So said John Maynard Keynes, but what did he know about the long run? He was a swell who spent his mornings in bed, trading international currencies over tea and crumpets.

Yesterday was my day for the long run: I ran a marathon for the first time in my life. The race was the Baystate Marathon in Lowell, Massachusetts. I finished in 5:11:23, good enough for 1,494th place. (Complete results here.) Except for the leaden skies, the blustery wind, the temperatures falling from the 40s into the 30s, and the steady rain that later turned to “wintry mix” and then a bit of snow, it was a perfectly lovely day for a run along the Merrimack River. I had a splendid time. Really. And yet it would have been even more fun if it hadn’t gone on quite so long.

Somewhere around mile 20, I began reflecting on the fact that the leaders of the race had already finished more than an hour earlier, and by now they were probably showered and dressed and having something hot to eat. I had to ask myself: Why hadn’t I been running faster, so that I too could now be sitting somewhere comfortable and dry and warm? A mile later, as I slogged on, it dawned on me that surviving a marathon is basically an optimization problem. The essential task is to balance the pain of running faster against the suffering of staying on your feet longer. There’s a mathematical function to be minimized here. But what’s the form of that function?

Splashing on through the puddles, I didn’t make a lot of progress on that question, but several hours later, wrapped in a blanket and enjoying a grilled-cheese sandwich, I realized that a simple candidate function is just 1/t + t, where t is the total time of the run. This function clearly has the right boundary behavior. It diverges at t = 0, reflecting the common-sensical notion that running infinitely fast is infinitely painful. The function is also unbounded as t goes to infinity, since taking forever to complete the course would also be very unpleasant. In between these extremes, there must be at least one t of minimal misery.

To get quantitative results from this function, we need to plug in some constants and coefficients. Marathon times near zero are unrealistic, so we ought to replace t with t–a, where a is the best plausible finishing time. Then we need a coefficient b that sets the scaling between the two kinds of discomfort. The full expression becomes:

\[f(t)=\frac{b}{(t-a)} + (t-a)\]

The value of a in this expression is somewhat arbitrary, but a good guideline might be the current world record marathon time of about 2:04. If we set a = 120 minutes, there’s no need to worry that I’ll ever do better than that (and thereby flip over onto the negative branch of the hyperbola, where everyone runs backwards). With this parameter fixed, the location of the least-arduous marathon time depends only on the value of b, which defines the cost of speed vs. the cost of endurance. If my run in Lowell was a correct solution of the optimization problem, then my personal value of b is 36,481.

This line of reasoning has a curious corollary. If I want to improve my marathon time, I don’t have to learn to run faster; all I have to do is become less tolerant of prolonged standing or walking. This impatience will shift the point of minimum pain leftward, toward higher speeds and shorter elapsed times. For example, if I can just get my value of b down to 14,400, I’ll be running four-hour marathons. I don’t think I’ve ever heard of a training program based on this principle.

Update 2009-10-20: This morning it occurred to me that the same graph, relabeled, could also serve to predict the future course of my geriatric athletic career. I’ve been running for only a few years, and I never attempted distances greater than 10K until this summer. Thus it seems reasonable to suppose that with further effort I might improve somewhat. On the other hand, I’m about 30 years beyond the age at which distance runners tend to reach their peak, which argues that I’m running against the wind, so to speak. (In the long run, Keynes was right after all.)

Here’s a fancifully relabeled version of the graph that I find rather cheering.

Unfortunately, there’s no good reason to believe that either of these phenomena are truly described by a law of the specific form 1/t + t. The second term could be t², or even e^t. (The marathon “ranking standards” published by USA Track and Field appear to increase quadratically with age.)

Update 2009-10-23: Perhaps someone at The New York Times reads bit-player. In today’s paper there’s a story under the headline “Plodders Have a Place, but Is It in a Marathon?” Juliet Macur writes:

Purists believe that running a marathon should be just that — running the entire course at a relatively fast clip. They point out that a six-hour marathoner is simply participating in the event, not racing in it. Slow runners have disrespected the distance, they say, and have ruined the marathon’s mystique.

