## 600613

Pick a number, N, then try searching for it on the web via Bing or Google (or maybe the leet version of Google). What can you expect to learn? I wasn’t quite sure of the answer, so I ran some experiments.

When N is a small positive integer—less than 100, say—the leading results tend to be mass-audience web pages that happen to display the numeral N in some prominent way, such as in a headline or a title. There are news stories (Packers 43, Falcons 37), TV stations (WXMI Fox 17), a few brand names (Motel 6), references to iconic events (9/11, Apollo 13), listings of Bible verses (Romans 3:23).

With somewhat larger integers—three or four digits—I see a lot of street addresses, area codes, tax forms, statutes and ordinances. With five-digit numbers, Zip codes become prominent. At six digits we enter the land of hex colors, accompanied by a baffling variety of part numbers, account numbers, serial numbers, patent numbers, error numbers, lottery numbers. With a search string of 8 to 10 digits, telephone directories dominate the results. Still further out on the number line, you eventually come to a numerical desert where Google and Bing usually come up empty.

To get a more quantitative sense of how numbers are distributed across the web, I decided to do some sampling. I randomly selected 2,000 positive integers of 1 to 12 decimal digits, and submitted them to Google as web search queries. To construct the query integers I started with 2,000 floating-point numbers selected uniformly at random (with replacement) from the range $$0 \le m \lt 12$$. For each $$m$$ I calculated $$N = \lfloor 10^{m}\rfloor$$, then ran a Google search for N. The work was done by a Python script with a politeness pause of one second between queries.From the results of each search I extracted $$H(N)$$, the number of hits, which Google reports near the top of the page. Here’s what I found, plotted on log-log scales:

What an intriguing graph! Over most of the range in the log-log plot, the broad trend looks very nearly linear. What does that mean? If the Google data accurately reflect the state of the web, and if my sampling of the data can be trusted, it means the number of web pages mentioning numbers of magnitude $$10^k$$ is roughly constant for all k in the range from $$k = 2$$ to $$k = 10$$. I don’t mean to suggest that specific large numbers appear just as frequently as specific small numbers. That’s obviously untrue: A typical two- or three-digit number might be found on a billion web pages, whereas a specific nine- or ten-digit number is likely to appear on only one or two pages. But there are only 90 two-digit numbers, compared with 90 billion 10-digit numbers, so the overall number of pages in those two classes is approximately the same.

Here’s another way of saying the same thing: The product of $$N$$ and $$H(N)$$ is nearly constant, with a geometric mean of roughly $$7 \times 10^{10}$$. An equivalent statement is that:

$\log_{10}{N} + \log_{10}{H(N)} \approx 10.86.$

You can visualize this fact without doing any arithmetic at all. Just print a series of $$N, H(N)$$ tuples in a column and observe that the total number of digits in a tuple is seldom less than 11 or greater than 13.

    N,        H(N)
96964835, 2120
2048, 164000000
476899618, 214
96416, 374000
75555964, 3020
171494, 182000
154045436, 2160
1206, 112000000
761088, 50200
7500301034, 24
13211445, 10900
1289, 77000000
1507549, 18100
3488, 3330000
7507624475, 10
17592745, 2830
1430187656, 30
691, 265000000
41670244642, 2
326, 52900000

Although the vast majority of the 2,000 data points lie near the 10.86 “main sequence” line, there are some outliers. One notable example is 25898913. Most numbers of this magnitude garner a few thousand hits on Google, but 25898913 gets 29,500,000. What could possibly make that particular sequence of digits 10,000 times more popular than most of its neighbors? Apparently it’s not just an isolated freak. About half the integers between 25898900 and 25898999 score well below 10,000 hits, and the other half score above 20 million. I can’t discern any trait that distringuishes the two classes of numbers. Sampling from other nearby ranges suggests that such anomalies are rare.

A straight line on a log-log plot often signals a power-law distribution. The classic example is the Zipfian distribution of word frequencies in natural-language text, where the kth most common word can be expected to appear with frequency proportional to $$k^{-\alpha}$$, with $$\alpha \approx 1$$. Does a similar rule hold for integers on the web? Maybe. I tried fitting a power law to the data with the powerlaw Python package from Jeff Alstott et al. The calculated value of $$\alpha$$ was about 1.17, which seems plausible enough, but other diagnostic indicators were not so clear. Identifying power laws in empirical data is notoriously tricky, and I don’t have much confidence in my ability to get it right, even with the help of a slick code library.

