Sunshine In = Earthshine Out

No computer simulations have ever had broader consequences for human life than the current generation of climate models. The models tell us that rising levels of atmospheric carbon dioxide and other greenhouse gases can trigger abrupt shifts in the planet’s climate; to avert those changes or mitigate their effects, the entire human population is urged to make fundamental economic and technological adjustments. In particular, we may have to forgo exploiting most of the world’s remaining reserves of fossil fuels. It’s not every day that the output of a computer program leads to a call for seven billion people to change their behavior.

Thus sayeth I in my latest American Scientist column (HTML, PDF). Climate is a subject I take up with a certain foreboding. The public “debate” over global warming has become so shrill and dysfunctional that I can barely force myself to pay attention, much less join the fray. As I worked on writing the column, I found myself tiptoeing through the polemical minefield, carefully avoiding any phrase that might be picked up by “the other side” and used against me. (“Even a writer for American Scientist has doubts about . . .”). When I handed in the text, my editors went over it with the same cautious anxiety—and they found a few places where they thought I hadn’t been careful enough.

This is not the kind of science I enjoy. I would much prefer to dwell in less-contentious corners of the cosmos, where I can play with my sticky spheres or my crinkly curves, and never give a thought to “the other side.” But now and then one must return to the home planet. Besides, the computer modeling that plays a major role in climate science is just my kind of thing.

So, for the past few months climate models have been my breakfast, lunch, and dinner, and my bedtime snack. I’ve been reading in the literature, and poring over source code; I’ve managed to get a couple of serious models running on a laptop. The American Scientist column says more about the rewards and frustrations of all these undertakings. Here I want to talk about a lighter side of the project: a little climate model of my own. The image below is just a static screen grab, but you can go play with the real thing if you’d like.

Screen grab of the energy-balance model user interface

Okay, it’s not really all my own. The physics behind the model was explored more than 100 years ago by Svante Arrhenius. The first computer implementations were done by M. I. Budyko and William Sellers in 1969, and a third version was published by Stephen Schneider and Tevi Gal-Chen in 1973. But the JavaScript is mine. (Any mistakes are mine, too.)

The model is a simple one. The planet it describes is not the Earth we know and love but a bald sphere without continents or oceans, seasons or storms, or even day and night. Temperature is the only climatic variable. The globe is divided into latitudinal stripes five degrees wide, and temperatures are always uniform throughout a stripe. Other than warming and cooling, the only thing that can happen on this planet is freezing: If the temperature in a stripe falls below –10 degrees Celsius, it grows an ice sheet.

Why pay any attention to such a primitive model? Although a zillion details are missing, some important physical principles emerge with particular clarity. At the heart of the model is the concept of energy balance: If the Earth is to remain in thermal equilibrium with its surroundings, then it must radiate away as much energy as it receives from the Sun. The planet’s temperature will rise or fall in order to maintain this balance. The governing equation is:

\[Q (1 - \alpha) = \sigma T^{4}.\]

Here \(Q\) is the average intensity of incoming solar radiation in watts per square meter, \(\alpha\) is the albedo, or reflectivity, and \(T\) is temperature in Kelvins; \(\sigma\) is the Stefan-Boltzmann constant, which relates thermal emission to temperature. The numerical value of \(\sigma\) is \(5.67 \times 10^{-8} \mathrm{W m^{-2} K^{-4}}\). In other words, sunshine in = earthshine out.

If we know the solar input, we can calculate the Earth’s temperature at equilibrium by solving for \(T\):

\[T = \sqrt[4]{\frac{Q (1 – \alpha)}{\sigma}}\]

This is exactly what the energy-balance model does.

Two more effects need to be taken into account. First, most of the sun’s energy is received in the tropics, but it doesn’t all stay there. On the Earth, heat is redistributed by the process we call weather—evaporation and condensation, winds, storms, etc.—and also by ocean currents. In the model, the effect of all that swirly fluid dynamics is crudely approximated by a diffusive flow that simply smooths out temperature gradients.

Finally, there’s the greenhouse effect. Solar radiation at visible wavelengths passes through the Earth’s atmosphere to warm the surface, but the Earth emits at longer wavelengths, in the infrared; radiation in that part of the spectrum is absorbed by water vapor, carbon dioxide, and other minor constituents of the atmosphere. The blocked infrared emission raises the temperature at the surface by 30 degrees Celsius or more. The Earth would be a very chilly place without this effect, but the burning issue of the moment is how to avoid overheating. The model sidesteps all the complexities of atmospheric chemistry and absorption spectra; a single parameter, H, simply determines the fraction of outgoing radiation trapped by greenhouse gases.

The interface to the model has four controls—sliders that adjust the incident solar flux, the albedo of land, the albedo of ice-covered regions, and the greenhouse parameter. While playing with the slider settings, it’s not hard to get the model into a state from which there is no easy recovery. That’s what the reset button is for. (Too bad the real planet doesn’t have one.)

Some experiments you might try:

  • The default greenhouse setting is H = 0.4, meaning that 40 percent of the outgoing radiation is intercepted. Lower the slider to below 0.2. The model enters a “snowball Earth” state, where ice sheets descend all the way to the Equator. Now return the greenhouse slider to the default setting of 0.4. The ice persists, and it will not melt away until the control is moved to a still higher setting. When the thaw does come, it is sudden and overshoots, leaving the planet in a much warmer state.
  • Raise the greenhouse slider to about 0.45, where the polar ice cap disappears. Returning the slider to 0.4 does not restore the ice cap, and the polar areas remain several degrees warmer than they were before the greenhouse excursion.
  • Raising the land albedo (so that more sunlight is reflected from snowfree surfaces, and less is absorbed) reveals another “ratchet” mechanism. Once the planet becomes totally icebound, further changes in the land albedo have no effect at all. The reason is simply that no ice-free land is exposed.
  • Push the greenhouse slider toward the top of the scale. Beyond H = 0.75, tropical temperatures approach the boiling point of water, and the planet has obviously become uninhabitable. Question: What will happen at H = 1.0, where no radiation at all can escape the atmosphere?

Some of the extreme parameter values mentioned above yield fanciful results that would never be seen on the Earth. But certain interesting aspects of the model seem quite plausible. I am particularly intrigued by the presence of abrupt changes of state that look much like phase transitions. If we consider just the value of the greenhouse parameter H and the average global temperature T, we can draw a two-dimensional phase diagram in the (H, T) plane. The plot below traces the system’s trajectory as H is lowered from 0.4 to 0.2, then raised to 0.5, and finally returned to the initial setting of 0.4.

Greenhouse hysteresis plot

That four-sided loop is the hallmark of hysteresis (a term first introduced in the study of magnetic materials). Initially, as the greenhouse effect weakens, temperature falls off linearly. Then, at about H = 0.3, the curve steepens. (The stairsteps represent the freezing of successive zones of latitude.) Just below H = 0.218, the temperature falls off a cliff, dropping suddenly by 20 degrees Celsius. On the next segment of the curve, as H increases again, the temperature again responds linearly, but in a much colder regime. Only when H reaches 0.459 is warmth restored to the world, and this time there’s an abrupt upward jump of almost 35 degrees.

It’s no mystery what’s causing this behavior. There’s a strong feedback effect between cooling/freezing and warming/thawing. When the temperature in a latitude band falls below the –10 degree threshold, the zone ices over; the ice-covered surface is more reflective, and so less heat is absorbed, cooling the planet still more. Going in the other direction, when the ice melts, the exposed land absorbs more heat and brings still more warming.

The sharp, discontinuous transitions in the graph above could not happen on the real planet. They are possible in the model because it has no notion of heat capacity or latent heat; after every perturbation, the system instantly snaps back to equilibrium. But if the hysteresis loop in the model has unrealistically sharp corners, its basic shape is not impossible on a physical planet. What’s most important about the loop is that over a wide range of greenhouse parameters, the system has two stable states. At H = 0.4, for example, the world can have an average temperature of either 14 degrees of –22 degrees. That kind of bistability may well be possible in terrestrial climate.

Snowball Earth is not a fate we need to worry about anytime soon, but there is evidence that the planet actually went through such frozen states early in its history. The runaway greenhouse effect that would boil away the oceans is also not an immediate threat. So can we rest easy about living on a sedate, linear segment of the H-T curve? That’s a question that climate models are supposed to answer, but it’s beyond the scope of this particular model.

Posted in computing, physics | Leave a comment

The pi man

Some weeks ago a reader sent me a note about an old article of mine, written back in 1989. The PDF, he reported, was missing a page. So I went scrounging through some untidy heaps of papers, found a copy, scanned it, and posted a new PDF. But along the way I stopped to reread my own words from 25 years ago.

The piece was a review of the first version of Mathematica, which debuted that summer of 1989. I began my review:

Sometime in the 1850′s William Shanks, a British schoolkeeper and amateur mathematician, began calculating the digits of the decimal expansion of π. Twenty years later, when he published his results in the Proceedings of the Royal Society, he had gotten as far as 707 digits. Today, with the aid of the computer program Mathematica, performing the same calculation requires no more effort than typing the expression:

N[Pi, 707] .

In response to this command, the labor that occupied Shanks for two decades is completed in less than eight seconds. Furthermore, Mathematica produces the correct 707 digits. Shanks made an error in the 528th decimal place that spoiled all the rest of his work.

