All those books that Google has been scanning for the past ten years are surprisingly rich in numbers as well as words. The Google Books data set released last December by a Harvard-Google team includes (by my count) 9,620,835,344 occurrences of 458,794 distinct numbers. (Plus another 31,293 numeric values that have dollar signs attached.)

In recognition of pi day, I want to zero in on some successive approximations to the world’s favorite irrational:

In tabular form here are the closest approximations found in the files, along with the abundance of each value:

The data set includes only items that appear at least 40 times in the collection of scanned volumes. Closer approximations to pi evidently fell below that threshold. In particular there is no sign of William Shanks’s famous 707-digit calculation, which was published in 1873. So, just for the sake of celebrating 3.14, here are 707 digits of pi—but unlike the product of Shanks’s many years of labor, I think these digits may be correct:

3.1415926535897932384626433832795028841971693993751058209
74944592307816406286208998628034825342117067982148086513
282306647093844609550582231725359408128481117450284102701
938521105559644622948954930381964428810975665933446128475
648233786783165271201909145648566923460348610454326648213
393607260249141273724587006606315588174881520920962829254
0917153643678925903600113305305488204665213841469519415116
09433057270365759591953092186117381932611793105118548074462
3799627495673518857527248912279381830119491298336733624406
5664308602139494639522473719070217986094370277053921717629
3176752384674818467669405132000568127145263560827785771342
7577896091736371787214684409012249534301465495853710507922
796892589235420200

I have two postscripts to my American Scientist column on memristors and the  related posting here on bit-player.

First up is a charming preprint by Yuriy V. Pershin and Massimiliano Di Ventra on “Solving mazes with memristors: a massively-parallel approach.” The moment I saw the title, I guessed the basic idea behind this work—and I was half right.

The part I got right is that you fill the maze with a network of memristors, wiring them together along the corridors but making no connections that cross walls; then you apply a voltage between the entrance and the exit nodes. Current flows only through those nodes that lie along the solution path; there’s no current in any of the dead-end branches. The current passing through the memristors along the path alters their resistance, creating a permanent record of the solution.

But there’s a complication I missed. A memristor is a polar device: In one direction, current decreases the resistance; but when current flows the other way, the resistance increases. This wouldn’t be a problem if you could just arrange the memristors so that all of them on the solution path are oriented the same way—but if you knew enough to do that, you’d already know a great deal about the solution. Pershin and Di Ventra found a way around this problem; for details, see their paper. (Hint: their strategy depends on the observation that any practical memristor has a maximum resistance, which can’t increase even when current flows the “wrong” way.)

The illustration above (reproduced courtesy of Pershin and Di Ventra) shows the path identified by a (simulated) grid of memristors for a maze that has two solutions. The red nodes, with the lowest resistance, are common to both paths; the blue and green sections differ in resistance because they differ in length.

I suppose you could also solve mazes with ordinary resistors. Just wire them up in the same way, run a healthy current from entrance to exit, and take a thermal infrared photograph.

•     •     •

My second postscript concerns an omission. A comment from Mark Myatt and a letter from James R. Biard both point out that I neglected to mention an important predecessor of the memristor. This was the memistor—note the ever-so-slightly different spelling—invented in 1960 by Bernard Widrow and Marcian E. Hoff at Stanford in connection with their Adaline system of neural networks. The memistor was the part of the artificial neuron that assigned a weight to each input signal. A Stanford tech report gives a detailed description.

The memistor is the only computational device I’ve ever heard about whose underlying technology is electroplating. In the prototype version, the core element was a carbon rod—specifically, a pencil lead. In its native state, this slender carbon cylinder has a resistance of about 5 ohms, but plating a thin layer of copper onto the surface reduces the resistance to about 0.5 ohm. The process is reversible and almost linear over a defined range: The conductivity is proportional to the total plating charge, which is the time integral of the plating current.

In the photo, the carbon rod is the thinner straight piece; it has insulated copper wires soldered to its ends, and those wires in turn are mounted to a heavier copper rod that both provides mechanical support and serves as the anode in the electroplating cell—the source and sink of deposited metal ions. In operation, the assembly fits inside the test tube, which is filled with a sulfuric acid–copper sulfate solution. In case you’d like to try building your own memistor, Widrow and Hoff offer some practical advice of the kind that’s all too scant in most recent computer engineering publications. They recommend a Fineline H (medium hard) pencil lead, and they suggest polishing it with steel wool. Adding a “brightener” agent to the plating bath eliminates hysteresis in the plating-deplating cycle.

The evolution of computer technology in the years after 1960 did not follow the path of electrochemistry. But it would be a big mistake to dismiss Widrow and Hoff as eccentrics off in the bushes exploring fringe ideas. At the time, Hoff was Widrow’s doctoral student. A decade later he was employee number 12 at Intel and one of the inventors of the first microprocessor.

