. For example, if local optima are the major impediment, some variation on simulated annealing might help.

A more radical possibility is to try to construct an instance rather than merely search for it. If we assume that the four colors are represented as evenly as possible, then the 17-by-17 grid must have 72 nodes in each of three colors and 73 nodes in the fourth color. Starting from a blank grid, it’s easy to mark off 73 nodes in a single color without creating a forbidden rectangle. Adding 72 nodes of a second color is only a little harder. But then the job gets tricky. When you try to fill in a third color, you also by default choose nodes for the fourth color at the same time, and conflicts pile up in a hurry. Some kind of backtracking approach is probably needed here. Gasarch links to a paper by Elizabeth Kupin of Rutgers that explores these ideas in more detail. (If you want to prove the nonexistence of a four-coloring, this is presumably the way to go.)

Gasarch mentions two other promising avenues: integer programming (the discrete variant of linear programming) and SAT solvers–algorithms for the satisfiability problem. Having spent some time hanging out with a few master SAT solvers, I’m intrigued by the latter possibility. You can almost encode the grid-coloring problem as an instance of NAE-SAT, or not-all-equal SAT. Each node of the grid is represented by a variable that can take on any of four values. We group subsets of variables four at a time into clauses, where each clause includes the variables representing the four corners of a rectangle somewhere in the grid. For the 17-by-17 grid there are \({17 \choose 2}^2=18496\) clauses of this kind. The entire formula is satisfied if we can assign values to the 289 variables in such a way that none of the 18496 clauses has all four of its variables with the same value. After 40 years of work on SAT, there’s a highly developed technology for solving such problems. However, there’s a hitch. SAT problems are formulated in terms of boolean variables, with just two values each, but the grid-color variables have four values. Thus a further layer of encoding is likely to be needed, bringing a further explosion in the size of the problem instance.

One final hackerish note: What’s the best way to detect the presence of a monochromatic rectangle in a grid? My candidate goes like this. We encode the rows of the grid in a set of bit vectors–four vectors for each row, representing the four possible colors. For example, the red vector for a row has a 1 at each position where the corresponding node is red, and zeroes elsewhere. The blue vector has 1s for blue nodes, and so forth. Now we can detect a rectangle merely by taking the logical AND of two rows (an operation that could be a single machine instruction). A rectangle exists if and only if the result of the AND is nonzero. at least two bits are set in the resulting vector.

[Thanks to all the commenters for corrections and elaborations.]

I’m still busy digitizing a lifetime’s accumulation of clippings from magazines and journals, along with heaps of old tech reports, memos, and miscellaneous other cruft. There’s something slightly eerie about the process. So far I’ve emptied out a dozen file drawers, run several hundred pounds of paper through the scanner, and created thousands of PDFs. Yet my laptop is not a gram heavier. The glib explanation is that I’m just scraping pure information off the pages, leaving behind the ink and cellulose; I’m saving the bits and recycling the atoms. But is information so readily dematerialized? One of the manila folders I have just dredged up out of a filing cabinet is bulging with publications by the late Rolf Landauer, including several papers on the theme “Information is physical!”

I first met Rolf circa 1980. I had written a brief Scientific American article about some recent developments in optical computing technologies, and Rolf called to tell me I should never do anything so reckless and foolish and tasteless again. He took a dim view of photonics. This initial encounter was not a promising start to a friendship, but we got over it. He put me on his mailing list, which meant that I got a fat envelope once or twice a year, with reprints or preprints of his own latest work and often copies of other papers he thought I should being paying attention to.

Four of the articles in my Landauer folder have very similar titles:

• Information is Physical (Physics Today, 1991).
• The Physical Nature of Information (Physics Letters A, 1996).
• Information is a Physical Entity (Physica A, 1999).
• Information is Inevitably Physical (In Feynman and Computation, 1999).

If Landauer had lived longer (he died in 1999), I like to think that the next installment in the series would have been titled even more emphatically: Information is Physical, Damn It!

In all of these essays, Landauer’s thesis is straightforward:

Information is inevitably tied to a physical representation. It can be engraved on stone tablets, denoted by a spin up or down, a charge present or absent, a hole punched in a card, or many other alternative physical phenomena. It is not just an abstract entity; it does not exist except through a physical embodiment. It is, therefore, tied to the laws of physics and the parts available to us in our real physical universe.

This notion is obvious and totally uncontroversial–except to those who think it’s totally wrong. Doubters tend to focus on mathematical entities. Surely the integers exist as abstractions, independent of stone tablets and punchcards, no? And triangles would have three sides even if all the matter in the universe were annihilated–right? When it comes to numbers like π and e, one might well argue that they can exist only as abstractions; they can never be given a complete physical representation.

Landauer did not argue strenuously for his constructivist position within mathematics itself, but he did take a hard line about mathematical methods in the physical sciences:

There is a tendency to think of mathematics as a tool which somehow existed before and outside of our physical world. Mathematics, in turn, allowed the formulation of physical laws which then run the world, much as a process control computer runs a chemical plant. Here, instead, we emphasize that information handling has to be done in the real physical world, and the laws of physics exist as instructions for information handling in that real world. It, therefore, makes no sense to invoke operations, in the laws of physics, which are not executable, at least in principle, in our real physical world.

Our accepted laws of physics invoke continuum mathematics, which is, in turn, based on the notion that any required degree of precision can be obtained by invoking enough successive operations. But our real universe is unlikely to allow an unlimited sequence of totally reliable operations. The memory size is likely to be limited, perhaps, because the universe is limited. Even in an unlimited universe it is a strong presumption to invoke the possibility of assembling an arbitrarily large organized memory structure. Furthermore, in a world full of deleterious processes including noise, corrosion, electromigration, incident alpha particles and cosmic rays, earthquakes and spilled cups of coffee, it would be unreasonable to assume that each step in an unlimited sequence of operations can be carried out infallibly.