I remember being wowed by the sub-eight-second running time of the π computation. (My notes say the actual time was 7.97 seconds.) That was with Mathematica 1.0 on a Macintosh SE/30 (16 MHz 32-bit processor, 5 MB of memory). How long would it take with more recent hardware and software? Naturally I had to try the experiment. Wolfram Research has just released version 10.0 of Mathematica. I ran it on the modern equivalent of the SE/30: a middle-of-the-road Mac laptop (2.6 GHz 64-bit processor, 8 GB of memory). The result: 190 microseconds.

I’m wowed again.

We don’t know how long Shanks actually spent noodling away at the digits of π. He did not—despite what I wrote in 1989—work continuously for two full decades. One known data point is that he extended his result from 530 decimal places to 607 places during March and April of 1853. Suppose he gave the task 60 hours a week during those two months. That’s roughly 500 hours for 77 digits, which we might extrapolate to 5,000 hours for 707. The average computation speed is thus 0.15 digits per hour, or 0.000043 digits per second. Compare this performance with later results:

Computer Year Speed
Shanks with a pencil 1873 0.000043
ENIAC with punch cards 1949 0.0079
16 MHz Mathematica 1.0 1989 89
2.6 GHz Mathematica 10.0 2014 3.7 × 106

What I find surprising in these numbers is that the ENIAC was only a few hundred times faster than a Victorian English gentleman wielding pencil and paper, whereas my current laptop is a billion times faster than the ENIAC. Much of this recent speedup is technological—a blessing of Moore’s Law. But algorithmics surely plays a role too.

As I mused about these developments, I realized that I had no idea how Mathematica computes the numerical value of π, or whether the method might have changed between version 1.0 and version 10.0. I also had only a vague notion of how Shanks did his computation. It turns out that both he and the ENIAC programmers used something called Machin’s formula:

\[\frac{\pi}{4} = 4\arctan\left(\frac{1}{5}\right) - \arctan\left(\frac{1}{239}\right)\]

Thus they found the value of π by first finding the value of two arctangents. That’s good to know, but it raises a further question: How do you evaluate the arctangent of 1/5 or 1/239? You can’t just look it up in a table if you need 700+ digits of precision. The solution, I learned, is to evaluate some terms of an infinite series, and then sum them up:

\[\arctan x = \frac{x^{1}}{1} - \frac{x^{3}}{3} + \frac{x^{5}}{5} - \frac{x^{7}}{7} + \cdots = \sum_{k=0}^\infty \frac{(-1)^{k}}{2k+1}x^{2k+1}\]

Mathematically, this is all you really need to know to understand what Shanks was doing in his computation of π. The whole job reduces to elementary arithmetic: Adding and subtracting, multiplying and dividing. But this was arithmetic at scale. I was having a hard time imagining the logistical and mechanical details of working with hundreds of numbers that run to hundreds of decimal places. “More is different” is a slogan from physics, but surely it applies to arithmetic too.

The last time I mentioned Shanks in these pages, I heard from Erwin Engert, who has done some thoughtful analysis of Shanks’s methods (and his errors). The documents on Engert’s web site include records of his own pencil-and-paper computations of π to 20 and 40 decimal places. I’m not brave enough to try that myself, but I thought I might be able to learn something useful by writing programs that follow some of the same steps a human solver might choose. The project would take a day or two, I supposed. Then one thing led to another (as it has a habit of doing).


For many of us, William Shanks is a figure to be pitied or ridiculed. What a sad waste of human potential to spend a major chunk of your lifespan churning out digits that no one would have any use for even if they were correct. What I’ve been doing over the past several weeks may be just as foolish and futile: I have expended a nontrivial chunk of my own lifespan tracking down all things Shanks—reading his publications, casting about for scraps of biographical detail, and most of all trying to reconstruct his work and figure out where his calculations went awry.

The main results of this dubious effort are in my new American Scientist column (HTML, PDF). I have also made available some supplementary material, including an annotated reading list and an IPython notebook with code you are welcome to explore in case you too would like to devote a portion of your life to following the Pied Piper.

In this posting I merely want to tie up a couple of loose ends: one item that wouldn’t fit in the magazine (it would have required a multipage foldout), and another development I stumbled upon after the article was sent to the printer.


In one phase of his computation, Shanks faced the task of adding up 506 numbers, each with 709 decimal places. Half of these terms were positive and half were negative, so the problem would naturally decompose into two parts. But half of the addition problem is still a tableau of 253 × 709 = 179,377 digits. What would such an array of numbers look like, if you wrote them all down on one sheet of paper? I don’t have a sheet big enough to answer that question directly. But here’s a hint—in very tiny type, with scroll bars. These are the 253 positive terms in the series for arctan 1/5. Feel free to add up the columns and write the sum at the bottom.

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 0.0000000568 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 8888888888 888888889
 0.0000000000 6301538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 8461538461 5384615384 6153846153 846153846
 0.0000000000 0007710117 6470588235 2941176470 5882352941 1764705882 3529411764 7058823529 4117647058 8235294117 6470588235 2941176470 5882352941 1764705882 3529411764 7058823529 4117647058 8235294117 6470588235 2941176470 5882352941 1764705882 3529411764 7058823529 4117647058 8235294117 6470588235 2941176470 5882352941 1764705882 3529411764 7058823529 4117647058 8235294117 6470588235 2941176470 5882352941 1764705882 3529411764 7058823529 4117647058 8235294117 6470588235 2941176470 5882352941 1764705882 3529411764 7058823529 4117647058 8235294117 6470588235 2941176470 5882352941 1764705882 3529411764 7058823529 4117647058 8235294117 6470588235 2941176470 5882352941 1764705882 3529411764 7058823529 4117647058 8235294117 6470588235 2941176470 5882352941 1764705882 352941176
 0.0000000000 0000009986 4380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 8095238095 2380952380 9523809523 809523810
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 0.0000000000 0000000000 0185127900 6896551724 1379310344 8275862068 9655172413 7931034482 7586206896 5517241379 3103448275 8620689655 1724137931 0344827586 2068965517 2413793103 4482758620 6896551724 1379310344 8275862068 9655172413 7931034482 7586206896 5517241379 3103448275 8620689655 1724137931 0344827586 2068965517 2413793103 4482758620 6896551724 1379310344 8275862068 9655172413 7931034482 7586206896 5517241379 3103448275 8620689655 1724137931 0344827586 2068965517 2413793103 4482758620 6896551724 1379310344 8275862068 9655172413 7931034482 7586206896 5517241379 3103448275 8620689655 1724137931 0344827586 2068965517 2413793103 4482758620 6896551724 1379310344 8275862068 9655172413 7931034482 7586206896 5517241379 3103448275 8620689655 1724137931 0344827586 206896552
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 0.0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 000000005  

Written at normal size, this 253 × 709 array would fill a sheet of paper roughly two meters wide by one meter high. I’m sure that Shanks never actually created such an unwieldy artifact. Instead he must have broken the process up into small sections, with some sort of hierarchical system of subsums copied over from one sheet to another. But I was curious: If he had decided to tackle the summation all in one go, could he have found paper in large enough sizes? The British Association of Paper Historians suggests not: The largest sheets in their list of Old English Paper Sizes are Imperial, Elephant, Double Elephant, Atlas, Columbier, and finally Antiquarian, which is 31 by 53 inches. But another source, paper-sizes.com, includes a sheet called Emperor, which is 4 feet by 6 feet—just about right. Of course even without the Emperor, Shanks could have pasted together smaller sheets. (Scotch tape was still in the future.) I like to imagine him taking over the family dining table with this huge army of numbers in tidy rows and columns.


Shanks’s 707 decimal places of π were accepted as correct for more than 70 years; his value was enshrined in carved wooden numerals on the ceiling of the Palais de la Decouverte in Paris. His errors were not discovered until the 1940s, when D. F. Ferguson, working with a mechanical desk calculator, was making rapid progress on a new computation of the constant. His value of π differed from Shanks’s starting just before decimal place 530. Many subsequent computations confirmed that Ferguson was right and Shanks was wrong.

But where and how and why did he go wrong? If we had all of his working papers, it would be easy to track down the mistakes. In 1853 Shanks published a small book that included the individual terms he summed to calculate arctan 1/5 and arctan 1/239. Unfortunately, the terms are given only up to the 530th decimal place—exactly where the trouble begins. For all the remaining digits—up to 607 in the book, then on to 707 in an 1873 publication—we have only the sums of the terms, that is, the computed values of arctan 1/5 and arctan 1/239, along with the resulting value of π.

In the absence of the working papers, we can never identify Shanks’s errors with complete certainty, but there’s hope of finding some plausible candidates. The basic idea is to take the difference between the true arctan value and the value that Shanks reported, then subtract this difference from each of the individual terms. If we get lucky, an error will show up as some sort of distinctive pattern.