I’m actually surprised that the pattern in the graph above looks so Zipfian, because the data being plotted don’t really represent the frequencies of the numbers. Google’s hit count $$H(N)$$ is an approximation to the number of web pages on which $$N$$ appears, not the number of times that $$N$$ appears on the web. Those two figures can be expected to differ because a page that mentions $$N$$ once may well mention it more than once. For example, a page about the movie 42 has eight occurrences of 42, and a page about the movie 23 has 13 occurrences of 23. (By the way, what’s up with all these numeric movie titles?)

Another distorting factor is that Google apparently implements some sort of substring matching algorithm for digit strings. If you search for 5551212, the results will include pages that mention 8005551212 and 2125551212, and so on. I’m not sure how far they carry this practice. Does a web page that includes the number 1234 turn up in search results for all nonempty substrings: 1234, 123, 234, 12, 23, 34, 1, 2, 3, 4? That kind of multiple counting would greatly inflate the frequencies of numbers in the Googleverse.

It’s also worth noting that Google does some preprocessing of numeric data both in web pages and in search queries. Commas, hyphens, and parentheses are stripped out (but not periods/decimal points). Thus searches for 5551212, 555-1212, and 5,551,212 all seem to elicit identical results. (Enclosing the search string in quotation marks suppresses this behavior, but I didn’t realize that until late in the writing of this article, so all the results reported here are for unquoted search queries.)

In the graph above, the linear trend seems to extend all the way to the lower righthand corner, but not to the upper lefthand corner. If we take seriously the inferred equation $$N \times H(N) = 7 \times 10^{10}$$, then the number of hits for $$N = 1$$ should obviously be $$7 \times 10^{10}$$. In fact, searches for integers in the range $$1 \le N \le 25$$ generally return far fewer hits. Many of the results are clustered around $$10^{7}$$, four or five orders of magnitude smaller than would be expected from the trend line.

To investigate this discrepancy, I ran another series of Google searches, recording the number of hits for each integer from 0 through 100. Note that in this graph the y axis is logarithmic but the x axis is linear.

There’s no question that something is depressing the abundance of most numbers less than 25. The abruptness of the dip suggests that this is an artifact of an algorithm or policy imposed by the search engine, rather than a property of the underlying distribution. I have a guess about what’s going on. Small numbers may be so common that they are treated as “stop words,” like “a,” “the,” “and,” etc., and ignored in most searches. Perhaps the highest-frequency numbers are counted only when they appear in an <h1> or <h2> heading, not when they’re in ordinary text.

But much remains unexplained. Why do 2, 3, 4, and 5 escape the too-many-to-count filter? Same question for 23. What’s up with 25 and 43, which stand more than 10 times taller than their nearest neighbors? Finally, in this run of 101 Google searches, the hit counts for small integers are mostly clustered around $$10^6$$, whereas the earlier series of 2,000 random searches produced a big clump at $$10^7$$. In that earlier run I also noticed that searching repeatedly for the same $$N$$ could yield different values of $$H(N)$$, even when the queries were submitted in the space of a few seconds. For example, with $$N=1$$ I saw $$H(N)$$ values ranging from 10,400,000 to 1,550,000,000. Presumably, the different values are coming from different servers or different data centers in Google’s distributed database.

I was curious enough about the inconsistencies to run another batch of random searches. In the graph below the 2,000 data points from the first search are light blue and the 2,000 new points are dark blue.

Over most of the range, the two data sets are closely matched, but there’s a conspicuous change in the region between $$10^2$$ and $$10^4$$. In the earlier run, numbers in that size range were split into two populations, with frequencies differing by a factor of 10. I was unable to identify any property that distinguishes members of the two populations; they are not, for example, just odd and even numbers. In the new data, the lower branch of the curve has disappeared. Now there is a sharp discontinuity at $$N = 10^4$$, where typical frequency falls by factor of 10. I have no idea what this is all about, but I strongly suspect it’s something in the Google algorithms, not in the actual distribution of numbers on the web.

