Yesterday’s mail brought the latest issue of Focus, the magazine of the Mathematical Association of America. On the cover is a photograph of a gold icosahedron offered for auction last year at Sotheby’s. One reason for the MAA’s interest in this artifact is that the association’s emblem is an icosahedron. But there’s also a mathematical puzzle connected with it. On each of the 20 triangular faces of the object is inscribed a number. The numbers are listed in the Sotheby catalogue (and in Focus) as follows: 11, 20, 21, 31, 41, 51, 61, 71, 81, 91, 101, 201, 301, 401, 501, 601, 701, 801, 901 and 202. Actually, there are two puzzles connected with this set of figures. The lesser puzzle is why the authors of the catalogue chose to list the numbers in this almost-but-not-quite ascending sequence. The greater puzzle is what the numbers mean in their original context.

What is that context? The Focus article, written by Harry Waldman, gives this account of the provenance of the gold polyhedron:

This 17th or 18th century piece of Islamic art had been part of the treasury of Tipu Sultan, who died at the hands of the British in India at the Battle of Seringapatam, in May 1799. The victors then claimed possession of this and other valuables of the vanquished army of Tipu.

Thus the object is presumed to be of Indian origin and a few centuries old. It is a hollow box, with a hinged lid comprising one pentagonal pyramid of five triangles. The numerals on the faces are written in a script that resembles the one Karl Menninger terms East Arabic, with a centered dot for zero, but there are some variations from the letterforms seen in modern Arabic scripts. Since I don’t read any of the languages that use such numerals, I found the inscriptions very difficult to interpret. (I’m particularly iffy about the 7s and the 4s.)

What could the numbers mean? I spent a goodly part of the night puzzling over this question, but I don’t have a clue. The numbers fall readily into two series of nine (11, 21, …, 91 and 101, 201, …, 901) plus two oddballs—or should we call them evenballs?—20 and 202. But that classification has no obvious significance. Ideally, one would like to identify a pattern related to the symmetries of the icosahedron—maybe 12 overlapping groups of 5 numbers each—but I can find nothing of the kind. I thought it might be helpful to know how the numbers are arranged on the icosahedron. Working from the two photographs in Focus and a couple more on the Net, I put together the map shown below. But I have to emphasize that this reconstruction is tentative; it could be full of errors; the two numbers in gray are totally conjectural; and the white question marks denote faces that I can’t see at all in any of the photos. Annoyingly, the hidden faces seem to be those with the evenballs 20 and 202. (The orientation of the map puts the five triangles of the arctic cap at the top; the blue line represents the hinge.) The arrangement of the numbers—if I have read them correctly—seems orderly enough, but also uninformative.

When I first looked at the list of numbers, I thought it might be more promising if only we could reverse the digits to get 11, 12, …, 19 and 101, 102, …, 109. And, I thought, with a flicker of hope: Since there’s an Arabic connection here, maybe that’s not so far-fetched. But Menninger’s book Number Words and Number Symbols puts the kibbosh on that idea. Even though Arabic script reads right to left, numbers embedded in it are written from left to right, and there’s evidence this has always been so. “Many a neophyte,” Menninger scornfully notes, “has tried to reverse the digits this way….”

Waldman in Focus is circumspect about the prospect of solving the mystery: “Research… has failed to offer clues as to the meaning of these inscriptions.” On the other hand, the National Gallery of Scotland, which has had the icosahedron on exhibit, apparently has access to deeper insights:

A considerable amount of research has been devoted to the interpretation of the 20 numbers on this box, which at first appear to be unrelated. However, they have all been shown to demonstrate accurate values of Pi and Pi squared, and Phi and Phi squared, as well as the roots of 2, 3, and 5. It is possible that other mathematical relationships are concealed within these numbers.

A final note: The MAA declined the offer to purchuse Tipu Sultan’s polyhedral trinket. The asking price was 200,000 pounds sterling. Presumably the item is still available, so if you have someone on your holiday gift list who might appreciate such a bauble, now’s the time to make a bid.

Update 2006-12-30: Thanks to information from “A Numbered Icosahedron from India: Mystery & Meaning,” by Paul Bien, I have filled in the remaining blanks in the folding net above.

Our story so far: Having stumbled upon the Jacobsthal numbers, 1, 3, 5, 11, 21, 43, 85, 171, 341,…, I idly asked, “Who was Jacobsthal?” Keith Matthews promptly responded with a wealth of biographical information, even arranging to have an obituary translated from the Norwegian. So I asked, “Where did Jacobsthal mention these numbers?” and Barry Cipra quickly supplied a reference: “Fibonaccische Polynome und Kreisteilungsgleichungen” in Sitzungsberichte der Berliner Mathematischen Gesellschaft 17 (1919-1920), 43-57.

I dare not ask another question, lest someone else go running off to do more of my library errands for me. Embarrassing.