So why didn’t I tell this wonderful story in my American Scientist article? Well, I can mumble various excuses. Only so much stuff will fit in any one article before it bursts at the seams. And the Widrow-Hoff memistor seemed marginal because it was a three-terminal device: Current through the carbon rod is controlled by a separate plating current between the copper anode and the carbon. The memristor devices invented later have only two terminals, which puts them in a different category of circuit elements. Roughly speaking, the memistor was a weird transistor, whereas the memristor is a weird resistor.

Anyway, that’s how I persuaded myself that it was okay to ignore the memistor. But the decision was a mistake, and I’m grateful to my readers for prodding me to correct it here.

Several weeks ago, on the morning after the first winter storm here in the Boston area, I wrote about some peculiar snow geometry on porch railings. Now, following another storm (which I wish I could believe might be the last of the season), I have more puzzling snow shapes to present.

What gives rise to the quasi-periodic undulations in the snow filling my neighbors’ rain gutters?

Note that on the roof above the gutter the snow depth appears to be perfectly uniform. The scalloped shapes appear only in the gutters.

Elsewhere on the same house, the waves are less regular in period and smaller in amplitude, but they are still clearly present.

On the opposite side of a different house I find more undulations, this time in the gutters draining a relatively flat porch roof.

All of the gutters shown above are oriented northwest-southeast, and were roughly parallel to the prevailing wind direction during the storm (although winds were very light). Gutters perpendicular to these revealed little or no evidence of wavelike disturbances. Waves were also absent from other linear surfaces, such as railings, regardless or orientation.

I’ve seen no sign of such wavelike formations in any previous snowfall this winter. To my eye, the waveforms are cusplike—cycloidal, perhaps, rather than sinusoidal. The snow was fairly heavy and sticky (note the coating on tree branches); temperatures were just a few degrees below freezing; total accumulation was about 10 cm.

Anyone care to propose a mechanism?

Update: After a welcome day of warm rain, the answer has slowly emerged from the wilting snow, and it has nothing to do with subtleties of fluid dynamics or particle deposition.

In the photo above, peeking out from behind the last remnants of the melting snowdunes, are the metal spikes and straps that fasten the gutters to the fascia board. Their somewhat irregular spacing is a good match for the pattern of cusplike peaks. Of course! How could I have failed to think of that?

Why did the dunes appear only in this storm? Earlier snows were fluffier and deeper, and probably less affected by the hanger straps. Why only on certain sides of the houses? I’m not sure about that, but the one duneless gutter I’ve been to able to examine closely has a different hanger system, without the obstructing straps.

It is said that to explain is to explain away. This maxim is nowhere so well fulfilled as in the area of computer programming, especially in what is called heuristic programming and artificial intelligence. For in those realms machines are made to behave in wondrous ways, often sufficient to dazzle even the most experienced observer. But once a particular program is unmasked, once its inner workings are explained in language sufficiently plain to induce understanding, its magic crumbles away; it stands revealed as a mere collection of procedures, each quite comprehensible. The observer says to himself “I could have written that”. With that thought he moves the program in question from the shelf marked “intelligent”, to that reserved for curios, fit to be discussed only with people less enlightened than he.

These words were written in 1966 by Joseph Weizenbaum, in a paper that explained—and explained away—his famous program ELIZA, the mock psychotherapist. ELIZA’s conversations could occasionally suggest a glimmer of understanding:

I need some help, that much seems certain.

WHAT WOULD IT MEAN TO YOU IF YOU GOT SOME HELP?

Perhaps I could learn to get along with my mother.

TELL ME MORE ABOUT YOUR FAMILY.

But Weizenbaum revealed that ELIZA was merely picking out a few keywords from the text and applying simple syntactic transformations, along with a dose of randomness. The program was manipulating symbols, but the symbols had no meaning attached to them.

What about Watson, the new Jeopardy champion? Watson gave a dazzling performance this past week, decisively winning a two-game match against the best human players in the history of the quiz show. But will the magic crumble if we look closely at how it works? Does the program really understand those quirky Jeopardy clues, or is it just pushing symbols around, in the manner of ELIZA?

The most detailed account of Watson’s innards that I’ve been able to find is an article published in AI Magazine last fall by David Ferrucci of IBM, the project’s lead engineer, and a dozen colleagues from IBM and Carnegie Mellon. (”Building Watson: An overview of the DeepQA project,” AI Magazine 31(3):59–79. The article is behind a paywall at the AI Magazine web site, but resourceful internauts may find it elsewhere.)

Here’s the overview:

The system we have built and are continuing to develop, called DeepQA, is a massively parallel probabilistic evidence-based architecture. For the Jeopardy Challenge, we use more than 100 different techniques for analyzing natural language, identifying sources, finding and generating hypotheses, finding and scoring evidence, and merging and ranking hypotheses.

Thus we learn that behind Watson’s calm, metallic voice is a clamor of 100+ agents doing their massively parallel probabilistic evidence-based thing. This is not one big brain but a society of mind. (By the way, I think “probabilistic” means simply that potential answers are scored by assigning them probabilities; as far as I can tell there is no randomness or indeterminacy in the algorithms, but I could be wrong about that.)

The first step in the run-time question-answering process is question analysis. During question analysis the system attempts to understand what the question is asking and performs the initial analyses that determine how the question will be processed by the rest of the system.