As far as I know, Engert was the first to identify a likely error in this way. Shown below are decimal places 520 through 580 in the true and the Shanks values for arctan 1/5, along with their difference. When we subtract this difference from the true value of term 248 in the arctan series, we get an adjusted value that immediately suggests a clerical blunder. If we omit the 0 at position 530 (marked in red), letting all the following digits slide left one place, the true value of term 248 is converted into the adjusted value. It’s just the kind of slip that might well happen when copying a number from one page to another. This anomaly popped up when I ran a program searching for shifts caused by omitted or repeated digits; then I discovered that Engert had already spotted it some time ago. There is also a single-digit substitution of 2 → 3 (flagged in green) that Engert apparently didn’t notice.

a5 true     0.197 ... 560518377574220877835853152464749330914587633823112490332030
a5 shanks   0.197 ... 560518377617781642423378303370181926488028277686119150985606
diff       -0.000 ... 000000000043560764587525150905432595573440643863006660653576

t248 true   0.000 ... 744466800804828973843058350100603621730382293762575452716297
t248 adj    0.000 ... 744466800848389738430583501006036217303822937625582113369874

The omission at decimal place 530 in term 248 “uncorrects” the value of arctan 1/5 through place 568; then something else goes wrong. In my American Scientist column I suggest the following explanation of this second error. Starting with the “uncorrected” value of arctan 1/5—that is, the true value with the error in term 248 already incorporated—we subtract the Shanks value of arctan 1/5 to get a second difference. Then, subtracting this difference from term 72 in the series, we observe this intriguing pattern:

a5 248ucx   0.197 ... 018192648802827768629156477871020728799805452914758511130462
a5 shanks   0.197 ... 018192648802827768611915098560675901213598556363034321005664
diff        0.000 ... 000000000000000000017241379310344827586206896551724190124797

t72 true    0.000 ... 413793103448275862068965517241379310344827586206896551724137
t72 adj     0.000 ... 413793103448275862051724137931034482758620689655172361599340

We can convert the true term 72 into the adjusted value by excising the five digits marked in red and letting all the rest of the digits shift to the left to fill up the void. Again, one can imagine the omission as a careless error, probably made while transcribing intermediate results for use in a later stage of the computation.

This much of the story is told in my American Scientist column. But something about those numbers bothered me, and I kept poking at them even after the magazine went to press. Here they are again with different highlighting:

t72 true    0.000 ... 413793103448275862068965517241379310344827586206896551724137
diff        0.000 ... 000000000000000000017241379310344827586206896551724190124797
t72 adj     0.000 ... 413793103448275862051724137931034482758620689655172361599340

The very same motif—the 28-digit sequence 1724137931034482758620689655—appears in all three numbers, shifted left or right by a few places. The presence of these digits in t72 true is easy to explain: This sequence is the repeating unit, or repetend, in the decimal representation of term 72, which is equal to \((1/5)^{145}/145\). If you were to compute still more digits of the term, you would find that the same sequence repeats endlessly. But seeing the same sequence turn up in the other two numbers seems a bit surprising.

Try an experiment (or consult my IPython notebook, which includes code for the experiment). Take the sequence 1724137931034482758620689655 and create a cyclic permutation: for any k between 1 and 27, lop off k digits from the front of the sequence and paste them onto the back. Now subtract the permuted sequence from the original one. Notice anything?

Here’s an example, with k = 7:

 1724137931034482758620689655
–9310344827586206896551724137
–7586206896551724137931034482

The digits in the result form another cyclic permutation of the same sequence. That can’t be a coincidence!

And indeed it’s not a coincidence. The sequence 1724137931034482758620689655 is a cyclic permutation of 0344827586206896551724137931, which is the repetend of 1/29. It’s also not a coincidence that 145, the denominator in our term, is equal to 5 × 29. And the secret behind all of this hocus-pocus is that 29 is a “full-repetend prime” in base 10: a prime p whose reciprocal 1/p has a repetend with p–1 digits, the most possible. For all such primes the repetend is a “cyclic number”: multiplying by any integer in 1, 2, 3 … p–1 yields a cyclic permutation of the same digits. And, as observed above, subtracting any two cyclic permutations also yields another cyclic permutation.

I was led to look at term 72 by this note at the end of the 1946 paper in which Ferguson announced the discovery or errors in Shanks’s work:

In calculations made since the preceding part of this Note was written, from the 569th decimal place onwards I have a further disagreement with Shanks’ figures, namely 6896551724137931034482758620 (recurring). These are precisely the figures which should occur at this exact spot (i.e. from the 569th d.p. onwards) in the term 145 which comes in the series for \(\tan^{-1}(1/5)\). This surely cannot be a mere coincidence. It looks very much as if Shanks omitted the whole of this term from the 569th place onwards.

I am deeply puzzled by this statement. The sequence of digits mentioned does indeed appear exactly where Ferguson indicates in term 72. (In Ferguson’s numbering this is term 145; 2 × 72 + 1 = 145.) But truncating that term at position 569 does not have the desired effect of transforming the true value of arctan 1/5 into the value given by Shanks. To accomplish that, as noted above, what’s needed is not truncating the term but shifting the digits left by five places.

At a loss to explain the discrepancy between Ferguson’s observations and my own results, I went looking for other occurrences of the same cyclic number. It has to appear in at least one other place, namely term 14, which has the denominator 2 × 14 + 1 = 29. Here’s what I found:

t14         0.000 ... 4137931034482758620689655172413793103448275862068965517241379310
t14-diff    0.000 ... 4137931034482758620689654999999999999999999999999999999999478062
diff        0.000 ... 0000000000000000000000000172413793103448275862068965517241901247

Hence in term 14 we see the very phenomenon that Ferguson reported in term 72: truncating the term beyond decimal place 568 would recreate Shanks’s erroneous value, through decimal place 601. (The truncation could be expressed by setting the sigits to 5000… or to 4999…; it doesn’t matter which.)

With this discovery we have an alternative theory of how Shanks goofed in the neighborhood of decimal place 569. Instead of skipping five digits while transcribing term 72, he might have neglected the trailing digits of term 14 while summing up the series of terms. Which mechanism is more likely? If we could diagnose the further error beginning at place 602 in the arctan 1/5 series, we might learn the answer to this question. Within the range 569–601, the shift in term 72 and the zeroing of term 14 have identical effects, but in the later digits they would yield different values. Unfortunately, my attempts to find the source of those further errors in the arctan 1/5 computation have come up empty.

There’s also an error in Shanks’s value for arctan 1/239 that remains unaccounted for. And there’s the further mystery of Ferguson’s note. Why did he point to term 72 instead of term 14? To explain away his comment as a result of simple confusion or a memory lapse, we need to assume that he was mistaken both about which term he was looking at and which permutation of the cyclic number he was seeing. That’s hard to credit in someone who was an exceedingly meticulous calculator. I can offer up another hypothesis, though it’s no less farfetched: Maybe Ferguson derived the error term not from arctan 1/5 but from 4 × arctan 1/5, which is how the number enters into Machin’s formula. It turns out that 4 × 1724137931034482758620689655 is the very permutation that appears in term 72: 6896551724137931034482758620.

Even if we solve this mystery, there’s another intriguing conundrum in Ferguson’s observation. I don’t understand how he could have spotted the error at position 569—whatever its cause—without having discovered the site of the earlier problem at position 530. The distinctive repetend patterns appear only after the single-digit omission in term 248 has been “uncorrected.” Ferguson knew there was an error somewhere near decimal place 530, but his publications on the subject (or at least the three known to me) say nothing about the cause of that error.

In his 1853 book on π, Shanks ends his introductory remarks by declaring that he is going to “take leave of the subject for the present.” I’m going to follow his lead and declare that I’ve had enough of Shanks for the present. Maybe, like him, I’ll return to the topic after a suitable interval—say 20 years.

Posted in computing, mathematics | 6 Comments

A Glitch in the Maptrix

We must always be on the lookout for glitches in the Matrix—anomalies that give us a fleeting glimpse into the algorithms and data structures of the computer simulation that we call Reality. But there’s also the Maptrix, the alternative reality supplied by online mapping services. I spend a fair amount of time exploring that digital terrain, and lately I’ve noticed a few glitches. Exhibit 1 is a tank farm in Bayonne, New Jersey, near where Kill Van Kull enters New York Bay:

Bayonne tank farm as seen in Apple Maps satellite view, showing tanks as irregular polyhedral appoximations to a cylinder

I’ve seen a lot of oil tanks over the years, but never before have I encountered such ragged, faceted approximations to cylindrical form. These lumpy, polyhedral tanks suggest that in this little corner of industrial New Jersey, π has a value somewhat smaller than 3.14.

But the π peculiarity is ephemeral. The image above was captured from a laptop screen at the instant the landscape was first rendered in the Apple Maps program. Moments later most of the defects were magically healed, and the illusion of solid, circular reality reasserted itself:

Bayonne tank farm as seen in Apple Maps satellite view, showing tanks as normal cylinder

Here’s another example, from a tank farm in Carteret, New Jersey, just across the Arthur Kill from Staten Island. This time we’re looking down on the tanks from directly overhead. The image at left was captured as soon as the scene appeared on screen; at right is the rounded-out version that emerged a few second later. The software, again, is Apple Maps.

Linden tanks before and after

It’s not just Apple’s version of reality that has such anomalies. Here’s a sample from another source:

topologically defective tanks in Kearny NJ as seen by  Google Maps

The mangled petroleum tanks in this image are in Kearny, New Jersey, a few miles north of Bayonne along the Passaic River. In this case the picture was taken by Google’s eye in the sky, not by Apple’s. The distortion is different, but no less disturbing. Now it’s not just the geometry of the cylinders that has gone goofy but also the topology. Some of those tanks won’t hold oil (or any other fluid); they have holes in them. And notice how the spill-containment walls surrounding the tanks also look moth-eaten.