Those alpha particles and spilled cups of coffee bring me back to my little document-scanning project–my kitchen-table version of Google Books. I am well aware that my digitized archives are not disembodied abstractions, that the information I’ve scanned from Rolf’s preprints is still physical even if it’s less tangible, and that the bits remain vulnerable to all the perils of a material world. Indeed, the thought of losing all the files I’ve scanned–now that the paper originals are beyond recall–makes me itchy to plug in the back-up drive.

But the process of replicating the bits–which is even easier than capturing them in the first place–sends my mind off on another tangent. As Rolf said, we can represent information in many physical forms: as marks on paper, as magnetized domains on a metal-coated disk, as packets of electric charge, as base pairs in a DNA molecule, as beads on an abacus. When we build machinery to process this information, we can choose among many different computing technologies: transistors, brass gears, neurons, rubber bands and tinker toys, quantum dots, even photons in optical waveguides (though Rolf despised that last possibility, and he was skeptical about the quantum dots).

Somehow, this proliferation of physical embodiments for information does not strengthen the conviction that information is subordinate to its physical representation. When we can write the same message in so many forms–everything from lines in the sand to holograms–the message itself begins to seem just as substantial as the physical medium, and perhaps more enduring. I have digital documents that began life on eight-inch floppy disks 20 years ago. The files have migrated a dozen times or more to other media: five-and-a-quarter-inch floppies, three-and-a-half-inch floppies, Zip drives, digital audio tapes, CD-ROMs, a succession of hard disks. Most of the physical objects making up that long chain of transmission have long since succumbed to coffee spills, corrosion and other hazards, or else they have simply gotten lost. Yet the data persists, a sort of standing wave in the river of hardware rushing toward obsolescence and oblivion. Under the circumstances, it can be hard to keep in mind that the information depends for its very existence on those delicate shards of matter. It goes against the grain of the whole apparatus of computer science, where automata theory, the Church-Turing thesis, and the Turing equivalence of programming languages all encourage us to think that abstractions come first, and implementation is secondary.

These musings are not meant as an attempt to refute Landauer’s assertion. I still have to concede that I cannot record or express a pattern of bits without resorting to some physical medium, if only the gray matter in my own head. But the notion sits uncomfortably; it’s a conundrum. I wish I had a chance to chat with Rolf about it. But, sadly, Rolf Landauer is no longer physical.

Note: As far as can tell, none of the four Landauer papers I mention above are available online without payment. I am therefore taking the liberty of posting my scan of Rolf’s preprint of the information-is-a-physical-entity paper. I would also like to call attention to two recent articles about Landauer: a discussion of his contributions to solid-state physics by Bertrand I. Halperin and David J. Bergman, and a biographical memoir by Charles H. Bennett and Alan B. Fowler.

Someone out there is being paid to post comments on bit-player.org–and doubtless on tens of thousands of other blogs as well. The comments are mostly bland and inoffensive, sometimes effusive, always hastily composed. “Thanks for article..good work,” they say. “Amazing!!” “i like your article and i will be wating your net article….”

The payload attached to each of these comments is a link to a web site that someone wants to promote. Some of the sites are selling goods or services; others are billboards full of pay-per-view ads; a fair number are mysterious to me, being written in languages I don’t understand. I would not be astonished to learn that some of the sites are distributing malware.

Years ago, the first wave of comment spam was powered by scripts that flooded blogs and wikis and forums with hundreds of postings full of program-generated gibberish and long lists of links. That abuse was stopped by captchas and other simple filters, like the one I’ve been using here on bit-player. Another important defense is the “nofollow” tag, which instructs search engines to ignore links in comments, thereby eliminating the incentive of gaining PageRank points.

The comment spam arriving now is not generated by a Perl script. Somewhere in the world a person is being paid to read these very sentences, then to prove his or her humanity to the Turing-test filter, and finally to write a few words in response and sneak in a paid link. I’m both fascinated and appalled to learn that the Internet economy can support this activity. What’s the going rate for writing comment spam? Is it worth a penny to get your link briefly exposed to the vast daily readership of bit-player.org? How about a tenth of a penny?

I have a sinking feeling that the people doing this work are themselves victims of a scam, and that they’ll never see even the tenth of a penny. They have probably succumbed to a 21st-century version of the ads I used to see on matchbook covers: “Work at home! Make $500 a week stuffing envelopes in your spare time!”

Of all the ways that poor and desperate people are exploited, this is not the worst. Presumably the work is safe and sanitary, and it even rewards literacy. Some of my comment spammers would surely have interesting ideas to contribute if only they had the luxury of time.

All the same, this kind of commercial graffiti is not something I want to encourage. The available countermeasures include prohibiting all links in comments, holding all comments until a moderator approves them, or requiring commenters to register with a verifiable email address. None of these options appeals to me, but I may have to consider them if the problem persists. For now, though, I’m going to continue the human approach–manually deleting spammy comments as quickly as I can get to them. I am also closing comments on all but the 10 most-recent items on bit-player; the spammers seem to favor older posts.

I have to add that spotting comment spam is not always as easy as you might think. Consider this comment, which came in response to a story about editorial changes at Scientific American magazine:

Many times, when i read your American Scientist columns, I have asked myself that is any other country’s scientist didn’t give anything to the world?

The text of the comment is pertinent to the topic; it raises a question that’s entirely appropriate in this context; and there’s clear evidence that the author has actually been reading bit-player (and even my American Scientist columns) rather than merely spewing comments at random. This is someone I would like to be able to welcome into the community. But the link associated with the comment was an ad for a web-hosting service, and another comment from the same IP number advertised a different service. Was I wrong to hit the delete button?

You’re welcome to comment below, but without spammy links, please.

As I have mentioned in the past, I’m fascinated by the acrobatics of bird flocks, especially the big congregations of European starlings that gather in the evening at this time of year. Evidently I’m not the only one with such an interest. In the past few years the subject has attracted the attention of quite a large flock of scientists, including not only biologists but also various luminaries in physics, mathematics and computer science.

Below are some notes on a few of the recent papers, but first I have to mention a classic from 20 years ago:

Reynolds, Craig W. 1987. Flocks, herds, and schools: a distributed behavioral model. Computer Graphics 21(4):25–33. Author archive.