The limitations of string matching—or even regular-expression matching—are more troublesome when you go beyond searching for simple positive integers. I’ve hardly begun to explore this issue, but the following table hints at one aspect of the problem.

N top hit
17.3 HP Anodized Silver 17.3″ Pavilion
17.30 17.30j Syllabus – MIT
17.300 Chapter 17.300 COMPLIANCE
17.3000 Map of Latitude: 17.3000, Longitude: -62.7333
17.30000 41 25 0 0 2.000000 4.000000 6.000000 8.000000
17.300000 17.300000 [initially -35.600000] gi_24347982 (+) RNA

Search queries that are mathematically equal (when interpreted as decimal representations of real numbers) yield quite different results. And 4.999… is definitely not equal to 5.000… in the world of Google web search.

It gets even worse with fractions. A search for 7/3 brought me a calculator result correctly announcing that “7/3 = 2.33333333333″ but it also gave me articles headlined “7^3 – Wolfram Alpha”, “Matthew 7:3″, “49ers take 7-3 lead”, and “Hokua 7’3″ LE – Naish”. (Enclosing the search term in quotation marks doesn’t help in this case.)

Before closing the book on this strange numerical diversion that has entertained me for the past couple of weeks, I want to comment on one more curious discovery. If you run enough searches for large numbers, you’ll eventually stumble on web sites such as numberworld, numberempire, numbersbase, each-number, every-number, all-numbers, integernumber, numbersaplenty, and numberopedia. A few of these sites appear to be created and curated by genuine number-lore enthusiasts, but others have a whiff of sleazy search-engine baiting. (For that reason I’m not linking to any of them.)

Here’s part of a screen capture from Numbers Aplenty, which is one of the more interesting sites:

Each of the numbers displayed on the page is a link to another Numbers Aplenty page, and the site is apparently equipped to display such a page for any positive integer less than $$10^{16}$$. A few years ago, Google reported that they had indexed a trillion unique URLs on the world wide web. Evidently they hadn’t yet worked their way through the 10,000 trillion URLS at Numbers Aplenty. (But I’m pretty sure the server doesn’t have $$10^{16}$$ HTML files stored on disk, patiently waiting for someone to request the information.

And, finally, a trivia question: What is the smallest positive integer for which a Google search returns zero results? The smallest I have recorded so far is 10,041,295,923. (Of course that could change after the Googlebot indexes this page.) Can anyone find an example with 10 or fewer digits?

Posted in computing, mathematics | 18 Comments

## Four Fifths = A Fifth

Browsing in the library stacks the other day, I came upon this passage in a 1970 paper discussing the uses of computers in mathematics:

Notice anything fishy about that equation? On the right side of the equal sign we have an odd number of odd numbers, and so their sum must be odd; but the power of 144 on the left side is even.

I’m not clever enough to have caught that parity violation on first glance; I noticed it only later. On the other hand, my experience as an editor has taught me never to trust an author’s arithmetic. I had a computer sitting in front of me, open to an IPython notebook, so I typed in the numbers:

Had I just discovered that a 50-year-old counterexample to a 250-year-old conjecture is not in fact a counterexample after all? Surely not. Someone must have erred in copying the equation from the original source.

So I decided to track down that source. The reference in the quoted passage directed me to a 1966 paper by L. J. Lander and T. R. Parkin, which I quickly found online. However, the reference is mistaken; the cited paper says nothing at all about fifth powers and has no equation resembling the one above.

A faulty equation and a wayward reference within a single paragraph: That’s not so good. But I’ve done worse myself, and my purpose here is not to criticize an errant author (whom I’m not going to name). What interests me is how one might go about correcting the error in the equation and finding the true counterexample to the Euler conjecture. Here are some strategies I might have tried that morning:

• I was sitting in a university library. I had all the apparatus of scholarly research at my fingertips. I could turn to MathSciNet, say, and identify the correct paper by Lander and Parkin.
• I was sitting at a computer with a wifi connection. I had all the resources of the Internet at my fingertips. I could Google it, or consult some more specialized database such as the OEIS.
• I could just guess the answer. On the assumption that exactly one of the five terms has been altered, I would expect to come up with the correct equation after just a few tries.
• I could start from scratch. I could write a program to repeat the computer search that Lander and Parkin evidently used to discover the equation in the first place.
• I could bring the might of mathematics to bear on the problem. That is, I could try to “solve the equation.”