Thick-skinned as I am, this stings. My apologies to the purists, and to the mystique. I’ll just note that the hundreds of volunteers who put on the race in which I “participated” were marvelously supportive of us plodders. The high school students at the water stations did not abandon their posts, or lose their enthusiasm, after the fleet-footed runners passed by. And when I arrived at the finish line, hours after the winners, there were still fans applauding in the grandstand, and volunteers to wrap me in a blanket, bring me water, offer congratulations. They had been out in the rain as long as I had, and they weren’t done yet. Three cheers for them.

The new issue of American Scientist is on the newsstands and on the web. My “Computing Science” column takes a stab at explaining the Hubbard model, a staple of condensed-matter physics that I’ve been struggling to understand for at least a decade. Have I finally figured it out? You can see for yourself.

The issue also includes an announcement that this new column will be my last for a year. I’m taking a sabbatical—or is it a dekasabbatical? (Both etymology and academic habit imply that a sabbatical is a break from the routine that’s supposed to come along every seven years or so. I’ve been writing the column for seventeen years.)

Friends ask what I’m going to be doing for the next twelve months. Well, one of the great advantages of my line of work is that you get to learn something completely different with every issue of the magazine—flitting from pseudorandom numbers to genetic codes to ichnofossils to interval arithmetic to ferromagnets to programming languages to the childhood of C. F. Gauss, and on and on, like some maniacal butterfly visiting every pretty flower in the field. One of the great disadvantages of my line of work is that you get to learn something completely different with every issue of the magazine—and as soon as you start to make a little progress, you have to set it aside and start all over on a whole nother subject. Thus I’m looking forward to being a little more single-minded for a while. Just one flower. Now if only I knew which flower.

I’ve been Way Out West for the past few days, exploring a desert landscape where everything bristles. Ros and I went walking in the foothills of the Santa Catalina range, north of Tucson. At one point along the trail we spotted a pair of tweezers someone had left behind near the base of a prickly pear. Those lost tweezers told their own story—a painful one. We hadn’t thought to pack such implements; later, a more experienced hiker told us that what you really want to take along are not tweezers but pliers.

The iconic cactuses of southern Arizona are the towering saguaros. They are magnificent plants, but I also find them a little clownish, with their limbs attached like sausages or knotted balloons.

On our stroll through the desert I was taking particular note of the saguaros’ rib patterns. As we later learned, the ribs work like accordion pleats, allowing the plant to expand its girth when it absorbs water after a rainfall. The ribs and their intervening grooves form vertical stripes, most of which extend the full length of the main trunk or an arm. Every now and then, however, an extra rib appears out of nowhere, like this:

What sort of developmental algorithm can generate a pattern like this? It looks like a textbook example of a mechanism discussed almost 60 years ago by none other than Alan Turing. Imagine clusters of specialized plant cells that secrete some chemical substance—a morphogen—that induces other cells to form a spine-crested ridge. Surrounding the growing tip of the plant is a necklace of such morphogen-secreting clusters, creating long parallel ridges as the trunk elongates, much like cake frosting extruded from a fluted nozzle. This is a self-organizing and self-maintaining process. The morphogen clusters (and the ridges) remain evenly spaced because each secreting group of cells inhibits all nearby cells from secreting the substance. Thus the morphogen sites act as if they have coiled springs keeping them equidistant. As the cactus grows taller, however, it also increases in circumference, spreading the clusters apart. When two adjacent morphogen sites are too distant to suppress activity between them—or in other words when the coiled spring no longer exerts a repulsive force—a new morphogen cluster can form spontaneously in the gap. The result is a new rib arising midway between two existing ribs.