Finally, returning to Apple Maps, and scrolling just half a mile northwest from the Carteret tanks, we cross the Rahway River into Linden, New Jersey, where we come upon this alarming scene:

Linden tank farms, some 3D but some flattened

Toward the right side of the image we see more cylindrical tanks, some with faint, remnant traces of polyhedral approximation. But when your glance wanders to the upper left, you find that the world suddenly loses all depth. The tank farm over there, and the water treatment plant at the top of the frame, are merely painted on the landscape—trompe l’oeil structures that don’t trompe anyone.

Linden low angle

This image offers another view of the same Linden landscape, looking obliquely to the west or northwest. The road that runs through the scene from foreground to background, crossing the New Jersey Turnpike at the very top of the frame, is Tremley Point Road. Suppose you were driving west along that road. Just beyond the row of lumpy trees that extends from the left edge of the image toward the road, you would cross a mysterious boundary, leaving behind the pop-up 3D world and entering flatland. What would happen to you there? Would you be pancaked like those tanks, reduced to a two-dimensional object painted on the pavement, with a painted shadow to accompany you?


Leaving behind tank farms but still poking around in the same general neighborhood of northern New Jersey, I was able to record four stages in the “construction” of the Bayonne Bridge, which crosses Kill Van Kull between Bayonne and Staten Island. These are images from Google Maps, the first three captured at intervals of about a second, the last after a delay of a few seconds more:

stage 1 in the

stage 2 in the

stage 3 in the

stage 4 in the "construction" of the Bayonne bridge at 4:08.32 PM


In calling attention to these oddities in online map imagery, my aim is not to mock or belittle. To me, these maps are one of the marvels of the age. A century ago, it was a huge novelty and liberation to soar above the earth’s surface for the first time, and see the landscape spread out below as if it were a map. It’s no less remarkable that we have now transformed the experience of looking at a map into something like flying an airplane.

The first “satellite view” maps were just that: montages of images made from hundreds of miles above the earth’s surface, looking straight down. They portrayed the territory as an array of pixels, assigning an RGB value to every (lat, lon) pair on the surface of a sphere.

The next step was to add a digital elevation model, giving each point on the surface a z value as well as a color. This scheme allows us to gaze obliquely across the landscape and see a realistic rendering of mountains, river valleys, and other natural landforms. It works well as long as you don’t try to get too close: the model is well-suited to forests, but not to trees. And it doesn’t work well at all for manmade artifacts.

In representing engineered structures, one weakness of elevation maps is that any reasonable scheme for interpolating between sample points will tend to round over corners and sharp edges, so that buildings become lumpish mounds. Symmetries are also lost: the sampling fails to preserve the rectilinearity or circularity of the various objects we strew around the inhabited patches of the world. And the biggest problem with an elevation map is that it’s a mapping in the mathematical as well as the cartographic sense. The surface of the planet is defined by a smooth, single-valued function, assigning a unique elevation z to every (lat, lon) point on the sphere. Any line radiating from the center of the earth must cross that surface in exactly one point. As a result, there can be no vertical cliffs and no overhangs. Also no bridges. The surface defined by the elevation model can go under a bridge or over it, but not both.

The latest mapping programs are apparently addressing these issues by building explicit three-dimensional models of selected landscape features. I see evidence of several different techniques. The Bayonne Bridge model that assembles itself in the four frames above is clearly based on a triangulated mesh: all the surfaces making up of the envelope of the structure are decomposed into elementary triangles. The cylindrical tanks in the Apple Maps images seem to grow their circular form through an Archimedean process, in which a circle is defined as the limit of an n-gon as n goes to infinity. Elsewhere, I think we may be seeing some kind of spline curves or patches.

Having constructed the model and embedded it in the landscape, the next trick is to project the pixel pattern from a photograph onto the model surface, as a texture. This process too has a certain comical potential:

Barge and tug near Bayonne containerport

What we’re seeing here is apparently a tugboat nudging a barge equipped with a crane. The matching of surface pattern to three-dimensional form has gone badly awry, which gives the whole scene a toylike quality. I find it charming. In future years, when the Maptrix has become a hyperrealistic, real-time virtual world with day and night, weather and seasons—maybe with inhabitants who wave back at you—we’ll wax nostalgic over such quaint foibles.

Posted in computing, modern life, photography | 9 Comments

Cosmic eggshells

In the creation myths of certain cultures, the world began as an egg, which broke open to form the earth and the sky. What if some intrepid explorer mounted an expedition to the ends of the earth and came back with fragments of the original eggshell, or maybe feathers from the bird that laid it? That’s what happened last week. Thanks to a group called BICEP2, which took a telescope to the South Pole and pointed it at a single patch of sky through three long Antarctic winters, we now have bits of eggshell providing strong evidence for our modern creation myth: the inflationary universe.

Alan Guth lectures at Harvard 2014-03-25In referring to cosmic inflation as a creation myth, I don’t mean to be dismissive—not in the least. It’s just that when you listen to this narrative of how the world began, the story is every bit as fantastical as any tale of hatching eggs or turtles-upon-turtles. Last Tuesday I heard Alan Guth, who dreamed up cosmic inflation in 1981, tell the story to an overflow crowd at Harvard. The video of the colloquium is on the net. Here’s my paraphrase of a few highlights:

Once upon a time, the entire universe we know today was packed into a tiny dot, much smaller than an atom—only a millionth of a billionth the size of a proton. With so much matter and energy confined in such a small space, gravity was a very powerful force, which you might think would hold everything together forever. But one day something odd happened. Inside the microdot universe, gravity suddenly changed its sign; it became a repulsive force instead of an attractive one. That repulsion is what put the bang in the big bang. The universe doubled in size in 10–37 second. Then it doubled again, and again. After about 100 doublings, the dot had expanded by a factor of 1028, and our universe had a diameter of 1 centimeter—the size of a marble. It was all over in a trillionth of a trillionth of a trillionth of a second, or thereabouts. Afterwards, gravity resumed its familiar, convivial, let’s-all-stick-together personality, and the rest of the history of the universe unfolded at a more stately pace over the next 13.8 billion years.

Three decades ago, when I first read about (and wrote about) the inflationary universe, physics itself was in a bubbly, inflationary mood. Theorists, it seemed, couldn’t dream up an idea too outlandish for experimentalists to confirm. Let’s have three quarks for Muster Mark, decreed Murray Gell-Mann; sure enough, dense little quarky pits were found inside the proton and the neutron. We need a fourth quark—and let this one have charm—said Sheldon Glashow and his friends; and there it was in the J/ψ particle, discovered in 1974. Give us supermassive analogs of the massless photon, said Steven Weinberg and Abdus Salam; nature obliged with the W± and the Z0. Now and then, the theorists had to scramble to catch up with experiments, as when Leon Lederman found evidence for a fifth quark. On the other hand, a few theoretical predictions were way out in front of experimental prowess: Finding the Higgs boson took 50 years.

Not every flight of fancy landed in Stockholm. Theorists asked experimenters to keep an eye out for decaying protons; they spent years looking and saw nothing. The magnetic monopole is still missing in action. Supersymmetry has not checked in yet. String theory remains… well, a theory. My own sentimental favorite was the preon or rishon model, a fantasy of recursive descent: If the proton is made of quarks, the quarks must be made of something still smaller. If only it were so.

In that giddy era, Guth’s inflation conjecture was far from the wildest or weirdest idea to come along, but it did seem to be among the proposals that would be most difficult to test. Yet here we are with solid observational evidence.


Guth’s talk on Tuesday was the first half of a double feature organized by the Harvard physics department. The second speaker was John Kovac, leader of the BICEP2 collaboration—the group that brought back the eggshells from the South Pole. Written on those eggshells was a distinctive pattern of polarization in the cosmic microwave background (CMB) radiation.

John Kovac lecturing at Harvard 2014-03-25The peculiar distribution of the CMB was one of the puzzles that led to the inflationary hypothesis in the first place. The CMB is al­most perfectly uniform across the entire sky, suggesting that whatever emitted the radiation must have been at a uniform temperature every­where. In a non-inflationary cosmology, that widespread thermal equilibrium seems mysterious. Distant regions of a very young universe are causally dis­connected: They have not yet had time to communicate with one another, even by signals moving at the speed of light. Some sort of miracle is needed to make all of these isolated regions so nearly uniform in temperature. Inflation dispenses with the miracle by squeezing the early universe into a much smaller volume, where all the parts can become throughly mixed.

Inflation predicts a CMB that is nearly uniform, but not quite. Quantum fluctuations introduce variations in density and temperature, which become slightly warmer and cooler spots in the CMB radiation field. (Later in the history of the universe, the denser spots evolve into clusters of galaxies.) A series of satellite observatories has mapped the CMB with high precision, finding splotchy patterns with an angular size scale of about 1 degree, and temperature differences of 1 part in 100,000. These results are in good agreement with the predictions of inflation. This is encouraging news for the theory; on the other hand, inflation was invented to explain these very characteristics of the CMB, so there’s some circularity in the argument.

Inflation makes another prediction. Quantum fluctuations in the high-density plasma of the inflating universe should give rise to powerful gravitational waves. These are propagating disturbances in the gravitational field, just as light waves are disturbances in the electromagnetic field. We are too late to detect the gravitational waves directly, but they should have left a distinctive signature in the microwave background—a swirly pattern of polarization, or wave alignment.