This is the paper that began the modern era of flocking studies by proposing that animals could coordinate and synchronize their movements without any need for a leader or external cues. Others were thinking along the same lines at about the same time, but it was Reynolds who attracted wide notice with his enchanting computer animations of “boids” soaring through an imaginary three-dimensional space. Each individual in the flock acts according to simple, local, fixed rules, and the synchronized maneuvers emerge spontaneously.

Reynolds suggested three particular rules that might guide the behavior of each bird:

• Avoid collisions.
• Try to match the speed and heading of nearby birds.
• Move toward the center of the group in which you are flying.

Reynolds was working in computer graphics, and his ideas were soon taken up by movie studios and by the makers of video games. In a sense, his simulations only had to look right; they didn’t have to reflect what actually goes on in a starling’s head. But whether or not the birds were paying attention, students of animal behavior certainly were.

Much of the recent activity arises out of new field studies, conducted mainly by physicists.

Cavagna, Andrea, Irene Giardina, Alberto Orlandi, Giorgio Parisi, Andrea Procaccini, Massimiliano Viale and Vladimir Zdravkovic. 2008. The STARFLAG handbook on collective animal behaviour. 1: Empirical methods. Animal
Behaviour 76:217–236. Preprint.

Cavagna, Andrea, Irene Giardina, Alberto Orlandi, Giorgio Parisi and Andrea Procaccini. 2008. The STARFLAG handbook on collective animal behaviour. 2: Three-dimensional analysis. Animal Behaviour 76:237–248. Preprint.

This group, coordinated by Andrea Cavagna and Irene Giardina of the University of Rome La Sapienza, has been photographing starling flocks near the city’s main railroad station (the Termini), which is just a few blocks from the university. Using pairs of synchronized cameras, the observers have captured stereoscopic images and then applied special image-analysis software to reconstruct the three-dimensional trajectory of each bird. Similar techniques have been tried in the past, but only with small flocks (a few dozen birds). The Italian group has traced the motions of individual birds in groups of up to 2,600. The two papers cited above give technical details on how the data were gathered and analyzed.

Ballerini, Michele, Nicola Cabibbo, Raphael Candelier, Andrea Cavagna, Evaristo Cisbani, Irene Giardina, Alberto Orlandi, Giorgio Parisi, Andrea Procaccini, Massimiliano Viale and Vladimir Zdravkovic. 2008. Empirical investigation of starling flocks: a benchmark study in collective animal behaviour. Animal Behaviour 76:201–215. Preprint.

Ballerini, Michele, Nicola Cabibbo, Raphael Candelier, Andrea Cavagna, Evaristo Cisbani, Irene Giardina, Vivien Lecomte, Alberto Orlandi, Giorgio Parisi, Andrea Procaccini, Massimiliano Viale and Vladimir Zdravkovic. 2008. Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study. Proceedings of the National Academy of Science of the USA 105:1232–1237. Open access.

And here the same authors (with a few additions) report their results and conclusions. They base their interpretation on a computational model that is recognizably a descendant of the Reynolds scheme, but with one crucial modification. Reynolds and others assumed that each bird is influenced by all other birds within some fixed distance (a “metric neighborhood”); Ballerini et al. get a closer match to the data by assuming that a bird attends to the motions of a fixed number of near neighbors, regardless of distance (a “topological neighborhood”). In other words, the graph of interacting birds has nearly constant vertex degree; the typical degree is probably six or seven. The main significance of this algorithmic change is that it helps maintain the cohesion of the flock in spite of large variations in density.

Hildenbrandt, Hanno, Claudio Carere and Charlotte K. Hemelrijk. 2009. Self-organised complex aerial displays of thousands of starlings: a model. arXiv:0908.2677v1

Those same flocks at Termini have a role in this study as well; the model presented here draws on data from Ballerini et al. as well as videotapes made at Termini by Carere. (Carere is another physicist at Sapienza; Hildenbrandt and Hemelrijk are biologists at the University of Groningen.)

The model works on the same essential principles, but it differs in intellectual style and emphasis. Hildenbrandt et al. want to account for specific details of a flock’s behavior—not just the general tendency to fly in close formation but also the particular shapes of starling flocks, the maneuvers they perform, the altitudes they prefer, and so on. Reaching for this verisimilitude leads to a rather complicated model with many parameters in need of fine tuning, such as aerodynamic properties of the bird’s wing and body and banking angles in turns. Hildenbrandt et al. report some success in explaining the geometry of flocks (they tend to be horizontally flattened rather than spherical). They do less well in an attempt to account for an extra-dense layer of birds observed at the periphery of a flock.

Cucker, Felipe, and Steve Smale. 2007. Emergent behavior in flocks. IEEE Transactions on Automatic Control 52:852–862.

Chazelle, Bernard. 2009. Natural algorithms. Proceedings of the 20th Symposium on Discrete Algorithms, pp. 422-431. Preprint.

Chazelle, Bernard. 2009. The convergence of bird flocking. arXiv:0905.4241v1

Leaving behind the breathy wing-beats of living starlings, we enter a world of mathematical abstractions.

Cucker and Smale, peripatetic mathematicians currently at the City University of Hong Kong, take a stripped-down model of flocking and ask this question: Is it guaranteed that all the birds in the flock will eventually settle on the same velocity, and thus fly together forever? Chazelle, a theoretical computer scientist at Princeton, asks a follow-on question: If the birds do converge on the same speed and heading, how long might it take for them to do so, in the worst case?

The answer to the Cucker-Smale question turns out the be yes: Given certain preconditions and parameter values, convergence is certain. But Chazelle shows that it can take quite a while for the flock to reach consensus. For n birds adjusting their velocities in discrete steps, the upper bound is 2 ↑↑ (4 log n) steps. As I was saying just the other day, this up-arrow notation denotes an exponential tower of 2s with, in this case, 4 log₂ n levels. In other words, in a flock of a thousand birds, the convergence time is roughly

\[2^{2^{2^{\cdot^{\cdot^{\cdot^2}}}}}\]

with 40 levels of exponentiation. This is a ridiculous number, far exceeding the lifetime of a starling (or of a universe, for that matter). As Chazelle notes: “Our bounds obviously say nothing about physical birds in the real world. They merely highlight the exotic behavior of the mathematical models.”