At this point you might want to stop reading, choose your own favorite method, and give it a try. The answer, once you have it, is not all that illuminating, but I found the process of searching had its moments of interest.

If guessing is one of your strategies, you had better try it first. I already had the equation entered into the IPython notebook, where I could easily alter a value and observe the result. (A spreadsheet would serve as well in this role.) I started with the $$27^5$$ term, reducing it to $$26^5$$, then $$25^5$$, and so on, but the effect was too small. Even setting that term to zero or to $$(-27)^5$$ left an excess, so I moved on to the $$85^5$$ term. Proceeding in this way, it didn’t take long to pinpoint the error.

The reference is: L. J. Lander and T. R. Parkin, 1966, Counterexample to Euler’s conjecture on sums of like powers, Bulletin of the American Mathematical Society 72:1079. The text of this paper consists of five lines, one of which is the equation.Googling also produced useful results, including pages on Wikipedia and MathWorld that give the correct equation and the correct reference. But finding those pages was not straightforward, and the experience has left me wondering how best to search for numerical or mathematical content on the web. You can always go after the metadata; in this case the names Lander, Parkin, and Euler are good choices, as well as the terms “nth power” and “conjecture.” You might also get lucky searching for character strings such as “144^5” and “27^5”, but here the outcome is hit or miss. There are many ways of encoding mathematics in digital documents—TeX, MathML, the ASCII pidgin that evolved years ago on Usenet, notations based on programming languages. Without some higher-level semantic processing, no single search string will find them all. Some web sites (including MathWorld) turn equations into tiny GIF images, making search even more difficult.

A question I haven’t been able to answer through web search is whether the erroneous equation from that 1970 paper has propagated into the subsequent literature. I have found no evidence of it, but I have little confidence in the ability of current search technology to answer the question reliably. Gathering quantitative data—how many times does the equation appear on the web?—also seems to be out of reach.

The issue goes beyond Google and the other web search engines. MathSciNet claims to accept search strings in TeX format, but I’ve never figured out how to make it work. As for the OEIS, the search function there is specialized for finding integer sequences—as one might expect in an Online Encyclopedia of Integer Sequences. Searching for an equation is an “off label” use of the archive. Still, I was sure I’d find it. I tried searching for 61917364224 (which is $$144^5$$), and got 47 hits. (It’s a popular number because it’s a product of many small factors, $$2^{20} 3^{10}$$, and is also a Fibonacci number.) However, none of those 47 sequences has anything to do with the Euler sum-of-powers conjecture or the Lander-Parkin counterexample. Yet the equation I was looking for does indeed appear in the OEIS, namely in sequence A134341.

The next approach—redoing the entire computation from scratch—turned out to be easier than I expected. Lander and Parkin ran their search on a CDC 6600, which was the most powerful computer of its time, the world’s first megaFLOPS machine, designed by Seymour Cray early in his career. I don’t know how long the computation took on the 6600, but there’s not much to it on a modern laptop. Here’s some Python code:

import itertools as it

def four_fifths(n):
"""Return smallest positive integers ((a,b,c,d),e) such that
a^5 + b^5 + c^5 + d^5 = e^5; if no such tuple exists
with e < n, return the string 'Failed'."""
fifths = [x**5 for x in range(n)]
combos = it.combinations_with_replacement(range(1,n), 4)
while True:
try:
cc = combos.next()
cc_sum = sum([fifths[i] for i in cc])
if cc_sum in fifths:
return(cc, fifths.index(cc_sum))
except StopIteration:
return('Failed')

The first step here is to precalculate the fifth powers of all integers less than a given limit, n. Then we generate all combinations of four integers in the range from 1 to n, starting with the 4-tuple (1, 1, 1, 1) and continuing to (n, n, n, n). For each combination we look up the fifth powers of the integers and check to see whether their sum is also present in the precomputed vector of fifth powers. Invoking this program as four_fifths(150) returns the correct answer in about 30 seconds. A little sad, isn’t it? This was once a research project fit for a supercomputer, and now it’s reduced to the level of a homework assignment.