If the inflationary scenario is correct, two modes of polarization should co-exist in the CMB; they are called the E-mode and the B-mode, after the E and B vector fields of electromagnetism. In the argot of vector calculus, the E field has divergence but no curl: Think of electric lines of force streaming radially away from a charged particle. The B field has curl but zero divergence, like the magnetic lines of force that wrap around a current-carrying conductor. E-mode polarization comes from a conventional process of photon-electron scattering. B-mode polarization is the mark of gravitational waves.

The search for CMB polarization takes us to a rough neighborhood of the electromagnetic spectrum: the millimeter band, in the no man’s land between radio and optical wavelengths. Nearly anything can emit and absorb at these wavelengths, so it’s a noisy place. But astronomical signals are very faint because moisture in the atmosphere soaks up millimeter waves and not much gets through. I gather that the best place on earth to do millimeter-wave astronomy is the South Pole, which has exceptionally dry air and dark skies. In the past decade half a dozen millimeter-wave telescopes have been set up at the Amundsen-Scott South Pole Station. E-mode polarization was first detected in 2002 by one of these instruments, called DASI. BICEP2 was installed in 2009 to look for B-mode signals.

BICEP2 and South Pole Telescopes

The BICEP2 telescope was operated at the Amundsen-Scott South Pole Station, on a platform located some 800 meters from the geographic pole. The telescope is inside the conical shield on the roof of the building; the shield blocks stray radiation from nearby sources. The instrument at right is the South Pole Telescope, which also operates at millimeter wavelengths. Just visible in the background at left is the IceCube neutrino telescope. Photograph by Steffen Richter, who spent three “nights” at the station (each night being six months long). Reproduced courtesy BICEP2 collaboration.

The BICEP2 instrument is a curious hybrid of optical and radio technologies. The telescope is a refractor, with objective and eyepiece lenses—a design known to Galileo. But the lenses are not ground from optical glass; they are molded of high-density polyethylene (the material used to make plastic milk jugs). And the detector in the focal plane of the instrument is not the kind of imaging device seen in optical instruments; it is an array of tiny crossed antennas, etched into silicon wafers. The whole instrument is evacuated and cooled to liquid helium temperature. The detector is colder still: about 0.25 Kelvin.

The telescope was installed at the South Pole in December of 2009 and recorded data almost continuously until December of 2012. (It was shut down briefly every three days to replenish the helium and brush off any snow, and there were slightly longer interruptions in summer.) Most of the observing time was spent repeatedly scanning the same patch of sky, an area known as the Southern Hole, which is unusually free of “nearby” dust—that is, schmutz from our own galaxy. In a sense, the entire three-year observing campaign was an effort to take a single picture of that patch of sky. Here it is:

BICEP2 polarization map for E mode and B mode

Polarization maps of the target area for BICEP2 reveal both E-mode and B-mode signals. E-mode polarization had been seen in earlier observations, but this is the first evidence of B-mode fluctuations, thought to result from gravitational waves in the inflationary era. Note that the intensity scale at right is six times larger for the E-mode map. Graphic reproduced courtesy BICEP2 collaboration.

The important features of these vector maps are the pinwheels in the B signal panel. They are interpreted as the telltale remnant of gravitational waves.

The BICEP2 group calculates that the B-mode signal is statistically significant at the 5σ level, which pretty much rules out the possibility of being fooled by mere random noise. More worrisome is the risk of some systematic error, such as a bias in the instrument, or polarization caused by non-cosmological sources, such as galactic dust. Kovac devoted much of his Harvard talk to these issues, and there is even more detailed discussion in two papers released on the BICEP web site. As far as I can tell, the checks for errors were very thorough and the evidence is very solid. In any case, we won’t have to wait another 30 years to see the result confirmed (or not). Another telescope, the Keck Array, is taking data now at the South Pole, and BICEP3 is being readied for installation.


In case it’s not obvious already, let me say that I’m really wowed by this story. I think it’s a big deal. The discovery of the CMB in 1964 transformed the big bang model from a plausible possibility to a prevailing theory, and the discovery of B-mode polarization will surely do the same for inflation.

To me it’s amazing that we can have such detailed knowledge of our most remote origins. A hundred years ago we knew nothing about the early universe. We didn’t even know that it had a beginning. We didn’t know how big it was or what it was made of. We had none of the conceptual tools that would prove essential to making sense of cosmic evolution: quantum field theory, general relativity, the nuclear chemistry that makes the stars shine. In one quick century it has all come together.

Or not quite all. We still don’t know what the universe is made of. Most of it is apparently “dark matter” and “dark energy” that we have yet to identify; all the stuff we can see is a minor constituent. And we still don’t have a satisfactory quantum field theory for gravity. Yet, somehow, we can reconstruct the dynamics of the exotic, singular event that created this place, and supply enough quantitative detail to detect the evidence 13.8 billion years later. It’s magnificent. It’s ridiculous.

Posted in physics, science | 3 Comments

Net pests (not)

Last night I posted this note:

Since Sunday afternoon bit-player.org has been under some sort of mysterious DDoS attack, with a rotating suite of IP numbers repeatedly downloading the same PDF files several times a second. For the time being, I’ve taken most PDFs offline. If you urgently need something and you get a permission-denied error, please send me an email. Sorry for the inconvenience. Back soon, I hope.

Turns out it was not net pests. It was my 15 minutes of fame. A link to an old story of mine found its way to the front page of Hacker News, and I misinterpreted the resulting net traffic jam. A couple of hours later (and after a couple of messages from helpful HN readers), I realized there was nothing malicious going on, and I put the files back on line.

I’m an amateur in all things, but I’m especially inept as a sysadmin. Looking at the server logs this morning, I’m still unsure of exactly what I’m seeing, but I understand a little more than I did 12 hours ago.

What set me off in the first place was seeing long lists of requests like these, all from the same IP number:

16/Mar/2014:14:17:25 "GET /AmSci-2005-11-Hayes-NewOrleans.pdf HTTP/1.1" 200 79640
16/Mar/2014:14:17:26 "GET /AmSci-2005-11-Hayes-NewOrleans.pdf HTTP/1.1" 206 65886
16/Mar/2014:14:17:26 "GET /AmSci-2005-11-Hayes-NewOrleans.pdf HTTP/1.1" 206 8781
16/Mar/2014:14:17:27 "GET /AmSci-2005-11-Hayes-NewOrleans.pdf HTTP/1.1" 206 65891
16/Mar/2014:14:17:27 "GET /AmSci-2005-11-Hayes-NewOrleans.pdf HTTP/1.1" 206 65892
16/Mar/2014:14:17:27 "GET /AmSci-2005-11-Hayes-NewOrleans.pdf HTTP/1.1" 206 65893

Good grief, I thought: Somebody is downloading the same PDF file six times within three seconds. I failed to notice (or appreciate the significance of) the response codes near the end of each line. The “200” code on the first line is the normal HTTP “OK” signal, but the “206” on the next five lines signifies “partial content.” What’s going on here—if I now understand correctly—is not one person downloading the same file six times; it’s one person downloading a file in six pieces. (The size of the file in question is 270,570 bytes. The byte counts at the ends of the six lines above add up to 272,343. I can’t account for the discrepancy; I’m still an amateur.)

The file mentioned in the six requests above is not the one linked to by Hacker News. That’s another reason for my confusion: I was seeing wholesale downloading of hundreds of different files. Apparently, when some people find an item that interests them on a web site, they wget -r the whole site. And I guess I understand why: If you don’t grab it immediately, the skittish site owner is likely to panic and take it offline. (But wouldn’t it be polite to throttle the request rate?)

The story that started all this fuss is a bit of whimsy I wrote almost 30 years ago for Computer Language, a magazine long defunct. In the past 18 hours the PDF has been successfully downloaded almost 12,000 times, which may be greater than the circulation of Computer Language.

Linode traffic 2014 03 17

Posted in meta | 4 Comments

Notes from the JMM

The annual Joint Mathematics Meetings are one of those vast smörgåsbord meals where you can’t possibly eat everything. I managed to sample 1 percent of the menu this year. By my count, the four days of meetings included 2,743 scheduled events (mostly talks, but also panel discussions, cocktail parties, poetry readings, film showings, and other diversions); I attended 28 of them. There are \[\binom{2743}{28} \approx 5 \times 10^{66}\] ways to choose such a sampling, so it’s unlikely any of the other 6,424 participants experienced exactly the same meeting I did. Technicalities: I came up with the number 2,743 by scraping the online program and counting all <li> tags that do not have a <ul> as one of their children. Counting links to abstracts gives an estimate for the number of talks alone: 2,569. Also, I should note that the binomial calculation exag­gerates a little: Some of those 1066 sets of 28 events are forbidden because of conflicts between simul­taneous sessions.

As a journalist and a generalist, I try to maintain a balanced mathematical diet, but this year I pigged out on number theory. A third of the talks I heard were in that area. There’s been a a lot of buzz lately about progress toward a proof of the twin-primes conjecture, starting with a surprise announcement last spring by Yitang Zhang of the University of New Hampshire. Twin primes, such as \(5\) and \(7\) or \(101111\) and \(101113\), are primes as closely spaced as two odd primes can be. The twin-primes conjecture says there are infinitely many of these pairs. Zhang didn’t prove there are infinitely many primes \(p\) and \(q\) with \(q-p=2\), but he came really close. He proved there are infinitely many primes \(p\) and \(q\) with \(q-p \le 70\,000\,000\). A few months later James Maynard, a postdoc at the Université de Montreal, brought the constant down from \(70\,000\,000\) to \(600\). (Maynard was inspired by Zhang’s work but used different methods. Terry Tao got the same result at the same time.) Then a Polymath collaboration organized by Tao made further progress. At last report, the minimum gap for which we can say with certainty there are infinitely many primes is 270.