It is rather wonderful to reflect—as you stand in a field of corn stubble admiring the flocks of birds wheeling overhead in the evening sky—that these avian entertainments should be the starting point for a line of reasoning that ventures so far into the wild blue yonder of inexpressible numbers.

Lebar Bajec, Iztok, and Frank H. Heppner. 2009. Organized flight in birds. Animal Behaviour 78:777–789. Preprint.

I mention this piece last, but it would actually be a good place to start if you want a primer on flocking. Frank Heppner, a biologist at the University of Rhode Island, is one of the pioneers of flocking-and-swarming studies; here, with a mathematical colleague from the University of Ljubljana, he reviews many of the recent contributions and puts them in historical context. The review includes a discussion of the more crystalline flying formations of large birds such as geese as well as the amorphous flocks of starlings.

A press release from the American Institute of Mathematics two weeks ago announced that all the congruent numbers up to 1 trillion have been enumerated. Two questions leap to mind. What the heck is a congruent number? And who cares?

I’ll return to those questions. But first I’d like to pause just a moment to marvel at the idea of calculating anything up to 10¹². A few decades ago, such a project would have been unthinkable. Today, counting to a trillion takes only an hour or so, even on plain vanilla hardware. This is truly one of the wonders of the age, and we shouldn’t grow too blasé about it. But the computation of all those congruent numbers involved a lot more than looping through “+1″ a trillion times, and it took considerably longer than an hour.

Okay, so what’s a congruent number? I would like to sidle up to that question rather than face it straightaway. The definition is not hard to understand—it’s all about right triangles with rational side lengths and integer areas—but when I started looking into this story, it took me a while to appreciate why those particular triangles might be worth thinking about.

Let’s begin with Pythagorean triples—sets of positive integers (a, b, c) that define a right triangle, with a and b giving the lengths of the legs and c the hypotenuse. The familiar (3, 4, 5) triangle is everybody’s favorite example. For which right triangles with integer side lengths is the area also an integer? That’s easy: All of them. The area of a right triangle is ab/2; in any Pythagorean triple either a or b (or both) must be an even number, which implies that ab/2 is a whole number. For the (3, 4, 5) triangle the area is (3 × 4)/2 = 6.

Knowing one such triangle, we can make more. Lots more. Just multiply a, b and c by any integer k, which has the effect of multiplying the area by k². If k = 2, we get the (6, 8, 10) triangle with area 24; k = 3 yields the (9, 12, 15) triangle with area 54, and so on. The resulting sequence of triangles is infinite but not very interesting; all the larger triangles are just scaled-up versions of the original, like photographic enlargements. We can view the entire infinite series as a single class of triangles, and take the smallest member of the series as the prototype. That smallest triangle is given by a primitive Pythagorean triple—one where a, b and c have no factors in common. It turns out there are also infinitely many primitive triples. Euclid bequeathed us an algorithm that will generate all of them, if you let it run forever.

Given this infinite set of infinite series of triangles, it’s clear that infinitely many integers can be the area of a Pythagorean triangle. But that certainly doesn’t mean that every positive integer has this property. For example, there’s no Pythagorean triangle with an area of 5. To convince yourself of this fact, just measure the area of every integer-sided right triangle with legs no longer than 10. None of those triangles has an area of 5, and no triangle with longer integer legs can have an area that small. In principle, the same laborious but reliable procedure could be applied to any integer N to determine whether or not it is the area of some Pythagorean triangle. Here are the first few integers that appear in such an enumeration ( sequence A009112.):

6, 24, 30, 54, 60, 84, 96, 120, 150, 180, 210, 216, 240, 270, 294, 330

Rational triangles. Suppose we relax the constraints a little, allowing the sides of a right triangle to be rational numbers—either fractions or integers but not irrational values such as the square root of 2. The area is still required to be an integer.

A first question is whether right triangles with fractional sides and integer areas even exist. Maybe the cupboard is bare; maybe there are no such triangles. But no, they do exist. Here’s a proof by example:

There’s no trickery here. If you’re in any doubt, do the arithmetic; you’ll find that the numbers define a genuine right triangle (the Pythagorean theorem holds) and the area really is exactly 7.

At last we have arrived in the realm of congruent numbers. A congruent number is an integer that’s the area of a right triangle with rational sides. All integer-sided right triangles are included in the category, along with those whose sides are rational fractions. Here’s a more formal definition:

A positive integer N is congruent if there exist rational numbers a, b and c such that a² + b² = c² and ab/2 = N.

All the same questions we were asking about Pythagorean triples also come up in this new context. Where do we find rational triangles with integer area, or how can we manufacture them? Which integers N can be the area of such a triangle? Answers aren’t quite as easy to come by in this case. In particular, the strategy of enumerating triangles from the smallest up won’t work, because there is no smallest rational triangle. There are other ways of imposing an ordering on the rationals, but none of them lead to a good algorithm for enumerating congruent numbers.

So where did I get the example illustrated above? I didn’t just stumble upon it by trying random rational triangles. Note that multiplying each rational side length by the common denominator of the fractions will return us to the land of integer-sided triangles. For the example shown, multiplying through by 60 yields a (175, 288, 337) triangle; these numbers have no factor in common, so they form a primitive Pythagorean triple. The area of the new triangle is 7 × 60 × 60, or 25,200. Thus we’ve recovered an integer triangle from a rational one; what’s more important, the process can be reversed to derive rational triangles from integer-sided ones.

I mentioned a Euclidean method for generating primitive Pythagorean triples. It goes like this: Take any two positive integers m and n that satisfy the following conditions:

• m > n;
• one of the numbers is odd and the other even;
• the numbers are coprime (no common factors other than 1).

Then (2mn, m² – n², m² + n²) is a primitive Pythagorean triple. By counting through all suitable values of (m, n) starting with (2, 1), all primitive triples are generated.