This description implies that the system is indeed making an effort to dig down into the semantics of natural language. But how does it attempt to understand the clue? The rest of the paragraph is more of a shopping list than an explanation:

The DeepQA approach encourages a mixture of experts at this stage, and in the Watson system we produce shallow parses, deep parses (McCord 1990), logical forms, semantic role labels, coreference, relations, named entities, and so on, as well as specific kinds of analysis for question answering.

The reference to (McCord 1990) is perhaps the most illuminating item in this list. The author is Michael C. McCord, who is at IBM’s Yorktown Heights lab where Watson was built. The phrase “deep parses” apparently refers to McCord’s idea of a slot grammar, which provides a single framework for combining the syntactic analysis of sentences (subject, predicate, object, etc.) with semantic features (word senses, logical relations, predicates). Unfortunately, the AI Magazine article gives no further hints about how slot grammars are used in the analysis of Jeopardy clues. (For some useful recent accounts of slot grammars, see the links in McCord’s publications list.)

The part of the question-analysis phase that Ferrucci et al. discuss at greatest length is a process they call “LAT detection.” LAT is “lexical answer type”: a word or phrase in the clue that specifies what kind of response is wanted—a person, a city, a book, a substance, and so on. Consider this clue, in a category titled “Oooh…. chess”:

Invented in the 1500s to speed up the game, this maneuver involves two pieces of the same color.

The LAT in the clue is “maneuver”: Whatever the answer is, it must be something that can plausibly be described as a maneuver. If you were to fixate on the wrong LAT—say “the game” or “two pieces”—you’d have no hope of coming up with the correct answer. Naming the two pieces “king and rook” would not score any points, even though that particular choice of pieces suggests you have the right idea in mind; to get credit for the answer, you need to give the name of the maneuver: “castling.”

Identifying the correct LAT is clearly important. It’s also clearly difficult. What’s not so clear is how Watson does it. In the chess example, does “maneuver” stand out from the rest of the words in the clue for grammatical reasons (it’s the subject of the main clause), or because it’s pointed to by the demonstrative adjective “this,” or for some other reason? How would you write a program to identify the LAT of an arbitrary Jeopardy clue?

Moving on from the analysis of questions to the finding of answers, the algorithmic details remain a little fuzzy.

Watson has access to various sources of “structured” knowledge: relational databases, taxonomies, ontologies. With such resources, retrieval is straightforward. Yet it turns out that few clues can be reformulated as database queries. Ferrucci writes: “Watson’s current ability to effectively use curated databases to simply ‘look up’ the answers is limited to fewer than 2 percent of the clues.” I suppose this is not really surprising. If the game could be reduced to database lookup, it wouldn’t be much fun.

For the other 98 percent of the queries, I gather that the retrieval process is more like Googling for the answer. The machine has no live internet connection during the Jeopardy contest, so it can’t actually search the web. But lots of free-form textual data was loaded into the Watson servers ahead of time, including all of Wikipedia and many other reference works. Using these documents as seeds, the system then trawled the web for other sources that might be useful, and cached copies of them for use offline. About four terabytes of material was available for query answering.

As for the search methods applied to this archive, the article by Ferrucci et al. offers another shopping list:

A variety of search techniques are used, including the use of multiple text search engines with different underlying approaches (for example, Indri and Lucene), document search as well as passage search, knowledge base search using SPARQL on triple stores, the generation of multiple search queries for a single question, and backfilling hit lists to satisfy key constraints identified in the question.

In a long series of training runs the system was tuned to balance the competing demands of coverage, accuracy and speed.

The operative goal for primary search eventually stabilized at about 85 percent binary recall for the top 250 candidates; that is, the system generates the correct answer as a candidate answer for 85 percent of the questions somewhere within the top 250 ranked candidates.

The trouble with free-form textual search is that you may very well identify relevant snippets of text but still have a hard time extracting the correct answer. Indeed, the same kind of analysis that goes into figuring out the question also has to be applied to candidate answers. For example, Ferrucci et al. discuss this clue: “He was presidentially pardoned on September 8, 1974.” Among the materials retrieved by the search algorithm was the text fragment: “Ford pardoned Nixon on Sept. 8, 1974.” For a human player with a little knowledge of U.S. history, this result would be more than enough to settle the matter, but a computer program still has some work to do. Suppose the program has correctly identified the LAT of the clue as “He,” and suppose further that it knows that both “Ford” and “Nixon” refer to male persons, perhaps even that they were presidents. Which of the two names is the right choice? Several of the tests that Watson applies are essentially string-matching algorithms, similar to those that search DNA sequences for genetic patterns. Those algorithms might count how often each name occurs in association with the given date, but that result will not resolve the ambiguity in this case. The correct answer comes from a program module that undertakes a deeper logical analysis and recognizes the difference between subject and object in the two statements.

•     •     •

Given this glimpse into how Watson works, do we deem its intelligence to be explained, or explained away? Personally, I have mixed feelings.