For more about the mathematics behind this work, I recommend two excellent articles by Erica Klarreich, published in the Simons Foundation’s Quanta magazine:

I’d like to add a few thoughts about the human side of Zhang’s story, as best I can piece that story together from published sources.

According to a profile by Virginia Stuart in UNH Magazine, Zhang was born in China in 1955 and was a boy when the Great Proletarian Cultural Revolution and the Red Guards movement convulsed the nation. With the rest of his family he was sent to the countryside for “re-education” through labor. The result of re-education was not much education, but he made up for lost time and in 1978 (at age 23) entered Peking University, earning bachelor’s and master’s degrees in mathematics.

In 1984 S. S. Chern took several American mathematicians to Beijing, where they spent a summer working with Chinese graduate students. Zhang was one of the students, and the following year he came to the U.S. to study at Purdue with T. T. Moh, who had been a member of the Chern delegation. Moh has recently written a short memoir on Zhang’s period at Purdue, from which I quote at length:

When he arrived, we had a cordial talk. Yitang expressed his desire to work in the field of Algebraic Geometry…. Yitang also mentioned that he wanted to study under my guidance…. I was surprised by Yitang’s next request of working on the Jacobian conjecture as his thesis topic. I felt it was odd to select such a difficult task.

The Jacobian conjecture is one of Steve Smale’s “problems for the 21st century,” which Smale describes as follows:

Suppose \(f : \mathbb{C}^n \to \mathbb{C}^n\) is a polynomial map with the property that the derivative at each point is non-singular. Then must \(f\) be one-to-one?

I can say nothing more about this question except that it remains open, and that Moh is the author of a long 1983 paper with computational results for all two-variable polynomials of degree less than 100. Moh continues:

Yitang spent all of his free time thinking of mathematics. After years, Yitang started to believe that he might have gotten a solution, one independent of my paper, to the Jacobian conjecture. As a gatekeeper of the palace of the Jacobian conjecture, I did my duty of examining every claim presented to me and denied the entrance of anybody (even if the claim has nothing to do with my work) if the proof was invalid. “Maybe the Jacobian conjecture is a problem for the future”, I thought.

By 1991 Zhang was approaching the seven-year limit for Ph.D. candidates at Purdue. He wrote a brief thesis on a specialization of the Jacobian conjecture, which earned him his degree. Then he took off. Moh writes:

Sometimes I regreted not fixing him a job. But really, who could tell whether it was a good decision or not? Maybe it was his destiny to endure and turn out to be great…. When I looked into his eyes, I found a disturbing soul, a burning bush, an explorer who wanted to reach the north pole, a mountaineer who determined to scale Mt. Everest, and a traveler who would brave thunders and lightnings to reach his destination. Yitang never came back to me requesting recommendation letters. Apparently, he did not seek a job.

An interview with Zhang, conducted by Michael Segal and published in Nautilus, puts a somewhat different spin on the matter of recommendations. In the interview Zhang remarks: “During that period it was difficult to find a job in academics. That was a job market problem. Also, my advisor did not write me letters of recommendation.”

Zhang spent most of the next decade without an academic position. Here’s the Wikipedia account of that period: “Prior to getting back to academia, he worked for several years as an accountant and a delivery worker for a New York City restaurant. He also worked in a motel in Kentucky and in a Subway sandwich shop.” The article by Virginia Stuart makes clear that the job as an accountant and the job at Subway were the same job. “In a pinch, he would help out behind the counter, a fact that has been exaggerated in the press and has inspired online banter about a mathematical genius making sandwiches for a living.”

In 1999 Zhang was hired as a lecturer in the department of mathematics and statistics at UNH. I haven’t been able to learn exactly how this came about. The chairman of the department of the time was Kenneth Appel, who had his own history of breakthrough proofs—in his case, the four-color theorem. Appel died just a few days after learning of Zhang’s triumph.

UNH saved Zhang from a life of making footlong sandwiches, but for 15 years his job was a nontenured lectureship, on a year-to-year contract. He apparently taught a lot of first-year calculus—and got rave reviews on ratemyprofessors.com. One review says, in its entirety: “HOT!!!” The UNH web site still lists him as a lecturer, but news reports this month say he has been promoted to full professor.


At the JMM, Andrew Granville of Montreal gave a big tutorial talk summing up the recent and ongoing work on bounded gaps between primes. But the main event Yiting Zhang speaking at the Joint Mathematics Meetings, 2014-01-16happened a day earlier, when both Maynard and Zhang spoke in the same session. The room was jammed. Maynard went first, and gave a lucid and precise account of his own work. When he finished, Zhang stepped forward with a sheaf of transparencies, which he’d been scribbling on with a Sharpie pen during some of the earlier talks in the session. There was a brief delay while the video projector was set aside and the overhead projector was moved into place and switched on. Still more people flowed into the room from the doors at the rear. (I had arrived early to get a seat up front.) The hubbub continued until Granville rose and bellowed out “Quiet!” Then Zhang turned to face the audience and announced: “I know you all expect me to talk about twin primes, as the abstract says, but I’ve decided to talk about something else.” I was mildly shocked. I don’t think I’ve ever seen anyone pull such a switcheroo at a major conference.

Zhang's transparencyThe JMM still provides an overhead projector in every lecture room. Among the talks I attended this year, Zhang was the only speaker who made use of this quaint device.

“Something else” turned out to be the Goldbach conjecture—the assertion that every even integer greater than 2 can be expressed as the sum of two prime numbers. Zhang wasn’t claiming to have proved the Goldbach conjecture; he wasn’t even laying out a clear pathway to a proof; but he did vaguely hint that he has some idea of how to proceed. We’ll see, I guess. For the moment he has a healthy balance of credibility.

An hour after he gave his talk, Zhang was awarded a share of the Frank Nelson Cole Prize in Number Theory.


Nonmathmatical geographic/autobiographic/photographic addendum. The JMM was held in Baltimore this year, a city I lived in from age 0 to age 4 and again from age 20 through 23. On this trip I spent my first few hours in town strolling around taking pictures, and belatedly discovered that my camera was set to one of the cute “illustration” modes that modern digital technology has to offer. I was shocked at this switcheroo, too, but after a few days the cartoonish pictures grew on me.

The Morris A Mechanic Theater, in ruins

The image above shows the Morris A. Mechanic Theater, which was the centerpiece of an urban redevelopment project in the early 1970s, when I was a young man about town. The theater was supposed to bring Broadway tryouts and other forms of stagecraft back to the heart of the city. It closed in 2004, and now even the posters promising another phase of redevelopment are looking tattered.

As for the Hotel Junker/Envy Hotel, below, I didn’t stay there, but it was just around the corner. Echt Baltimore.

Hotel Junker Envy Hotel

Posted in mathematics | 4 Comments

The rhodometer

Years ago I lived in a house with a rhododendron outside the kitchen window, which served as our family thermometer on winter mornings. If the leaves were tightly rolled, you bundled up in gloves and scarf and ski cap; if not, just a jacket would do.

nearly flat rhododendron leaves in warm weather, and tightly rolled ones in the cold

Rhododendron leaves with the temperature at 35º Fahrenheit (left) and 17º Fahrenheit (right).

Where I live now, we have a rhododendron by the front porch, and I’ve been keeping an eye on it for the past two winters. The qualitative relation between temperature and leaf-rolling is just as I remembered it, but I’ve been trying to add some quantitative precision. Can a rhododendron bush serve as a reliable temperature-measuring instru­ment? To find out, I bought a set of calipers, and I’ve been making daily measurements whenever the early-morning temperature is near or below freezing.

rhododendron leaf diameter as a function of temperature

Here’s what the data look like (right). Each dot gives the diameter of a leaf, measured at its widest point, as a function of ambient temper­ature. The four colors distinguish measurements of different individual leaves. Last year I followed a single leaf (green dots); this winter I’ve been tracking three leaves I marked by wrapping twist-ties around the stems (tan, blue, magenta). If you’d like a look at the raw data, grab the csv file.

The overall shape of the relation is clear from the graph: Measured diameter increases monotonically with tempera­ture, as expected. But there’s a fair amount of noise, as well as a sharp kink in the curve. If I had continued measurements to higher temperatures, we would surely see a second bend in the trend line. The full curve must be S-shaped, because the diameter of a rolled leaf has a finite maximum and minimum. No matter how high the temperature, the diameter can’t be greater than the width of the flattened leaf, and no matter how low the temperature, the diameter is bounded above zero.

Given the sigmoid shape, the right model for this data set is presumably a logistic curve, but I have done something simpler. I’ve partitioned the data into cold and warm subsets fitting two linear functions to the colder and warmer parts of the leaf-diameter dataand fitted two piece­wise linear functions. In the figure at left the data points are recolored to indi­cate which points go with which line. The crossover between the two functions is at 24.8 degrees, which max­imizes the total R2 goodness-of-fit for the two lines.