A postprocessing step built atop Euclid’s procedure will generate congruent numbers. The plan is to produce a primitive Pythagorean triple and calculate the area N = ab/2 of the corresponding triangle. If the area is “square-free”—that is, it has no pairs of repeated prime factors and thus cannot be divided evenly by a perfect square—then we’re done; that value of N is a congruent number and cannot be reduced to a smaller congruent number. But if N is not square-free, we can divide out each square factor, leaving a smaller triangle with rational sides and integer area, and thereby generating another congruent number.

A worked example: The (m, n) pair (5, 4) produce the primitive triple (9, 40, 41), with area N = (9 × 40)/2 = 180. Thus 180 is identified as a congruent number, but it is not square-free; it factors as 2² × 3² × 5. We can therefore divide the area by 4 and the sides by 2, producing a shrunken (9/2, 20, 41/2) triangle of area N = 45. In this way we learn that 45 is also a congruent number, but again it is not square-free. Dividing the area by 9 and the sides by 3 yields the (3/2, 20/3 and 41/6) triangle with area N = 5. And so 5, too, is congruent; it’s also square-free and therefore cannot be reduced further. There is no smaller triangle similar to the (9, 40, 41) triangle that has integer area.

This scheme can produce an unlimited supply of congruent numbers. Unfortunately, it’s not so well suited to answering the question of whether a particular integer is congruent. The problem is that the congruent values are not generated in order from smallest to largest. If we turn the crank for a while and discover that a certain integer N appears in the algorithm’s output, then we know for sure that N is congruent. But if N has not shown up, we can’t conclude that it is not among the congruent numbers; we might simply have to wait longer for N’s turn to come.

When I turn the crank on my own little program for generating congruent numbers, these are the first 101 values to pop out, in order of appearance:

6, 30, 60, 15, 84, 21, 210, 180, 45, 5, 330, 630, 70, 924, 231, 546, 504, 126, 14, 1320, 1560, 390, 840, 1386, 154, 2340, 585, 65, 1224, 306, 34, 990, 110, 2730, 3570, 1710, 190, 2574, 286, 4620, 1155, 5610, 5016, 1254, 2310, 1716, 429, 7140, 1785, 7980, 1995, 3036, 759, 4290, 7956, 1989, 221, 10374, 10920, 8970, 3900, 975, 39, 7854, 11970, 1330, 14490, 1610, 11550, 462, 4914, 6630, 12540, 3135, 19320, 4830, 6090, 4080, 1020, 255, 11856, 2964, 741, 18480, 23184, 5796, 1449, 161, 25200, 6300, 1575, 175, 7

Here’s the same list sorted into ascending order of magnitude:

5, 6, 7, 14, 15, 21, 30, 34, 39, 45, 60, 65, 70, 84, 110, 126, 154, 161, 175, 180, 190, 210, 221, 231, 255, 286, 306, 330, 390, 429, 462, 504, 546, 585, 630, 741, 759, 840, 924, 975, 990, 1020, 1155, 1224, 1254, 1320, 1330, 1386, 1449, 1560, 1575, 1610, 1710, 1716, 1785, 1989, 1995, 2310, 2340, 2574, 2730, 2964, 3036, 3135, 3570, 3900, 4080, 4290, 4620, 4830, 4914, 5016, 5610, 5796, 6090, 6300, 6630, 7140, 7854, 7956, 7980, 8970, 10374, 10920, 11550, 11856, 11970, 12540, 14490, 18480, 19320, 23184, 25200

The problem, again, is that we can’t conclude anything about the numbers that don’t appear in this collection. For example, 13 is absent, even though it is in fact a congruent number; it just doesn’t turn up until we get well down the list—it comes from the 49,485th triangle examined, which has sides (780/323, 323/30, 106921/9690). On the other hand, 8 is unlisted because it is not congruent; it will never show up in the output hopper no matter how long we keep cranking away.

Here are the first 101 true congruent numbers (sequence A003273):

5, 6, 7, 13, 14, 15, 20, 21, 22, 23, 24, 28, 29, 30, 31, 34, 37, 38, 39, 41, 45, 46, 47, 52, 53, 54, 55, 56, 60, 61, 62, 63, 65, 69, 70, 71, 77, 78, 79, 80, 84, 85, 86, 87, 88, 92, 93, 94, 95, 96, 101, 102, 103, 109, 110, 111, 112, 116, 117, 118, 119, 120, 124, 125, 126, 127, 133, 134, 135, 136, 137, 138, 141, 142, 143, 145, 148, 149, 150, 151, 152, 154, 156, 157, 158, 159, 161, 164, 165, 166, 167, 173, 174, 175, 180, 181, 182, 183, 184, 188, 189

Ancient history. The search for congruent numbers does not stretch back all the way to Pythagorus or Euclid, although Diophantus apparently considered a couple of special cases. L. E. Dickson’s big history of number theory attributes the first full statement of the problem to “an anonymous Arab manuscript, written before 972.” Other sources cite the works of Abu Bakr al-Karaji, a mathematician who worked in Baghdad at the end of the 10th century. A couple of centuries later Fibonacci, who straddled the Arabic and European worlds, had more to say about the problem in his Liber Quadratorum. Later still, Pierre de Fermat proved (in a marginal note, for which he had sufficient room!) that 1 is not a congruent number. (The proof extends to 4 and 9 and 16 and all other square numbers, none of which are congruent.)

These early writers formulated the problem in a somewhat different way than the rational-triangle scheme explained above. Given an integer N, they asked whether it’s possible to find three perfect squares in arithmetic progression with the interval between the squares equal to N. In other words, they sought a number s² such that s² – N, s², and s² + N are all perfect squares. For example, if N = 24, then the solution is s = 5; the three perfect squares are 25 – 24 = 1, 25, and 25 + 24 = 49.

The two versions of the problem—the integer-area right triangles and the squares in arithmetic progression—are equivalent, although that’s not exactly obvious. Here’s the connection: If a² + b² = c² and ab/2 = N, then setting s = c/2 guarantees that s² – N, s², and s² + N are all perfect squares. (For the algebraic exercise proving this, see the book by Neal Koblitz cited below.)