I admit to a sentimental fondness for what John Haugeland called Good Old-Fashioned AI, or GOFAI—the ambitious kind of artificial intelligence that aspired to build a true thinking machine, a system with some deep internal representation (a mental model) of the world in which it functions. The outstanding example of this style is Terry Winograd’s SHRDLU program, written in 1970, which conversed about objects in a world of toy blocks on a tabletop. At the time, Winograd firmly asserted that the program was able to “understand discourse,” and he meant by this that the program understood not only the words but also the objects and relations the words referred to.

The promise of SHRDLU was that we could extend the same methods to broader domains of discourse, steadily building toward a general-purpose, human-like intelligence, with the same kind of carefully planned knowledge representation. But that never happened. Later in the 1970s, AI entered a time of troubles. When it came back in the 80s, the emphasis had shifted, and the technology had diversified. The new AI focused on expert systems, on data mining, on statistical rather than deductive methods; another branch of AI turned away from the human cerebral cortex in favor of the motor neurons of the cockroach. Overall, the field took a more pragmatic turn, with less concern for understanding the ultimate nature of intelligence and more energy invested in getting useful results, whatever the methodology.

Watson is in this latter-day pragmatic tradition, with its 100+ agents and its massively parallel probabilistic evidence-based architecture. Compared with SHRDLU, it’s all so messy, so ad hoc, so opaque. But it works, doesn’t it.

And I suppose my own mind is not quite as tidy as I would like to believe.

•     •     •

Although Watson won its Jeopardy match by a wide margin and made very few mistakes along the way, the moment everyone will remember is the program’s spectacular flub of a Final Jeopardy question on the second night. The category was “U.S. Cities,” and the clue was:

Its largest airport is named for a WWII hero, its second largest for a WWII battle.

Watson replied “Toronto.” As it happens, I got that question right; just seconds after the clue was revealed, I called out “Chicago.” Later, though, when I thought about the mental process that led to my answer, I realized that this was not at all a product of well-focused deductive reasoning. I was doing the same kind of scattershot, parallel, probabilistic groping in the dark that I frown on in a machine.

My “reasoning” went something like this: If it has two airports, it must be a pretty big city…. New York has three airports…. There’s Dallas, with DFW and Love—but no heroes or battles there. Chicago has two. Oh! Midway—that must be the battle of Midway.

That’s when I pressed the buzzer.

Note how sketchy my thinking was. I had no idea O’Hare was named for a war hero. As a matter of fact, I had no idea that Midway was named for the naval battle. If I had been asked in a more straightforward way, “Why is Chicago’s second airport named ‘Midway’?”, I would have guessed that it lies halfway between Point A and Point B. The Pacific island would not have entered my consciousness. And I never bothered to dig any deeper into the catalogue of multi-airport cities—Washington, San Francisco, Houston (isn’t G. H. W. Bush a WWII hero?).

So messy and ad hoc.

Two weeks ago my wife told me about her new Googling strategy: She ignores the top-ranked items and immediately clicks through to page six of the results. All the earlier pages, she says, are larded with SEO spam—links whose ranking has been artificially inflated by some nefarious form of search-engine optimization.

When I heard this, my first thought was: Well, there’s a business opportunity! Let’s set up a search engine—we’ll call it page6.com, or maybe goooooogle.com—that will pass each query along to Google, collect the results, discard the first five pages, and return the rest. But that daydream didn’t last long. It’s not just that Google would slap a cease-and-disist order on me. More important, there’s no need for anything as elaborate as a pass-through search engine. It can all be done with some simple scripting within the browser. The following XML, when loaded into Firefox, creates a search-bar plugin that returns Google results starting with page six.

<OpenSearchDescription
xmlns="http://a9.com/-/spec/opensearch/1.1/"
xmlns:moz="http://www.mozilla.org/2006/browser/search/">
<ShortName>Goooooogle</ShortName>
<Description>Get page 6 from Google</Description>
<InputEncoding>UTF-8</InputEncoding>
<Url type="text/html" method="GET"
template="http://www.google.com/search?q={searchTerms}
&ie=utf-8&oe=utf-8&aq=t&start=50"></Url>
</OpenSearchDescription>

The crucial bit is the phrase “start=50,” highlighted in red, which skips over the first five pages of the results.

Problem solved! Mission accomplished! But then over the weekend the business section of the Times ran a long story by David Segal that both validated and undermined the page-six strategy. The validation came from the revelation that SEO manipulation of search results is blatant and widespread and all too effective. Segal revealed that J. C. Penney, the retailer, has been finagling their way to the top of the Google lists for dozens of search terms, such as “dresses,” “area rugs,” “home decor” and “furniture.” Penney’s method is to buy lots of inconspicuous links on “innocent” sites, all pointing to Penney’s pages and thereby raising their Google PageRank. I shouldn’t be surprised to learn of this practice. I get offers every week or two to place such paid links on bit-player.org. “Who couldn’t use an extra $100, $2,000, $10,000/month or more in passive advertising income?” asked one recent enticement. The surprise, I guess, is that such a clumsy and cloddish manipulation of the search engines actually works.