The kink in the curve is quite sharp. It remains visible after a log transformation, or even log log. Per­haps there’s a real, physical discontinuity there. After all, it’s not unreasonable to suppose that the freezing point of rhododendron sap might be somewhere around 25°F—although I don’t actually know that a phase transition has anything to do with the behavior of the leaves.

As a practical matter, the nonlinearity is an inconvenience if we want to use the leaves as a thermometer. It’s probably best to regard the rhododendron as a simple binary indicator: Either it’s cold out, or it’s not.

My hiking friend Dinosaur has told me about a wilderness school for teenagers with a rule that the kids have to stay in camp whenever the temperature is below 16°F. When this rule was announced, the camp leader complained: “We’re in the wilderness; we don’t have a thermometer.” The reply was: “Just use the rhododendrons.” That’s a wonderful suggestion. However, even with careful measurements of the leaves, it might be hard to gauge the temperature accurately in the range between 10° and 20°. And I suspect that a hiking group without a thermometer also lacks vernier calipers.


In the course of my morning measuring-the-rhodo rituals, I became curious about the mechanism of this temperature-sensitive leaf rolling. An analogy that immediately comes to mind is the bimetallic strip, often used as a temperature sensor in thermostats and oven thermometers. The idea is to bond layers of two metals (typically brass and steel) that have different coefficients of thermal expansion. With heating or cooling, one side expands or contracts more than the other, and so the strip curls up.

detail of a dissected leaf showing how layers overlapMaybe the rhododendron leaf works this way too, with materials on the upper and lower surfaces that differ in thermal coefficient. But there are problems with this hypothesis. In the first place, a bimetallic strip curls end to end, not side to side. In other words, the axis of curva­ture is parallel to the shorter dimension; indeed, the device is fab­ricated as an elongated strip to avoid compound curvature, which could lead to buckling or other complex dynamics. If the same mechanism were at work in the rhododendron leaf, it would curl up from tip to stem, but in fact the curling axis is parallel to the long dimension of the leaf. It rolls up like the wrapper of a cigar. Another objection to the bimetal analogy is that really hot weather ought to cause curling in the opposite direction, with the underside of the leaf outermost. I’ve never seen that happen. Still another issue: The amount of curvature implies a huge difference in coefficient of expansion. In the photo above, a dissected leaf in near-zero-degree weather is rolled up tightly enough that opposite edges have passed each other once and are almost meeting for the second time.

The magnitude of the curling effect suggests there may be something more than just thermal expansion and contraction going on. Likewise the kink in the temperature-diameter curve hints at some discontinuous process. And the axis of curvature has me wondering about the possibility of a structural asymmetry—something that inhibits lengthwise curling and favors crosswise.

30x micrographs of the upper and lower surfaces of a rhododendron leaf

In the search for asymmetry, I pulled out the microscope. The images above show the upper (left) and lower (right) surfaces of a leaf at 30 times magnification. I see no network of crosswise contractile fibers. The only distinctively asymmetric structure is the leaf’s stout central vein—a continuation of the stem—but maybe that’s all we need. The central vein might be strong enough to inhibit lengthwise curling in the initial stages of the process, at temperatures near freezing, when the curling forces and the resulting curvature are slight. Later, the crosswise curling creates a tube, which offers mechanical resistance to lengthwise bending as the curling continues. But all this is just speculation.

A little casual Googling turns up lots of stuff on the folklore of rhododendrons as weather indicators, but the only scholarly discussion I’ve stumbled on is a 1990 article by Erik Tallak Nilsen of Virginia Polytechnic in Blacksburg (where lots of rhodos grow). Nilsen doesn’t have much to say about how the leaves curl; he’s interested in why. The widely prevalent theory—often stated as established fact—argues that leaf-rolling helps prevent water loss by covering the pores (called stomata) on the underside of the leaf. Nilsen doesn’t think much of this idea. He points out that the stomata close in cold weather, sealing off the moist internal milieu. And he backs up this assertion with his own experimental results.

Among several alternative explanations, Nilsen favors the following idea: Rhododendrons live in the understory of hardwood forests, where they are shaded in summer but exposed to full sunlight after the trees drop their leaves in winter. The photosynthetic membranes of shade plants are susceptible to damage from ultraviolet light, especially in freezing weather. Rolling the leaves reduces the exposed area.

Posted in modern life, science | 2 Comments

My First 1000000 Years

Limites
Hay una línea de Verlaine que no volveré a recordar,
Hay una calle próxima que está vedada a mis pasos,
Hay un espejo que me ha visto por última vez,
Hay una puerta que he cerrado hasta el fin del mundo.
Entre los libros de mi biblioteca (estoy viéndolos)
Hay alguno que ya nunca abriré.
Este verano cumpliré cincuenta años:
La muerte me desgasta, incesante.
                      —Jorge Luis Borges, 1923 

Passing half a century is a landmark for those who count on 10 fingers, but as a bit-player I celebrate powers of 2. Today I turn 26, which is a mega-milestone: 10000002 years. It’s a special occasion in several ways. Need I point out that it’s the last power of 2 I’ll ever see? Also, other than 0 and 1, it’s the only sixth power I’ll visit in my lifespan, and hence it is the only age that’s both a square and a cube.

As a young man—well shy of 50—I admired that poem by Borges, but as I grow older I find its mood of lugubrious foreboding less attractive. Yes, it’s true: There is a line of Borges I’ll never read again. But if my time is so limited, I shouldn’t be spending too much of it stuck in a tight loop, rereading the same lamentations over and over. Let the last time pass; I’ll move on to something new—something I haven’t yet done for the first time.

Life is all too full of never-to-be-repeated moments, right from the outset. Even the silly numerology of birthdays makes this apparent. Long before I reached my last power of 2, I celebrated my last Bell birthday (52), my last Catalan birthday (42), my last perfect birthday (28), my last factorial birthday (24), my last pair of birthdays satisfying \(x^m – y^n = 1\) (\(2^3 = 8, 3^2 = 9\)). Never again will my age be an even prime number, a multiplicative identity, an additive identity. But I can live with that!

And I still have a few treats to look forward to. In a couple of years I come to my next triangular birthday—and that one may not be my last. Then of course there’s \(3^4 = 9^2\). At the end of my eighties there’s a Fibonacci waiting for me, if I can make it. And, nearer at hand, I have marked my calendar for May 22, 2018, when I will celebrate my 25,000th spin around the earth’s axis. Maybe I should try to spend the day in orbit.

Posted in mathematics, modern life | 7 Comments

Boldface names

names in stone: Newton, Tycho Brahe, Galileo, Kepler, Euler, D'Alembert, LaGrange, LaPlace, Herschel, Adams, Hill, Poincare

Strolling across Killian Court, the only sizeable patch of public grass on the MIT campus, I glanced up at the entabulature pictured above. Chiseled into the marble façade are the names of eleven celestial mechanics: astronomers and mathematicians who figured out how the solar system works. I could attach identities to nine of those names—or thirteen of them if you count all the astronomical Herschels (William, Caroline, John, Alexander). Two names stumped me. Do you know who Adams was? How about Hill?

In this age of Google and Wikipedia, it doesn’t take long to fill in the blanks. But tracking down Hill did require two searches because my first source—a 1922 article in The Tech, the MIT student newspaper—omitted that name. Maybe the Tech author was stumped, too.

Apart from Newton, who stands alone, the names are in birth order (assuming that Herschel is either Caroline or John). It’s not really surprising that the two names I didn’t know are among the later ones. When the buildings around Killian Court were erected in 1916, Galileo and Kepler had been heroes of science for more than 300 years; they had already passed the test of time. But Adams, Hill, and Poincaré were figures of the very recent past. Carving their names in stone amounted to making a bet on the judgment of posterity.

In the case of Henri Poincaré, the bet paid off. His reputation is soaring even higher now than it was at his death in 1912. (There’s a hefty recent biography by Jeremy Gray that I mean to read one of these days.)

Adams is a name that I suppose I should have recognized, because I’ve read his story several times, going back to childhood. But the bell didn’t ring. John Couch Adams (1819–1892) was one of two astronomers who predicted the existence of a planet beyond Uranus; the other was Urbain Le Verrier, who published first and whose work led directly to the discovery of Neptune. In choosing Adams and excluding Le Verrier, MIT was taking sides in a priority dispute. Some recent historical scholarship suggests they took the wrong side.

As for George William Hill (1838–1914), the MacTutor biography suggests he was an interesting character, and I am glad to have finally made his acquaintaince. Apparently he spent his entire career—a lifetime, really—wrestling in solitude with the three-body problem. He didn’t win the match.

I don’t know enough about the history of celestial mechanics to say with any authority that Adams and Hill don’t merit a place on this list. On the other hand, I can state with firm conviction that there’s a scandalous omission from the MIT pantheon. We have Euler (or EVLER, in monumental capitals), but where oh where is Gauss? By modern measures he ranks among the very greatest mathematicians, and he first achieved fame for work in celestial mechanics—predicting the position of the planetoid Ceres. Yet he is not listed here, nor does his name appear on any of the nine other honor-roll tablets surrounding Killian Court.

(Another conspicuous absence in the photo above has an easy explanation. Along with Tycho, Galileo and Kepler, surely one should mention Copernicus, no? In fact he has a place in the court of honor. Like Newton, Copernicus is on the A-list. He gets sole possession of an architrave, with his name writ in double-size letters.)