By the way, the squares-in-arithmetic-progression version is where we get the term “congruent numbers.” The three numbers s² – N, s², and s² + N are all congruent modulo N. For example, 1, 25 and 49 are all congruent to 1 modulo 24. If you ask me, “congruent numbers” is a poor excuse for a name, and is particularly confusing in the context of triangles, where “congruent” has another meaning altogether. But after a thousand years I suppose it’s too late to fix it. “Karaji numbers,” anyone?

By 1915, all the integers up to 100 had been classified as either congruent or noncongruent. But as recently as 1980, when Ronald Alter wrote a brief review of the status of the problem, there were still 189 square-free numbers less than 1,000 for which the question of congruence remained unsettled. Then everything changed in 1983. In that year the nature of the congruent-number problem was transformed by Jerrold B. Tunnell of Rutgers, who not only found a better way to calculate congruent numbers but also showed why it’s worth the effort to do so.

From right triangles to elliptic curves. Before I go on, a warning: I am about to walk up to the blackboard with a piece of chalk in my hand and impersonate one of those masterful, self-confident lecturers who rattle off long trains of equations, invent lemmas on the fly, and always come to QED just as the blackboard fills up and the class ends. In my case any such air of mastery is a complete illusion; I’m just learning this stuff as I go along. But I’ll do my best to make it an entertaining illusion.

So here goes. It’s not hard to see that any product of perfect squares is also a perfect square: a²b²c² = (abc)². Therefore, if N is a congruent number, and s² – N, s², and s² + N are all squares, we can set the product of these three factors equal to another square; call it y². Thus we get the equation:

y²  =  (s² – N) s²(s² + N).

Multiplying the three terms on the right, this becomes:

y²  =  (s²)³ – N²s².

Now perform a simple substitution of variables, setting s² = x:

y² = x³ – N²x.

This is the equation of an elliptic curve—another mathematical object with a really unfortunate name. An elliptic curve looks nothing like an ellipse. Here’s the graph of a particular elliptic curve, the one for N = 6:

The two pieces of the blue line mark the locus of all points (x, y) that satisfy the equation

y² = x³ – 36x.

And the hot pink dot? That marks a rational point on the curve—a point where x and y both take on rational values. (Insiders, I’m told, call it a ratpoint.) Specifically, the dot identifies the point x = 25/4, y = 35/8. If you care to plug those numbers into the equation, you’ll find that indeed 1225/64 = 15625/64 – 225, and so the point does lie on the curve.

Tunnell, elaborating on earlier work of Kurt Heegner, showed that N is congruent if the elliptic curve y² = x³ – N²x passes through a certain kind of point on the (x, y) plane. Specifically, we have to search for points whose x and y coordinates are both rational numbers, but we ignore the three points with y = 0. Then the x coordinate of the point has to satisfy three more conditions. Letting x = u/v, we require that:

• u and v are perfect squares,
• v is even,
• u has no factors in common with N.

If we can find just one point on the curve that matches all these properties, then we can generate an infinity of other rational points. Moreover, the existence of such a point implies, according to Tunnell’s theorem, that N is congruent. (The hot pink point above clearly qualifies.) Thus the congruent-number problem appears to be solved: We have an unambiguous criterion for deciding if a given N is congruent or not. Just construct the corresponding elliptic curve and check the ratpoints.

Regrettably, we’ve been set up for yet another disappointment. Searching for rational points on elliptic curves is a task for which we have no efficient general method. We’re really no better off than trying to generate all Pythagorean triples.

But wait! We’re not done yet. From the properties of elliptic curves, Tunnell derived yet another criterion—and this one turns out to be the key to a simple and practical test. It all hinges on the number of integer solutions to some quadratic equations that on first glance appear to be arbitrary strings of symbols plucked out of thin air. The criterion is this: If N is a square-free congruent number, and if N is odd, then the number of integer solutions to the equation N = 2x² + y² + 8z² must be exactly double the number of integer solutions to N = 2x² + y² + 32z². (If both equations have no integer solutions, the condition is satisfied, since 2 × 0 = 0.) For even N, the two equations are slightly different, but the test works in exactly the same way.

The graphic above shows the 361 square-free numbers up to 1,000 that pass the Tunnell test. The height of each dot indicates the number of integer solutions to the equation N = 2x² + y² + 8z² (or the corresponding equation for even N). The two values of N with 160 solutions are 689 and 761. (It’s curious that in most cases—308 out of 361—the number of solutions is actually zero. I don’t know what this means or whether that pattern continues with larger N.) [See Update 2009-10-10, below.]

What sets the Tunnell criterion apart from all the others is that we can actually carry out the test in a reasonable and predictable amount of time. For any given N, counting the solutions should be doable in time proportional to N^3/2, simply by trying all integer values of x, y and z less than the square root of N.

So that’s it, right? Problem solved at last? Well, no, there’s still a small hitch—a bit of awkward fine print. Tunnell proved that if N is square-free and congruent, then the criterion on the number of integer solutions must be satisfied. This much is unconditionally true. But what about the converse? If the criterion is satisfied, can we be certain that N is a congruent number? In this direction, the proof is not unconditional. It’s contingent on a proposition known as the Birch–Swinnerton-Dyer conjecture, which is widely believed, and supported by an abundance of empirical evidence, but still unproved. Which leaves just enough doubt to make the game still interesting.

Why should anyone care about this stuff? On first acquaintance, the search for congruent numbers sounds like one of those mathematical pastimes that appeal to amateurs (like me!) but don’t really command the attention of the research community. There are so many kinds of cutely named numbers out there—perfect, amicable, lucky, happy—and not all of them carry deep significance. The congruent numbers might well be just another amusing sideshow. But it turns out they’re not. There’s serious mathematics here, enough to engage the interest of serious mathematicians.