So that explains why Google’s page-one results are all crap, and we’re better off skipping to page six. Unfortunately, toward the end of Segal’s story we learn that page six isn’t safe either. When Segal went to Google with the evidence of these shenanigans, Google took corrective action.

On Wednesday evening, Google began what it calls a “manual action” against Penney, essentially demotions specifically aimed at the company.

At 7 p.m. Eastern time on Wednesday, J. C. Penney was still the No. 1 result for “Samsonite carry on luggage.”

Two hours later, it was at No. 71.

At 7 p.m. on Wednesday, Penney was No. 1 in searches for “living room furniture.”

By 9 p.m., it had sunk to No. 68.

In other words, all the cruft that used to be on page one is now on page six or seven or eight. My Goooooogle trick is hosed.

The memristor is a new electronic component, an addition to the family of “passive” circuit elements—a family that for well over 150 years had just three members: the resistor, the capacitor and the inductor. Enthusiasts for memristor technology argue that it will bring higher-density, lower-cost, nonvolatile computer memory, with chips that hold terabits rather than mere gigabits. There’s also talk about “neuromorphic” computer architectures, where the memristor would play the role of a synapse in a nerve-like network. Of course these wonders are still in the future. While you’re waiting, you could visit the memristor web site, where you could buy a memristor tee shirt. Or you could read more about memristors in my new “Computing Science” column in American Scientist.

Will the memristor turn out to be a transformative technology, the key to putting hundreds of trillions of devices in the palm of your hand? Or will we be asking, a few years from now, “Whatever happened to the memristor?”

You won’t find an answer to that question in the column. You won’t find the answer here, either. Instead of trying to predict the future, I’m going to tell a story from the remote past, having to do with a different electronic device, the thermistor. There’s a connection with memristors, although it may take a while to emerge.

•     •     •

Once upon a time I was a nerd with a soldering iron who knew the resistor color code by heart. I had an older friend, Dave, an electrical engineer, whose confidence in my ability was inspiring and empowering even though it was also misplaced. Dave’s company had built an oven for drying the ink on silk-screened fabrics. The customer wanted to monitor the temperature of the cloth as it moved through the oven. Dave gave me the contract to design and build a suitable instrument.

I considered doing it with thermocouples, which I had played with in a physics class. But thermocouples measure the temperature difference between two points, and I had no way to provide a stable reference temperature for the cold junction. Besides, thermocouples produce signals in the millivolt range, and I worried about stray currents in an environment where electric heating elements were drawing about 20 kilowatts. So I turned to a different temperature-sensing device, the thermistor, which has a history going back to Faraday but still seemed to be slightly exotic in the 1960s.

A thermistor is a resistor with an unusually large temperature coefficient—that is, the electrical resistance changes dramatically with temperature. Furthermore, in all the thermistors available in those days, the coefficient was negative, meaning that the resistance decreased as the device was heated. (This is the reverse of the relation in most materials.)

I designed a bridge circuit something like the sketch at right, where the T labels the thermistor and the other resistances are ordinary, fixed resistors. Component values are lost to memory, but I think the supply voltage was 12 or 18 volts and the voltmeter in the middle of the bridge read 5 or 10 volts full scale. The baseline thermistor resistance was a few hundred ohms—dropping from maybe 500 ohms to 250 over the temperature range of interest. I chose the fixed resistance values so that the bridge would balance somewhere below room temperature and produce full deflection of the voltmeter at about 300 degrees Fahrenheit.

The components were ordered by mail from the Allied or the Lafayette Electronics catalog, whose long lists of part numbers and tables of specifications I drooled over in those days. When the package of hardware arrived, I wired up the shiny new parts. Amazingly, it worked. I did a two-point calibration by immersing the thermistor in an ice bath and in boiling water, and I drew a custom scale for the meter. I also lavished attention on a logotype identifying the manufacturer of the instrument; I think I created it with leftover decals from a model-airplane kit.

When it came time to install the gadget, all went well at first. We got a room-temperature reading, then we turned the oven on and watched the needle move slowly across the scale. But later I began to notice a troubling anomaly. The needle kept rising even when the fabric temperature should have stabilized. A few more experiments showed that even with the oven cold, the indicated temperature would very slowly climb to 100 degrees or so, then accelerate on the way up to 150 degrees, and after 10 minutes would be at the top of the scale.

It was a wrenching moment when I realized what was the matter. In designing the bridge circuit I had successfully worked out various applications of Ohm’s Law: E = IR. But I had forgotten all about W = I²R, which defines the power dissipated by current flowing through a resistance. The 10 or 20 milliamps flowing through the thermistor were enough to heat it, thereby lowering its resistance and increasing the current further, bringing still more heating and more current, in a positive feedback loop. I had not observed this behavior in the calibration runs because the water bath efficiently carried away the heat. But the self-heating effect rendered the whole apparatus useless in the fabric-drying oven.

•     •     •

In retrospect, I can look upon this phenomenon and say: How interesting! What a pretty demonstration of nonlinear dynamics. Maybe one could even find a way to put the nonlinear effect to good use, for example by creating a bistable system with low- and high-resistance attractors.