After examining a few dozen saguaros we passed along the trail, I thought I understood the biology of the rib patterns. New ribs arise like mountains out of a valley floor, and so the valley itself must bifurcate, passing to the left and right of the new ridge. The fork where the valley splits always has its tines pointing upward. Thus when you trace along the epidermis of the cactus from ground level to the top of the plant, new ridges appear spontaneously from time to time, but existing ridges never disappear.

Those hypotheses lasted a kilometer or two. Then we came upon the cactus in the photograph below:

This saguaro has both bifurcating valleys and bifurcating ridges (along with a couple of doubtful cases where it’s not entirely clear whether the ridge or the valley is doing the splitting). The idea of morphogen centers separating and allowing a new cluster of cells to form between them cannot explain everything we’re seeing here; it appears that morphogen clusters are undergoing their own fission events as well. It’s not just the valleys that bifurcate but the ridges too.

A hundred meters farther along the trail, we found another exceptional cactus:

Here we have both forks and antiforks: As we proceed upward along the body of the cactus, ridges are both created and annihilated. Near the top of the image is a confused ring that includes several initiation and termination sites. Perhaps there was some disruptive event at that moment in the history of the plant’s growth. (Saguaros live for 150 or 200 years, so there’s plenty of history to play out.)

Bifurcating ridges can be explained in the same general way as bifurcating valleys, and so I suspect the saguaros are still good candidates for a model based on Turing’s ideas. But maybe the model is not quite as simple as I first thought. And the explanatory challenges get harder still. Elsewhere in the Tucson area, a fellow hiker drew my attention to this cactus:

Or how about the unfortunate specimen pictured below?

As the detail at left reveals, this one has a fork with its tines pointing neither upward nor downward. How the vandals managed to impale the cactus at such an elevation—the dinner fork is four or five meters off the ground—remains an unanswered question. But in the end it’s also not the most interesting question about these curious plants, which wear their algorithmic structure on their skin.

It’s the custom among mathematicians that when you reach age 60, you’re put on an ice floe with a day’s supply of walrus meat and set adrift.

But I’m not a mathematician.

Computer scientists get a few years’ grace. Because they count in binary, for them the dreaded end of productive intellectual life is put off until age 2⁶.

I’m not a computer scientist either. I’m a writer. This is no dispensation, however; the literary world also worships precocity. Byron said it: “Is there anything in the future that can possibly console us for not being always twenty-five?” I took an even harder line in my own youth. At age 16, when I ran away from home to write a novel, I vowed that I would be a famous author by 22 or find an ice floe and make an end of it.

And today I am 60–1, and I find I am not yet quite ready to be set out to sea. I look back to the poets who seemed so intensely young when I was young, and I find they have aged well. Here’s George Herbert:

I’m afraid. I’m afraid, Dave. Dave, my mind is going. I can feel it. I can feel it. My mind is going. There is no question about it. I can feel it. I can feel it. I can feel it. I’m a… fraid.

I’m about to dig into the innards of the software that runs this blog (updating WordPress from 1.5 to 2.6, installing some plugins, fiddling with CSS and PHP). So, just in case I can’t get the pod doors open when I’m done, I want to say it’s been fun, and thanks for tuning in.

Update: It seems I’m still here. Much remains to be done, but I’m hoping the site will remain functional while under reconstruction. Comments and complaints are welcome.

Update 26 October 2008: It’s like gardening, or dentistry, or cleaning out the attic: Once you start mucking about, you notice more and more that needs attention, and there’s no end to it. It’s clear now that I’m going to be fussing with the presentation of this site for weeks to come. Expect something different every time you visit. Those who read via RSS will probably be seeing the same posts flagged as ‘new’ over and over again. Sorry about that. (Does anyone know a way to avoid it, other than shutting down the feed altogether?)

Anyway, don’t be shy about letting me know what doesn’t work or what looks awful. One of the main motives for this housecleaning was to provide a less-lousy interface for leaving comments. I’ve already experimented with three approaches to that problem, and I’m not terribly happy with any of them. I’m likely to change again. If you’d like to test the system, this post would be a good place to leave witty remarks such as “Hello, world!” and “Testing, one, two, three.”