Elliptic curves have been a focus of intense scrutiny for decades. Henri Poincaré studied them in the early years of the 20th century. In the 1920s Louis Mordell proved a theorem about the rational points on elliptic curves: Even when there are infinitely many points, they all come from a finite set of “generators”; the number of generators and hence the number of infinite families is called the rank of the curve. In the 1950s Yutaka Taniyama and Goro Shimura, with later refinements by André Weil, worked out a conjecture about elliptic curves and another class of mathematical objects, called modular forms. Andrew Wiles and Richard Taylor proved part of that conjecture in the course of settling Fermat’s Last Theorem in the 1990s; the rest of the Taniyama-Shimura conjecture has since been proved as well.

And now we have the Birch–Swinnerton-Dyer conjecture, one of the famous million-dollar math problems. I’m not going to try to explain the conjecture in any detail—I’ve used up all my chalk, and probably my readers’ patience, too—but I think the basic idea goes something like this. On the one hand we have an elliptic curve, drawn in the (x, y) plane. On the other hand we have something called an L-function, which is defined on the plane of complex numbers. Think of the L-function as an undulating landscape with mountains rising above the plane and submarine canyons deep under it. The topography of this surface is determined in part by points selected from the elliptic curve, so there’s a connection between the two objects. The conjecture formulated in the 1960s by Bryan Birch and Peter Swinnerton-Dyer says that the shape of the L-function in the neighborhood where it passes through zero gives us information about the rank of the elliptic curve and thus about the number of rational points.

There’s an analogy between the BSD conjecture and an even more celebrated problem on the million-dollar list, the Riemann hypothesis. The zeros of the Riemann zeta function (which is much like an L-function), carry information about the distribution of prime numbers. The Birch–Swinnerton-Dyer conjecture invites us to suppose that the zeros of L-functions of certain elliptic curves tell us something about the distribution of congruent numbers.

Incidentally, the BSD conjecture must be one of the earliest products of computer-driven experimental mathematics. The numerical explorations that led to the conjecture were done with the EDSAC, the pioneering electronic computer built at the University of Cambridge.

The trillion triangles. Hand-waving about elliptic curves and L-functions might provide vague a rationale for interest in congruent numbers, but one part of the story I still didn’t get was why anyone would bother to compute mass quantities of congruent numbers. So I asked Michael Rubinstein of the University of Waterloo. Rubinstein had earlier computed the congruent numbers up to 10⁹, and it was his challenge that provoked the recent thousandfold expansion of the inventory. Rubinstein explained:

I’m interested in the statistical distribution of congruent numbers. A few years ago, Brian Conrey, Jon Keating, myself, and Nina Snaith came up with a prediction for the asymptotic number of congruent numbers up to x, akin to the prime number theorem. This prediction grew out of remarkable models created by the same researchers and David Farmer for related elliptic curve L-functions that were inspired by random matrix theory and based on a number of unproved assumptions.

The prediction is that the number of congruent numbers less than x, arising from even-rank elliptic curves, is asymptotically

\[c x^{3/4} \log(x)^{11/8}\]

where c is a constant. Our model doesn’t let us completely nail down the constant c, and the data will help us understand what the correct constant is. The asymptotic behaviour stabilizes at a logarithmic rate, so going to 10¹² is only 50 percent better than going to 10⁸.

It will also provide a good amount of data for studying the statistics of ‘higher rank elliptic curves’ in the family of elliptic curves related to congruent numbers. The correct model to use for higher rank elliptic curves is not really understood, and the billions of congruent numbers found will end up yielding several million higher rank curves with which we hope to gain insight into the statistics of higher rank curves (compared to just thousands of higher rank curves from earlier computations).

The enumeration of the trillion triangles was done by two teams. William Hart of the University of Warwick and Gonzalo Tornaria of the Universidad de la Republica in Uruguay ran their program on a computer called Selmer at Warwick. Mark Watkins of the University of Sydney, David Harvey of the Courant Institute and Robert Bradshaw of the University of Washington used the SAGE computer at Washington.

The strategy of the computation is based on the Tunnell criterion, but it turns out there’s a better way to go about it than explicitly counting the number of integer solutions to those various quadratic equations in (x, y, z). Tunnell showed that all the information about the number of solutions could be encoded in a formal power series, which looks like this:

\[A(q) = a_{1}q^{1} + a_{2}q^{2} + a_{3}q^{3} + a_{4}q^{4} + ... \]

It’s a “formal” series in the sense that we don’t actually care about evaluating the sum for any particular value of q; instead, we just want to know the various coefficients a_i. This is how the Tunnell criterion gets encoded in the series: If a_i is zero in this series, and if i is a square-free number, then i is congruent. (Actually, there are two separate series, one for odd i and one for even i.)

The odd and even power series are generated by a couple of formidable-looking products. Here’s the one for the odd series:

\[A(q)=q\prod_{n=1}^\infty(1-q^{8n})(1-q^{16n})\left(1+\sum_{n=1}^\infty 2q^{2n^2}\right)\]

Since both the overall product and the sum in the third factor call for infinitely many values of n, we can’t multiply out this whole expression. Fortunately, though, it turns out that larger n contribute only to higher coefficients, and so we can compute early terms in the series without worrying about what might happen later. Kent Morrison notes that taking just the factors

\[q(1-q^8)(1-q^{16})(1-q^{16})(1+2q^2+2q^8+2q^{18})\]

is enough to generate all the odd congruent numbers up to 25. Those five factors expand to:

\[A(q)=q^1+2q^3+q^9-2q^{11}-4q^{17}-2q^{19}-3q^{25}+...\text{ higher terms}\]

The terms missing from this series—those with a zero coefficient—are just the odd square-free congruent numbers in this range: 5, 7, 13, 15, 21, 23.

The big computation by Bradshaw et al. used essentially this same scheme, with a few refinements and optimizations. The challenge is that when you’re trying to compute all the terms of the series out to a_{1,000,000,000,000}q^{1,000,000,000,000}, you wind up multiplying some enormous numbers. I don’t mean just that the numbers are too big to fit into the registers of a 32-bit or a 64-bit computer. They’re too big to fit into the main memory of a computer with 128 gigabytes of RAM. Moreover, the arithmetic has to be done exactly; approximations are useless. Thus a major part of the effort in setting up the computation was devising efficient “out of core” multiplication routines for ridiculously large numbers. The basic algorithm was a fast-fourier-transform method.