A year or two before these events, a comprehensive review article, “Theory and Application of Self-Heated Thermistors,” by M. Sapoff and R. M. Oppenheim, had appeared in Proceedings of the IEEE (Vol. 51, October 1963, pp. 1292–1305), which set forth exactly these possibilities. If I had read and understood that paper, I would have saved myself a lot of bother. But I knew nothing of the article at the time; indeed I learned of it only a few weeks ago, when I found it cited in various works on memristors.

The foundation paper on the memristor itself appeared just a few years after my mortifying experience with the fabric-drying oven. The article is: “Memristor—the missing circuit element,” by Leon O. Chua, IEEE Transactions on Circuit Theory Vol. 18, September 1971, pp. 507–519 (available online through a link here). Chua, then and now at UC Berkeley, conceived of the memristor and gave it a name almost 40 years before a hardware implementation was discovered and understood in 2008 by R. Stanley Williams and several colleagues at Hewlett-Packard. In the interim, Chua described several memristor-like circuit elements, the first of them being the thermistor.

A memristor is a special kind of nonlinear, variable resistor—one whose resistance at any given moment depends on the entire history of currents and voltages it has experienced in the past. Under conditions where self-heating is important, a thermistor has this very property. As Chua put it in a 1976 paper: “[A] thermistor is in fact not a memoryless temperature-dependent linear resistor—as is usually assumed to be the case—but rather a first-order time-invariant current-controlled memristive one-port.” If only I had known.

So, in the light of history, I return to that teenage kid with good soldering skills but a somewhat shaky grasp of circuit theory. Was he on the verge of inventing the memristor ahead of Chua? No, sadly, not at all. This was not a near miss. At the time, I saw nothing in the outcome of my experiment but a humiliating failure. Looking back, though, I’m inclined to go easy on myself for that missed opportunity. I did manage to understand where I’d gone wrong; I just couldn’t see how to turn my goof into a gold mine.

By the way, the silk-screening outfit did eventually get their temperature monitor. Dave found an off-the-shelf solution, which turned out to work just like my design, but with higher resistances and lower currents, ducking below the thermistor’s region of nonlinear self-heating.

Over the weekend, identical twin sisters Inez Harries and Venice Shaw both celebrated their 100th birthday in California. I heard about this on the TV news, where it was the human-interest teaser story. “What are the odds of that?” the anchorman asked. Then he promised: “We’ll do the math.”

I stayed through the whole broadcast just to see them do the math, but in the end all they gave was an answer, without showing where the number came from: The odds are 1 in 700 million, they said. On the web I found other accounts of the Harries-Shaw birthday party that quoted the same figure, but they were no more helpful about the details of the calculation. One version cited “family members who researched the question.” Another story, discussing another pair of centenarian twins, attributed the number to a spokeswoman for Guinness World Records. I found no supporting information on the Guinness web site. Nevertheless, I suspect that Guinness is indeed the source of the number. An Amazon page for a 2002 edition of Guinness World Records includes the following statement:

The chance of identical twins both reaching and surpassing the age of 100 is about one in 700 million.

What does that mean, exactly? What the words seem to say is this: In a population of 700 million pairs of identical twins, we should expect to find approximately one pair in which both members survive to age 100 or more. (Or should it be 700 million twins, and thus just 350 million pairs?)

I suspect that the author of the sentence actually meant something different: In a human population of 700 million, we should expect to find about one pair of individuals who are identical twin siblings and who also both live to be 100 or more.

Even under the latter interpretation, I was skeptical of this number. My back-of-the-envelope estimates of the probability differed from the Guinness value by orders of magnitude. However, the back of my envelope is notoriously unreliable, especially when it comes to calculating probabilities. I might well have blundered.

Here are some numbers pertinent to the calculation:

• According to twins.com, one out of 250 live births produces monozygotic twins. (I think that means that the proportion of people who have an identical twin sibling is 2/251. Gotta watch out for those pesky factors of 2-ish.)
• According to the Centers for Disease Control, there were 2,809,000 births in the U.S. in 1911. (It is the survivors of that cohort who are turning 100 this year.)
• According to the U.S. Social Security Administration, the survival rate for reaching age 100 or more in the U.S. is 657/100,000 for men and 2,223/100,000 for women.
• According to Wikipedia, which in turn cites the Bureau of the Census, the U.S. had some 70,490 centenarians (age 100 or more) in September 2010.

And here’s how I figured it. Among the 2.8 million births in 1911, there should have been about 11,000 pairs of monozygotic twins. We want to know how many pairs have survived to 2011. We can get an individual survival rate either from the Social Security actuarial table or from the Census Bureau’s count of surviving centenarians. The two coefficients differ substantially—0.014 vs. 0.025—probably because the Census count includes the effect of net immigration. Let’s split the difference and say that 0.02 of the cohort has survived to age 100. Since we are tracking the simultaneous survival of pairs of individuals, we want the square of this number—0.0004. Multiplying the initial number of twin pairs by this factor suggests there should be four or five pairs surviving today. The odds I calculate are roughly 1 in 600,000, not 700 million.