Here’s a specific question: All of the schemes for providing a preview of comment formatting rely on Javascript. I read RSS feeds with a program that doesn’t do Javascript (NetNewsWire 2.0). How about others? Will a Javascript solution actually solve anything?

Tardy announcement: I’ll be giving a talk tomorrow at Harvard. Details. If you’re in the neighborhood, please look in.

Update: Harvard has posted video of the talk; please have a look, if you’re interested, and you’re equipped to play RealMedia files. I’m not equipped, so I haven’t seen it myself. But I’m sure it’s wonderful (thanks to the AV expertise of Bill Countie and Geoff Maness).

Slides are here (PDF), though they won’t make much sense without the narration.

This morning I’m about to get on the T and go to Cambridge City Hall for a significant personal event. I’d like to celebrate the occasion with a few lines of verse by the great mathematician (and lesser poet) James Joseph Sylvester. (A tip of the top hat to Jerry Alexanderson for bringing Sylvester’s versifying to my attention.)

Fairest, O ! of lily-kind,
Perfect pearl and priceless find !
Pure as poet’s milk-white hind,
Spirit ! from all dross refined,
Hearts to ravish Heav’n-designed ;
Fresh as rills that sparkling wind
On their way the sea to find,
[O'er smooth pebble-bed patíned.]
Dew-drop ! on which sun hath shined,
Shedding glamour undefined,
And a light that doth remind
of the cloudless heav’n behind.
Whose light laugh, like bells sweet-chimed,
[Or hushed trill of bow refined
Touched with fingers taper-tined,]
Piercing feeling’s inmost rind,
Tone-spray tossing on the wind
Leaves no smart unmedicined.

  With fond folly all untined,
  To each duteous law resigned ;
    (Round her finger she can wind
    All to obey her disinclined :
    Who so wilful or so blind
    To say nay to Rosalind !)

Scott Aaronson, at Shtetl-Optimized, blogs:

To those of us who can’t tell a hypotenuse from a rhombus, the phrase “math journalism” sounds like an oxymoron. It brings to mind boring pedants like Martin Gardner, Sara Robinson, and Brian Hayes….

Thanks, Scott! Can I get that on a teeshirt?

Dear Readers,

You may have noticed something weird in these pages over the past few days. When you roll your mouse cursor over a link to an external web page, a thumbnail preview of the page pops up like a thought balloon. For example, try it with This Link. If any of you have strong opinions about this sort of thing, one way or the other, I’d be grateful to hear them. Myself, I’m undecided.

Note: As far as I know, the previews will appear only if you’re reading these pages with an ordinary web browser, not with an RSS reader.

Update 2006-12-08: The verdict is in. Bit-player is unsnapped. While I’m at it, I’m also turning off the little quiz that greets commenters. At some point the quiz may come back, but I’m looking at other ways to deal with the problem of comment-spam.

Whether or not anyone else is reading the stuff I’m posting here, the spammers have discovered me. And I’m afraid that “comment spam” is not as amusing as the e-mail variety. I have therefore installed one of those annoying sentry programs that will ask for the answer to a trivial question before allowing you to post a comment. I’m choosing questions that I’m sure every reader of this blog will be able to answer instantly — things like, “Who published the first proof of the impossibility of trisecting an angle?” and “True or false: The present king of France is bald?” Please give it a try. If you have any trouble — and you’re not a spambot — e-mail me at brian@bit-player.org.

Update 2006-12-14: A few days ago I thought I had solved the comment-spam problem by other means, and so I turned off the interrogation. But the spammers are back, and so I have put the sentry on duty again. This time the questions all take the same form: a sequence of positive integers. You’ll be asked to name the next element of the sequence. None of them are tricky (no lists of subway stops), or even interesting. If the spambots learn to solve the problems in this set, well, good for them; at that point we can think about escalating to the next level.