The final tool we might apply to this problem is the hammer of mathematics. I would put the question as follows. We are looking for integer solutions to this equation:

$a^5 + b^5 + c^5 + d^5 - e^5 = 0.$

In that quest, do we gain any significant leverage by knowing that:

$27^5 + 85^5 + 110^5 + 133^5 - 144^5 = 254933701 ?$

Can we take the number 254933701 and apply some algebraic or analytic hocus pocus that will lead us directly to the correct values of a, b, c, d, e? If we make no further assumptions or guesses, I believe the answer must be no. After all, there are innumerable values for a, b, c, d, e that we can plug into our equation to get something other than 0. Consider this example:

$30^5 + 69^5 + 82^5 + 86^5 - 100^5 = 43.$

That’s a very near miss—it comes within 5 parts per billion of being a true solution—and yet it tells us nothing about the set of correct solutions except that this isn’t one of them.

An erroneous equation seems to be useful only if we know quite a lot about the nature of the errors. For example, in the faulty equation from 1970, if someone tips you off that all the terms are correct except for $$85^5$$, you can form the equation

$85^5 - x^5 = 254933701$

and solve for $$x$$. If $$x$$ turns out to be an integer, you have your answer. And indeed this is the case:

$x = \sqrt[5]{4182119424} = 84.$

But is this procedure better than guesswork? Is it different from guesswork?

Just for the record, the correct Lander-Parkin equation is:

$27^5 + 84^5 + 110^5 + 133^5 = 144^5 = 61917364224.$

After discovering their counterexample to the Euler conjecture, Lander and Parkin, together with John Selfridge, made some conjectures of their own. Suppose an nth power can be partitioned into k smaller nth powers. They conjectured that k is never less than $$n-1$$. And they asked whether there always exists an example where $$k = n-1$$. You might try to settle the latter question for the case of $$n = 6$$. Can you find a set of five 6th powers that add up to another 6th power? For this problem, Google won’t help you, because the answer is unknown. No one has even identified a set of six 6th powers that sum to a 6th power. If you approach this task as a computational challenge, it’s rather more than a homework assignment.

Posted in computing, mathematics | 8 Comments

## Lopsided

Sifting through the election results last week, I noticed that the precinct where I used to live in Durham, North Carolina, voted 620 to 40 in favor of the Blue candidate in the U.S. Senate race. That’s a margin of just under 94 percent. A few nearby polling places were even more lopsided: One score was 288 to 7, which works out to 97.6 percent. Statewide, however, the contest was quite close; the Blues lost with 49.13 percent to the Reds’ 50.87 percent. (The percentages I’m giving here are based on votes for the two major parties only; published results are slightly different because they include votes for third-party candidates and write-ins.)

The combination of one-sided local results and a nearly even split in the statewide totals left me curious about the distribution of vote margins, or what we might call the political polarization spectrum. Are extreme ratios, like those in my old neighborhood, rare outliers? Or have we become a nation of segregated political communities, where you live with your own kind and stay away from places where the other side dominates?

The infographics group at The New York Times has produced a series of election maps that offer much geographic insight.

Portion of a New York Times map showing voting patterns in the 2014 U.S. Senate election in North Carolina. Durham is the blue inkblot near the right edge. Graphics by Amanda Cox, Mike Bostock, Derek Watkins, and Shan Carter.

The North Carolina map shows deep blue city cores surrounded by ruddy or purplish suburbs. Rural areas, with lower population density, range from pink to baby blue. The maps are lovely—they transform ugly politics into luscious cartography—but they can’t give a quantitative answer to my question about political segregation and integration.

So off I went to the web site of the North Carolina Board of Elections, where I pulled down the latest report (still unofficial), giving vote totals for the state’s 2,726 precincts. After some vigorous data wrangling (Excel, Textmate, Python, Lisp, eyeballing), I had a histogram with 20 bins, classifying precincts according to their percentage of votes for the Red candidate.