Unfortunately, though, when I look into the Jacobsthal paper, further questions are inescapable. Nowhere in the article, as far as I can tell, does Jacobsthal list any of the numbers in the sequence that now bears his name. Admittedly, I don’t actually read German; I just make it up as I go along. But, ungebildet as I am, I can at least recognize numerals, and 1, 3, 5, 11, etc. are not to be found. The closest approach is in this passage:

Here we see the recurrence relation f(n+1) = f(n) + xf(n–1), which produces the Jacobsthal sequence in the case when x = 2. But nowhere does Jacobsthal mention the specific case of x = 2, or any other specific example for that matter, except for noting that x = 1 corresponds to the “so-called” Fibonacci numbers.

So once again I’m left wondering: Exactly how did Jacobsthal’s name get attached to the numbers 1, 3, 5, 11, 21, 43…?

Incidentally, Google Language Tools offers a wonderful translation of Jacobsthal’s title: “Fibonacci polynomials and circling hurrying equations.”

The new issue of American Scientist is on the Web and will soon be in the mail. My “Computing Science” column begins disarmingly enough, “I was a teenage angle trisector,” but shortly descends into the usual boring pedantry. On the other hand, I do answer the question that has been on every lip these past few weeks: Who was Pierre Laurent Wantzel?

The mathematics section of the arXiv archived 989 preprints in October. Why is that fact worth noting? Because arXiv papers are identified by numbers of the format YYMMNNN, with two digits for the year, two digits for the month, and a three-digit sequence number. Ten more papers and all the world’s mathematicians would have been put on involuntary furlough, forbidden to sum another series or solve another equation until month’s end. It would have been rather like the state of New Jersey shutting down when the legislature fails to pass a budget bill. (I realize there are people who would find neither event regrettable.)

The arXiv is now introducing a new numbering scheme, with room for 9,999 papers per month. “If current growth rates continue,” says the announcment, “we expect to change the sequence number to 5-digits NNNNN in 10 to 15 years.” The new format continues to allocate two digits to the year. The arXivists seem to have no worries about disambiguating centuries. They weren’t panicked by Y2K, and apparently they’re not afraid of 9108 either, when the arXiv will have its own centenary.

Trivia note: Curiosity moved me to look up the very first paper issue by the arXiv (although the site was not then arxiv.org but rather xxx.lanl.gov). Paper 9108001 is “Exact Black String Solutions in Three Dimensions,” by James H. Horne and Gary T. Horowitz; the abstract begins, “A family of exact conformal field theories is constructed which describe charged black strings in three dimensions.” I can all too easily imagine that paper 9108.00001 in the new series will be on the same topic.

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?

This just in from the BBC…. Maths boffin from Berkshire solves 1200-year-old problem of division by zero.

Dr James Anderson, from the University of Reading’s computer science department, says his new theorem solves an extremely important problem - the problem of nothing….

The theory of nullity is set to make all kinds of sums possible that, previously, scientists and computers couldn’t work around.

“We’ve just solved a problem that hasn’t been solved for twelve hundred years - and it’s that easy,” proclaims Dr Anderson having demonstrated his solution on a whiteboard at Highdown School, in Emmer Green….

Despite being a problem tackled by the famous mathematicians Newton and Pythagoras without success, it seems the Year 10 children at Highdown now know their nullity.

Anybody who can solve a problem that was tackled by Pythagoras 1200 years ago must be onto something.

(Thanks to reader Frank Tamborello for the timely tip.)

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.

The U.S. Supreme Court heard arguments the other day on programs meant to maintain racial diversity in the public schools of Seattle and Louisville. Listening to accounts of the debate put me in mind of Thomas C. Schelling’s elegant mathematical model of race relations. The model suggests that extreme segregation can arise spontaneously even when no one in the population desires that outcome. I wish I could submit a friend-of-the-court brief (although I’m not feeling very friendly toward this particular court).

The panel at left shows four stages in the evolution of a simple version of Schelling’s model. Initially, equal numbers of individuals from two groups—they’re shown here as blue and tan, but I’m going to refer to them as black and white—are dispersed at random. Neither race seeks to escape or exclude the other; the only racial preference is of the mildest sort—a desire not to be entirely cut off from people of one’s own group. But that desire is enough to cause the two populations to separate like oil and water.

The simulation takes place on a two-dimensional lattice with N sites, occupied by M agents, where M < N, so that at least a few sites remain vacant. At each time step each agent examines the eight sites surrounding its position, calculating the proportion of neighboring agents that match its own color. If the proportion of same-race neighbors exceeds some given threshold β, the agent is contented and stays put. On the other hand, if the fraction of like-colored neighbors falls short of the threshold, the agent moves, choosing its new home at random from among the N – M vacant sites; in the process of moving, of course, the agent creates a new vacancy in its former position. The process is then repeated, stopping only if and when there are no more discontented agents.