Do you see my error? Yes, I’ve goofed again. But as far as I can tell I’m off only by a factor of 2, not a factor of 1,000. [Please see the comments.]

Lacking all faith in my own competence to do the math, I checked my calculation with a simple-minded computer run. I set up a vector of 2,809,000 bits, designated 11,191 pairs of them as identical twins, killed off bits at random until only 2 percent remained, and finally counted the pairs left standing. In a thousand runs I came up with a mean of 2.23 pairs (and a standard deviation of 1.45). The result of my earlier calculation was 4.48 pairs—just double the simulation outcome. Where did I go wrong? I had been careful to count pairs rather than individual twins in the hope of avoiding just this kind of confusion. But then I went and counted each pair twice! In effect, I was counting Venice and Inez as well as Inez and Venice. Sigh.

Even after correcting my flub, this computation should not be taken too seriously. It neglects a bunch of not-so-subtleties. In particular, I am pretending that the probabilities of all events are independent, when in fact the longevity of identical twin siblings is doubtless very highly correlated. But taking those correlations into account would make the discrepancy between my result and Guinness’s even wider.

My arithmetic—if I have finally done it right—suggests twin-survival odds of roughly 1 in 1.2 million. How did Guinness (or whoever?) come up with 1 in 700 million? My best guess is that they based their calculation on global demographic estimates rather than regional or national statistics. By tweaking the numbers, I can come fairly close to their result.

A U.N. report on aging estimates there are 455,000 centenarians worldwide. The global population in 1911 was apparently somewhere near 1.76 billion. To estimate the cohort size in 1911 we’d need to know the crude birth rate, which I have not been able to ascertain, but extrapolating backward from estimates for later years suggests something in the range of 50 births per thousand people per year. Running these numbers through the mill yields an estimate of 4.7 twin pairs worldwide, or odds of 1 in 375 million.

If I were to boost the 1911 birth rate a little higher, I could arrive at the curious prediction that there are more centenarian twins living in the U.S. than there are on the entire planet. This result does not inspire confidence in the methodology, but it’s also not really surprising. We’re dealing with events far out in the upper tail of a normal distribution. Including populations where the mean age at death is 48 rather than 78 adds more noise than information.

In any event, I toast Inez and Venice on this occasion. And I am happy to note that they are not alone. In rooting around on the web, I’ve found another way to put a lower bound on the twin survival rate. News reports published within the past few months mention other pairs of twins celebrating 100th birthdays in Alabama, Florida and Rhode Island. That makes at least four 700-million-to-1 events in a country of 300 million. Farther afield, there are also recent stories about pairs of centenarian twin sisters in Belgium and the U.K. And Wikipedia has a list of 16 such pairs (not all monozygotic) thought to be still living. Double congratulations to all of them.

Santa and alligators in the hotel lobby: This must be New Orleans. Laptoppers and iPhoners paying no attention whatever: This must be the Joint Mathematics Meeting.

It was a good meeting. I learned stuff. My take-home impression: The world of mathematics is thriving, and everybody seems to be having fun. There’s adventure to be found if you go seek it.

A week later, though, as I mull over the lectures I heard and try to make sense of my notes, not all my thoughts are so jolly and upbeat. One subject in particular leaves me feeling vaguely uneasy, and I can’t seem to let it go.

The seed of doubt was sown in the annual Current Events session organized by David Eisenbud. In this special series of talks, always one of the highlights of the meeting for me, “the speakers do not report on their own work, but survey some of the most interesting current developments in mathematics, pure and applied…. Excellence in exposition is a prime consideration.”

The final talk of the session was given by David Nadler of Northwestern University; he spoke on “The geometric nature of the Fundamental Lemma.” The subject is an apt one for a Current Events talk: The lemma in question was recently proved by Ngô Bảo Châu, who was just named a Fields medalist for this work.

So what is the Fundamental Lemma? Nadler began his talk by saying he wasn’t going to answer that question. He would not work through the proof. He would not even state the lemma. In an hour’s talk, it can’t be done, he said. The slide below is as close as he came.

(The blurriness of this image is entirely my fault, not Nadler’s. But perhaps the fuzziness is not altogether inappropriate. I begin to think that the lemma is so abstruse it can’t even be photographed clearly.)

My aim here is not to gripe about Nadler’s talk, which was in fact a masterful presentation; I found it engaging and entertaining even when the material soared by miles over my head. (A writeup of the talk is available online (3 MB PDF), although it’s quite different from the oral presentation.)

In any event, no one else has done better with this topic. The Wikipedia page isn’t much help. When Paul Basken wrote an account for the The Chronicle of Higher Education last fall, he started out bravely but punted when it came time to state the lemma:

It is an element of an overall structure known as the Langlands Program, a “grand unifying theory” that shows common links between number theory and harmonic analysis, said Edward Frenkel, a professor of mathematics at the University of California at Berkeley.

Thomas Hales has written “A Statement of the Fundamental Lemma” for the Clay Mathematics Institute, but I’m afraid it leaves me utterly unenlightened.