Before I show the data, it’s worth pausing to think about what the distribution might be expected to look like. Suppose the state’s population were thoroughly mixed, with Red and Blue voters scattered at random. Then the precinct voting margins would follow a normal distribution, with most of the precincts near the 50-50 mark and only a few out in the far-right and far-left tails. The curve might look something like this:

Actually, that’s not quite what the curve would look like; it would be much narrower. Suppose you take 3,000 precincts with 1,000 voters each, and you paint the individual voters Red or Blue at random with probability 1/2. You would get a normal distribution with a mean of 500 and a standard deviation of about 16 voters (i.e., $$1/2\sqrt{1000}$$). Almost all of the precincts would lie within three standard deviations of the mean, or in other words between 450 and 550 Red voters. The histogram would be more like this:

Of course nobody who reads the newspaper would suggest that random mixing provides a good model of the American political landscape. Given the evidence of increasing polarization in the public, we might expect to find a two-humped camel, with a Red peak and a Blue peak, and not much of anybody in the middle:

Or maybe the distribution is even more extreme: an empty bowl, with every precinct drifting centrifugally toward the outer fringes of the political spectrum:

All right, I’ve stalled long enough; no more playing with toy distributions. Here’s the real deal—the histogram based on North Carolina precinct data for the recent Senate election:

The shape of this curve took me by surprise, mostly because of its strong asymmetry. Remember, the total numbers of Red and Blue voters in the state are roughly equal, but the apportionment of Reds and Blues in precincts turns out to be quite different. At the left edge of the graph there are 150 precincts where at least 90 percent of the voters chose the Blue candidate, but over on the right side there are only 8 precincts that voted at least 90 percent Red.

Please don’t misinterpret this graph. It doesn’t say that Blue voters are more extreme or more radicalized than Red ones. Nothing of the sort. It says that Blue voters tend to huddle together in more homogeneous communities. If you’re Blue and you want to be surrounded by like-minded voters, there are hundreds of places you can live. If you’re Red, you have lots of choices where you’ll be in the majority—more than 60 percent of the precincts satisfy that criterion—but you would have a hard time finding areas where you have few or no Blue neighbors.

The pattern seemed peculiar enough that I began to wonder if it might be unique to North Carolina. So I took a look at the Senate race in Virginia, which was even closer than the North Carolina contest. The Virginia pattern differs only in detail. Again we have an abundance of solidly Blue precincts, but very few solidly Red ones:

Could it be something about the South that produces this skew in the curve? I wanted to check the closely contested governor’s race in Massachusetts (where I live now), but the state hasn’t yet posted precinct-level results. Instead I looked at the Minnesota Senatorial election:

Here the raised shoulder on the Blue side is not nearly as dramatic, but the asymmetry is present. (Note that this race did not have a close finish; the Blues won by 10 percentage points.)

There are several ways we might try to explain these patterns—or explain them away. A glance at the Times maps, with their inky blue urban neighborhoods, suggests it’s all about city living. But this impression is partly an artifact of the maps’ graphic scheme, which uses color to encode not just voting margin but also population density. As a result, strongly skewed votes in rural areas are not nearly as conspicuous as those in cities. In North Carolina there are several pale blue precincts in the countryside that have vote totals above 85 percent Blue. Minnesota exhibits the same pattern; indeed, one Iron Range precinct, in the far north of the state, recorded a clean sweep for the Blues, 218 to 0. In Virginia, on the other hand, all of the most strongly biased precincts seem to be urban. And it remains true that the vast majority of voters in lopsided precincts are city dwellers.

It’s also important to note that city districts are geographically much smaller than rural or suburban ones. Perhaps this fact alone could explain much of the Blue-Red difference. If we carved up the rural districts into areas the same size as the city precincts, would we find homogeneous clusters of voters? I have no data to answer that question one way or the other. But even if such clusters do exist, the situation remains asymmetric because the Blue voting blocks are larger in population by an order of magnitude.

Race is another factor that can’t be ignored. Those strongly Blue precincts in rural North Carolina are also largely black precincts. But race is not the whole story. Many of the Bluest urban districts are racially and ethnically diverse.

In the end, what’s most puzzling about the histograms is not just the existence of many pure Blue precincts but the near absence of pure Red ones. What accounts for this imbalance? It’s not hard to imagine social mechanisms that would separate people into affinity groups, but the simplest models are symmetrical. We need to find a factor that acts differently on the two parties.

I have not settled in my own mind what I think is going on here, but I would like to offer three possible mechanisms for consideration.