And the result? Up to 10¹², the computation identified 3,148,379,694 square-free congruent numbers number candidates with N in the residue classes 1, 2, and 3 modulo 8. I look forward to reading more about the analysis of their distribution. [See Update 2009-10-10, below.]

References and resources. Two weeks of reading and tinkering have not made me the master of this subject. For those who would like to explore further I can offer some resources I’ve found helpful, listed in no particular order.

• Ed Eikenberg has posted some lecture notes from a talk on elliptic curves and congruent numbers given in 2000.
• William Stein offers slides from a Harvard lecture in 2001.
• The web page for another William Stein course (this one at the University of Washington in 2006) provides notes and handouts and a collection of function definitions that will let you play with congruent numbers and elliptic curves in the Sage computer mathematics system.
• Koblitz, Neal. 1993. Introduction to Elliptic Curves and Modular Forms. Second edition. New York: Springer-Verlag. Some kind soul has posted page images for the first chapter online.
• Guy, Richard K. 2004. Unsolved Problems in Number Theory. Third edition. New York: Springer. (Section D27, p. 306, discusses congruent numbers.)
• Tunnell, J. B. 1983. A classical diophantine problem and modular forms of weight 3/2. Inventiones Mathematicae 72:323–334. (Available online via the DigiZeitschriften.)
• Barry Cipra writes on the recent trillion-trangle computation: Tallying the class of congruent numbers, ScienceNOW, September 23, 2009. (Get it now, while it’s free.)
• Dickson, Leonard Eugene. 1952. History of the Theory of Numbers. New York: Chelsea Pub. Co. (For congruent numbers see Vol. 2, Chapter 16, p. 459.)
• Brown, Ezra. 2000. Three Fermat trails to elliptic curves. The College Mathematics Journal 31:162–172. Free PDF available online (unlike most CMJ articles), apparently because it won a Polya prize.
• Rubin, Karl, and Alice Silverberg. 2002. Ranks of elliptic curves. Bulletin of the American Mathematical Society 39:455–474. Online.
• Silverberg, Alice. 2001. Open questions in arithmetic algebraic geometry. In Arithmetic Algebraic Geometry, Institute for Advanced Study/Park City Mathematics Series 9, pp. 83–142. Providence: American Mathematical Society. Postscript preprint.
• The American Institute of Mathematics web page on congruent numbers, which includes several helpful appendices, such as a discussion of FFT multiplication, an essay on congruent numbers by Kent Morrison and a link to the paper by Bradshaw, Hart, Harvey,
  Tornaria and Watkins reporting the trillion-triangle computation. (At this writing, the posted version of the paper is still an incomplete draft.)
• Conrey, J. B., J. P. Keating, M. O. Rubinstein and N. C. Snaith. 2000. On the frequency of vanishing of quadratic twists of modular L-functions. arXiv preprint. (This paper and the one listed below discuss conjectures about the distribution of congruent numbers, among other topics.)
• Conrey, J. Brian, Jon P. Keating, Michael O. Rubinstein and Nina C. Snaith. 2004. Random matrix theory and the Fourier coefficients of half-integral weight forms. arXiv preprint.
• Brown, Jim. 2007. Congruent numbers and elliptic curves. PDF (Brown refers to this text as lecture notes for an undergraduate course, but in fact it is a very polished and well-organized expository article. Includes some Sage code for working with rational points of elliptic curves.)
• Alter, Ronald. 1980. The congruent number problem. American Mathematical Monthly 87:43–45. (Tables of what was known and unknown shortly before Tunnell’s breakthrough.)

Special thanks to David Farmer for alerting me to this story in the first place and to Michael Rubinstein for help understanding what it all means.

Update 2009-10-10: In the comments, Barry Cipra points out that the quadratic equations in the Tunnell criterion necessarily have no solution whenever N is congruent to 5, 6, or 7 modulo 8. In fact all of the zero-solution points in the graph I presented above come from such N. Numbers in the residue classes 0 and 4 modulo 8 cannot be square-free. If we plot only those N that are square-free and congruent to 1, 2 or 3 modulo 8, we are left with just 53 square-free congruent-number candidates up to 1,000:

Only the numbers in these three residue classes were included in the trillion-triangle computation.

Sometime this morning, a web page that I visit occasionally changed its appearance from this:

to this:

The images are grabbed from Firefox on OS X; same effect in Safari. Some CSS forensics reveals what’s gone wrong here: Google has added a “height:28px” property to the style for the two buttons, and boosted their type size from 13 pixels to 16. (Type inside the search box is also given the giant billboard treatment.) Apparently the height property prevents the browser from using those cute jelly-bean widgets for the buttons.

Can we strike a bargain, Google? I’ll let you take over the world and track my every movement; I’ll sign over all my copyrights and promise never to block your advertising, if you’ll just give me my widgets back. Go ahead and Be Evil; just Don’t Be Ugly.

Or am I going to have to override you in userContent.css?

Update: Marissa Mayer, Google’s Vice President for Search Products & User Experience, explains:

For us, search has always been our focus. And, starting today, you’ll notice on our homepage and on our search results pages, our search box is growing in size. Although this is a very simple idea and an even simpler change, we’re excited about it — because it symbolizes our focus on search and because it makes our clean, minimalist homepage even easier and more fun to use. The new, larger Google search box features larger text when you type so you can see your query more clearly. It also uses a larger text size for the suggestions below the search box, making it easier to select one of the possible refinements.

For Firefox users who find the “supersized” page a nonimprovement, I offer the following quick hack:

@-moz-document domain(google.com) {
  .lsb, .gac_sb {
	font-size:13px ! important;
	height:inherit ! important;
	}

  .lst, .gac_m {
	font-size:14px ! important;
	}

}

Add these lines to your userContent.css file (see the Customizing Mozilla site for further instructions). This seems to fix the widget problem and the input-field text size, but all the drop-down gadgetry is still a mess and maybe the dropdown menu of suggestions. This will break if Google’s next whim is to change the class names “lsb,” “lst,” etc. If anyone has a cleaner solution, please pass it on.