The model has two parameters: the threshold of contentment β and the filling factor (the fraction of occupied sites, M/N). As each of these parameters increases from 0 toward 1, agents have a harder time finding a satisfactory home. The interesting region in parameter space is where the threshold β lies between about 0.25 and 0.5, and the filling factor is close to 1, so that there are few vacancies on the lattice. The high filling factor excludes solutions in which people get along by living in isolation. As for the threshold, it’s easy to find a pattern acceptable to all if no one minds being in the minority, but the challenge naturally gets harder as the agents become more discriminatory. Still, it’s important to note that even at a threshold of 0.5 (i.e., each agent wants to have at least half of its neighbors match its own color, and no one is willing to be in a minority) there is an obvious solution that makes everyone happy. That solution is a checkerboard pattern, which happens to represent a maximum of racial integration. But the checkerboard is not the pattern that typically emerges when the model is run from a random initial state. Instead, within a few hundred time steps, the two populations segregate themselves, forming large, amoeboid areas of racial uniformity. The sobering lesson of this simple model is that it doesn’t take deep-seated and vitriolic racism to produce a stark pattern of segregation; it’s enough that each of us feels uncomfortable when outnumbered. And the sharp color lines develop even though those who move don’t take race into account when choosing a new location.

For anyone who might care to experiment with the Schelling model, there are Java applet versions here and here (and probably more). The model is also among the demo programs included with several versions of the StarLogo and NetLogo programming languages.

Schelling’s model deals with residential racial patterns, but it’s not hard to adapt the same basic ideas to the study of segregation and integration in school systems, which was the issue before the Supreme Court this week. The main difference is that in the school case we need to look at a few large subsets of the population (the schools) rather than many small neighborhoods. In the scholastic variant of the Schelling model, a student attending a school where the proportion of his or her own race falls below the threshold β would transfer to a randomly chosen vacant place elsewhere in the system. Although I have not yet tried to build and run a computational version of this model, I think it’s possible to see what kind of behavior to expect, simply by considering the smallest nontrivial model: two schools of equal size, shared by black and white populations that are also equal. In this situation, if the fraction of white students in one school is r, then the fraction of black students in the same school must be 1–r, while in the other school the black fraction is r and the white fraction 1–r. Suppose r ≤ 1/2. Then any allocation of students for which r ≥ β will be agreeable to everyone. At the same time, there is another possible solution, namely r = 0, or total segregation, with one all-white school and one all-black school. As β approaches 1/2, the integrated solution becomes increasingly fragile and hard to maintain; at β = 1/2, the equilibrium is instantly destroyed by the slightest perturbation (such as someone moving into or out of the district); beyond β = 1/2 the shared-schools solution disappears, and only total segregation is feasible.

Does this simple-minded model have anything to do with the real-world plight of kids going to school in Seattle and Louisville? Both of those cities once had highly segregated schools and achieved a better racial balance only by means of deliberate incentives and policies, applied steadily over more than 30 years. In the case of Louisville, the desegregation plan was mandated and supervised by the courts from 1973 until 2000; since then, aspects of the program have been continued voluntarily as a means of maintaining racial diversity. In Seattle the measures were adopted by the school board without outside compulsion. Currently, Seattle’s 10 high schools have a citywide open-enrollment policy, so that any student can choose any of the schools. Those schools that have more applicants than places use several factors to determine which students will be admitted, including geography and the presence of siblings already attending the school. Racial balance is the final tiebreaker.

The students and parents who have challenged these policies (supported by an amicus brief from the Bush administration) argue that race should not be a factor at all in assigning children to schools—that it’s no more legimate to consider race as a way of achieving integration that it would be in maintaining segregation. In other words, policies must be colorblind, even though people aren’t.

We won’t know for months how the supremes will rule on this issue, but savvy court-watchers say the outlook is dim for the diversity policies; and if they are overturned in Seattle and Lousiville, similar programs will have to be dismantled in many other cities as well. What’s the appropriate response? One might well try to out-Herod Herod, taking the position that all school-assignment decisions (and analogous choices both within the schools and everywhere else in American life) should be conducted as a pure lottery. After all, if race is a forbidden criterion, how can we allow accidents of residence or wealth or personal history to intrude? This strategy has the virtue that it would make pretty much everyone fiercely unhappy—and so it probably wouldn’t last long.

For another approach to the problem, I would like to ask this: What could we change about Schelling’s model in order to get a somewhat brighter vision of the future? The obvious candidate is the model’s one-sided choice function. In the world of the Schelling model, we all want to avoid being too much in the minority, but no one worries about being too much in the majority. I’d like to think, on the contrary, that we might have at least a few people who go out of their way to seek diversity, who pull up stakes and move not only if they have too many neighbors of another race but also if they have too few. It would be interesting to explore the phase diagram of a model that included such a subpopulation.

A few references:

Schelling, Thomas C. 1971. Dynamic models of segregation. Journal of Mathematical Sociology 1:143–186.

Schelling, Thomas. C. 1978. Micromotives and Macrobehavior. New York: W.W. Norton.

Granovetter, Mark S., and Roland Soong. 1988. Threshold models of diversity: Chinese restaurants, residential segregation and the spiral of silence. In Sociological Methodology (edited by Clifford Clogg), pp. 69–104. PDF

Clark, W. A. V. 1991. Residential preferences and neighborhood racial segregation: a test of the Schelling segregation model. Demography 28:1–19.

Bruch, Elizabeth E., and Robert D. Mare. 2004. Neighborhood choice and neighborhood change. PDF

Aguilera, Antonio, and Edgardo Ugalde. 2006. A spatially extended model for residential segregation. http://arxiv.org/abs/nlin/0607026