First, maybe Red voters and Blue voters differ in the criteria they apply when choosing a place to live. We might test this idea in a variant of the Schelling segregation model, in which people tend to go elsewhere when they have too few neighbors of their own kind. Perhaps Red voters refuse to live where the proportion of Reds is less than 1/3, but Blues are content to stay even where they are a tiny minority. Alternatively, we might suppose that when Blues reach a 2/3 majority, they drive out the remaining Reds, whereas Reds are willing to tolerate a Blue minority in their midst. In either case, the result is “Red flight” from strongly Blue areas but no countercurrent of Blue voters fleeing Red areas. I think this model might be capable of explaining the observations, but I have no idea how to explain the model. Is there any evidence for such an asymmetry in personal preferences and behavior?

The second possibility is that Blues are more effective than Reds in persuading their neighbors to support the local political majority. In other words, it’s not that Reds flee or are expelled from Blue areas; rather, they are converted into Blues. Then we have to ask: How come the Reds are unable to win over their own Blue neighbors?

My third candidate explanation is gerrymandering. Maybe what we’re seeing is not some natural tendency to form uniformly Blue communities but rather an attempt to draw the boundaries of precincts in a way that concentrates Blue voters in certain districts, leaving the rest of the precincts with a Red majority. To test this hypothesis we might compare states in which different parties dominate the state legislature and thus control the redistricting process. If the asymmetry really is caused by an attempt to hem in the minority party, we should see a mirror image in states where the other party is in power. The three states I’ve looked at so far are not a useful sample in this respect: North Carolina and Virginia both have Red legislatures, and Minnesota’s was also controlled by the Reds during the most recent redistricting cycle. Massachusetts will be a good test case when the numbers come out.

One last thought: In this essay I have written about Reds and Blues rather than Republicans and Democrats in an attempt to keep the focus on a mathematical question and to keep my distance from partisan passions. For the record, however, I don’t actually believe that politics is a game played by brightly colored teams. And I do take sides. Last Tuesday, in my opinion, the stinkers won.

Posted in social science, statistics | 19 Comments

## Revised and Updated

The new edition of my big picturebook, Infrastructure: A Guide to the Industrial Landscape, was published October 27th and is now on display in better bookstores everywhere. Above is the cover; below is the book as seen in the wild—on the shelves of the Harvard Coop and the MIT Press Bookstore. And, yes, you can also get it at Amazon.

I am late in posting this announcement because I wanted to redo the book’s web site, and that took a week longer than expected. (I’m still not entirely happy with the design and the user interface, but that’s a story for another time.)

So what’s revised and updated in the revised and updated edition? It’s been almost a decade since the first edition came out, which is ages in Internet years. And, as a matter of fact, the chapter on communication technology was the one most in need of updating. No one seems much interested in fax machines and beepers these days; brushing those topics aside made room for a new section on data centers. The petroleum industry has also changed dramatically: Fracking is no longer a novelty, and worries about “peak oil” seem to have been displaced by “the new Saudi America.” Meanwhile aviation is on the cusp of a great change, as the whole network of ground-based navigational beacons is about to be dismantled in favor of GPS.

On the other hand, I couldn’t find much of anything new to say about American railroads. They still represent the pinnacle of development of 19th century technology.

It’s not only the landscape that has changed; I have changed too, and I would now offer a somewhat more cautious assessment of how technology serves human needs. Not that I’m about to join the Luddites; I still believe that if we’re going to accommodate 10 billion people on this planet, we need all the technological help we can muster. But I’m not greatly impressed with our social and economic mechanisms for choosing which technologies to pursue.

The new edition gave me a welcome opportunity to correct some errors, to add a few new photos and to improve the presentation of all the photos. (There are more than 750.) And I’m pleased to report that the publisher (W. W. Norton) and the printer (a branch of R R Donnelley in China) have done quite a spiffy job of bookmaking. They’ve also held the price at \$35, the same as the older paperback.

If you read the first edition, should you run right out to buy a copy of this new one? I won’t try to talk you out of it. Admittedly, this is not a new book; it’s an improved version of the old one. But I’d like to think it’s worth reading again, and you might as well have the benefit of the latest word.

Posted in books | 2 Comments

## 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.

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.

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 | 3 Comments

## 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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0.0000640000 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 000000000
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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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


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


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


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:

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:

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.

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

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:

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.

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:

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:

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.

In 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.

The 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.

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:

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.

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 happened 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.

The 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 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.

Posted in mathematics | 4 Comments