The MIT archives have published a few of the letters from MIT faculty suggesting names for these tablets. I detect no evidence that any of the nominators were bearing in mind that their selections would still be on exhibit a century hence. No one suggested leaving a few blank spaces where future generations might express their own preferences. I guess that would go against the whole spirit of stone inscription, where the idea is to make a commitment, stake out a position, proclaim eternal truths. You just have to be bold—and foolish.

If the same exercise were carried out today, the roster of names would be different. For one thing, we’d want to make room for people who were still living—or not yet born—in 1916: Einstein, Hubble, Turing, Gödel, Bohr, Dirac, Crick…. And it’s not just the names that would change; the categories have shifted, too. Celestial mechanics does not figure quite so prominently in modern thought as it once did, and the close ties between mathematics and astronomy have frayed somewhat. New fields are being cultivated, such as computation and information theory, as well as nuclear and quantum physics.

My question is: Would the faculty today acknowledge that the measure of greatness is certain to change again over the next 100 years, in ways we can’t predict? Maybe they would be shrewd enough to choose some medium more fluid than stone—perhaps a plasma display with a Twitter feed showing how various scientists are trending, moment by moment.

In the meantime, nature has overgrown and obscured some of those early 20th century judgments. Trees planted 75 years ago around the perimeter of Killian Court are now taller than the buildings, with the result that some of the panels can be seen only through the bare boughs of winter. In the photo below another list of names is glimpsed through an opening in the leafy canopy. The thread binding together the members of this group seems to be mechanical engineering and invention. Could you pass a quiz on those dozen names?

MIT panel listing inventors and mechanical engineers: Archimedes, Gutenberg, Watt, Arkwright, Whitney, Perkins, Fulton, Fairbairn, Froude, Otto, De Laval, Wright

Posted in mathematics, modern life, science | 1 Comment

Try a little tendrilness

I was in Florida for the New Horizons in Science briefings of the Council for the Advancement of Science Writing. On my way home I took a detour to hike a few miles through the Green Swamp, west of Orlando. The place lived up to its name: There’s a lot of chlorophyll out there. There are plants, and plants that live on the plants, and plants that live on the plants that live on the plants…. I had the feeling that I had better keep moving or the vines and epiphytes might grow over me, too.

Beards of Spanish moss in a live oak near the Mabel end of the Van Fleet Trail, Florida.

Among the epiphytes, the one everybody knows is Spanish moss (Tillandsia usneoides). In the photo above the long gray beards of Spanish moss droop from the limbs of a roadside live oak tree. Spanish moss is not really moss. (It’s not Spanish, either.) It’s an angiosperm—a flowering plant—in the bromeliad family. As parasites go, it seems to be reasonably well-behaved, draping itself from the major limbs of a tree but seldom blocking the foliage. The parasites have no roots and take nothing from the host but mechanical support.

detail of an aerial tuber

The air potato, left and above, is a relative of the yam that propagates by means of “aerial tubers.”

Air potato vine climbing on another vine

Vines are another matter. For these guys it’s all about upward mobility, and they’ll step on anybody to get up into the sunshine zone. The bright green leaves in the foreground of the image above belong to a vine called the air potato (Dioscorea bulbifera). Strictly speaking it’s a yam rather than a potato, but it does produce above-ground spuds (see the detail above at right). A student I met at the CASW conference in Gainesville told me there’s a spring-loaded mechanism that launches the potatoes into the treetops. I’ve not found any confirmation of that story in published sources; perhaps the student was having some fun with a city boy from up north. D. bulbifera is native to Asia and Africa and is not well liked in Florida; the student also told me about air potato roundups in Gainesville, and that part of the story is definitely true.

The inch-thick, cable-like vine in the photo above is not the stem of the air potato plant. Rather, it is an older, woody vine—probably poison ivy—that was once attached to the tree trunk at right. (The scar is still visible on the bark.) The air potato vine is slender, green and herbaceous. It twines counterclockwise around the larger vine, thereby parasitizing another parasite.

Passion vine tendrils

The plant in the photo above is a passionflower vine, Passiflora suberosa. I first noticed this species infiltrating a hedge on the University of Florida campus in Gainesville; in the Green Swamp I found it everywhere I looked. Like the air potato, the passionflower vine makes a living by climbing toward the light on the backs of other plants, but it goes about it in a different way. Instead of spiralling around a stem, it puts out tendrils that lash themselves to whatever supporting structure they happen to touch. The photo above shows several new tendrils at the growing tip of a vine. Note that a few of them have gotten tangled up among themselves; I’ll have more to say about this phenomenon.

In the photo below, a P. suberosa vine has successfully seized upon a twig of a nearby shrub. The tip of the tendril has tied itself into a knot resembling a clove hitch, and the rest of the tendril has become kinky, reducing its length and drawing the vine closer to the host plant. Other nearby tendrils may also latch onto the twig, strengthening the attachment.

Passiflora tendril lashed to twig


Before we continue with this subtropical expedition, please allow me a digression. Last year Edward O. Wilson published a grumpy essay arguing that mathematics is of little use to many scientists, especially in Wilson’s own field of biology. Members of the mathematical community took umbrage, of course. (The Notices of the American Mathematical Society reprinted Wilson’s essay with a rebuttal by Edward Frenkel.) As a nonmathematician and nonbiologist I don’t have standing in this contest, but I have an opinion all the same. It strikes me that learning mathematics not only supplies tools for getting answers but also cultivates habits of mind that can lead to interesting questions. Take this equation:

\[z = \frac{a + b}{a - b}\]

If you are the least bit math-minded, you are already asking yourself, “What about the case where \(a = b\)?” Similar issues come up in computer programming. When you write a procedure to select two items at random from a sequence, you need to think about the possibility that the two items are in fact the same item.

What does all this have to do with Florida vines and tendrils? Let’s plunge deeper into the jungle.


I was walking a section of the General James A. Van Fleet State Trail, a former rail line that cuts through the swamp straight as a laser beam. (It was once part of the Seaboard Air Line Railroad—“Air Line” meaning “as the crow flies.”) Along the way, I stopped to admire the occasional butterfly. I saw birds, too, but I didn’t get the camera out in time. I spoke with other walkers, and I dodged bicycles. I wandered off the trail, got my feet wet, picked up a cargo of burs in my pant cuffs.

Gulf fritillary and zebra heliconia butterflies

Left: Gulf fritillary (Agraulis vanillae). Right: zebra heliconian (Heliconius charitonius). As it happens, both of these butterflies spend their caterpillarhood munching on P. suberosa.

All the while I was puzzling over that vine whose modus vivendi is to send up a shoot, wave around in the breeze, and blindly grab hold of the first thing it touches. What bothered me about this strategy for cadging a free ride to the forest canopy was the hazard of grabbing hold of yourself—or of another vine of the same species, just as wimpy as you are. This is a case where if \(a\) leans on \(b\) for support, it’s important that \(a \ne b\). To put it another way, self-reliance is not a virtue when you’re unreliable.

Young P. suberosa vines tend to sprout in fairly dense clusters; I saw lots of them with stems a meter long and only a few centimeters apart at ground level. Very likely the stems are connected by roots or runners and are all parts of a single individual plant, or a clone. For many of the shoots, their closest neighbors were other P. suberosa vines. Under the circumstances, it appeared that the likeliest outcome is that all the vines would glom onto one another and collapse in a tangled heap.

I wondered if the plant might have some means of self-recognition that prevents or inhibits such an outcome. Animals put a lot of effort into identifying individuals, relatives, and conspecifics. We have both nervous systems and immune systems for the purpose. Why couldn’t plants do the same thing? All it would take is a substance in the cuticle of the stem that inhibits the grabbing action of the tendrils.

I can’t rule out the existence of such a mechanism. However, after some more poking around in the underbrush, I found evidence that the vines do indeed attack their own kind. In the photo below, two vines cross, and one of them has seized hold of a leaf stem on the other. Several dried and broken tendrils suggest this was not the first encounter between the two vines. I found other examples where three or more P. suberosa vines were all lashed together.

Self attack

Time for a new theory, eh? One possibility is that I’m just plain wrong about the futility of vines clinging to one another. Each vine has many tendrils, and each tendril has its own chance of finding a ticket to the sky. Maybe all the vines in a connected cluster can benefit if any one of them hits the jackpot. That is, if one member finds a path upward, all the other members can follow along as well. That would totally alter the calculus of fitness. In the simplest model we might assume that each vine makes \(n\) contacts with other plants, and each contact has probability \(\alpha\) of connecting to an upward path. Further assume that all of the probabilities are independent. Then the probability that everyone gets to heaven is \(1 – (1 – \alpha)^n\), which converges to 1 as \(n\) increases, no matter how small \(\alpha\) might be.

This scheme may well explain what’s going on with P. suberosa, but I’m not entirely satisfied with it. In the first place, the assumption that all events are independent is surely bogus. Plants are not like atoms in an ideal gas, where you might reasonably assume that any two atoms have the same probability of colliding. Plants have roots. They don’t roam at random.

Second, in the probability calculation above I have implicitly assumed that \(a \ne b\), and this assumption is unjustified. A single vine can vainly try to support itself by attaching itself to itself. I submit my final photograph, documenting an act of botanical suicide in which the tendrils of a P. suberosa plant strangle one of the plant’s own leaves:

Suicide vine

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