Of course it doesn’t really matter much whether I grasp these ideas. I’m merely an eavesdropper, and I have no right to demand that the conversations of professional mathematicians be tailored to my needs. But in the Current Events session Nadler wasn’t addressing an audience of amateurs like me. He was speaking to a roomful of his peers and colleagues. When even an audience of mathematicians can’t understand the statement of a problem—much less the solution—then I begin to worry about the sustainability of the enterprise.

In his history of 19th century mathematics Felix Klein wrote:

When I was a student, abelian functions were … considered the uncontested summit of mathematics and each of us was ambitious to make progress in this field. And now? The younger generation hardly knows abelian functions.

How did this happen? In mathematics, as in other sciences, the same process can be observed again and again. First new questions arise, for internal or external reasons, and draw researchers away from the old questions. And the old questions, just because they have been worked on so much, need ever more comprehensive study for their mastery. This is unpleasant, and so one is glad to turn to problems that have been less developed and therefore require less foreknowledge—even if it is only a matter of axiomatics, or set theory, or some such thing.

There is some dispute about whether Klein’s obituary for abelian functions might have been premature. But the general process he describes, in which a field of study sinks under the weight of accumulated scholarship, seems like a real hazard in some areas of mathematics.

It may well be that the Fundamental Lemma is so truly fundamental that students will continue to make the necessary investment of “ever more comprehensive study.” But I do hope someone will figure out how to communicate the meaning of this work to the rest of us, or at least to the rest of the mathematical community.

Here in New Orleans there’s nothing to make a snowball out of—not even a semisnowball—but I did hear several talks on snow and balls during the Joint Math Meetings.

An invited address by Yuval Peres was followed by a session of six talks related in one way or another to the theme of diffusion-limited aggregation. This is a subject dear to me; the graphic in the bit-player heading is a product of a DLA simulation (see my old column on the topic).

The snowiest of the DLA talks was by Janko Gravner, whose work on snowfakes I have discussed before. I won’t say more here except to mention one illuminating offhand remark. In discussing Norman Packard’s cellular automaton for modeling snowflakes, Gravner quoted Steven Levy’s book Artificial Life:

An elementary schoolchild could look at any of the gorgeous pictures of computer screens in Packard’s collection and instantly identify it as a snowflake.

What the schoolchild would actually recognize, Gravner pointed out, is not a snowflake but “Christmas iconography.” And surely it’s true: We’ve all seen so many of those lacy doily patterns with hexagonal symmetry that we believe snowflakes really look like that. Some do, but many others are plates or needles or even wheel-and-axle assemblies.

DLA is one way of modeling snowflake growth: Water-vapor molecules drift in from some distant source, wander randomly, and stick fast when they touch any part of the solid snowflake. The feathery patterns in the bit-player header were created by such an algorithm. But most of the talks in the JMM session concerned an inside-out version of this process: Particles (or “chips”) are released at the center and wander across the structure formed by the deposition of earlier chips, stopping as soon as they find a vacant site. What’s intriguing about this “interior” DLA is that the resulting shapes are not at all feathery or lacy. They are compact “blobs.” And if you let the process run long enough, the blob evolves into a ball, with a boundary approximating a circle.

Jim Propp invented a derandomized version of interior DLA called the rotor-router model. Each site in the lattice is equipped with an arrow, and when a chip reaches the site, it exits in the direction indicated by the arrow; then the arrow rotates by some fixed rule, such as turning 90 degrees clockwise. This scheme has many properties of a random walk—the chips are sent in all directions with equal probability—but it is fully deterministic. (Propp was one of the speakers in the DLA session. He promises to put his slides online, presumably here, but when I checked this morning they weren’t yet up.)

The rotor-router patterns are also round blobs, but in addition they have an intricate internal structure that becomes visible when the sites are colored according to the rotor-arrow direction. Tobias Friedrich, who gave another of the talks, has some spectacular examples on his web site. Using the technology of the Google Maps API, Friedrich’s site allows you to zoom into images of billion-pixel DLA patterns.

The big question is: Why so round? It’s not really surprising that the forms created in this was are generally compact, without the voids or branched structures seen in “external” DLA experiments. But the pure circular shape remains unexplained. In these models all the action takes place on a lattice—usually square, but triangular and hexagonal lattices have been tried as well—and you might expect some remnant of the lattice symmetry to show through in the deposited pattern. There’s no sign of it.

On the other hand, Matthew Cook, in the final talk of the session, revealed that although the blobs are very round indeed (within about a third of a pixel, on average), they are apparently not perfect circles. There are persistent bulges, not quite aligned with the lattice axes, that don’t seem to be smoothed away as the number of chips increases.

This is posted from the New Orleans airport, on my way back to the land of real snowflakes.

At most conferences, the sign above might provoke a moment of puzzlement. But this is the JMM, the Joint Mathematics Meetings, being held this year in New Orleans. As far as I can tell, the 5,400 mathematicians gathered here are quite untroubled by these instructions. And after all, they know perfectly well that a whole uncountable continuum lies between floors 2 and 3.

Onward and upward, indeed. And downward. And everything in between.