Quantum computing gets a lot of attention, but we don’t hear much about quantum mathematics. The very idea is an affront to Platonist thinkers everywhere—those of us who consider the elements of mathematics to be independent of the physical universe. Is the truth of the Pythagorean theorem subject to the same uncertainty as the fate of Schrodinger’s cat? Surely the counting numbers 1, 2, 3,… should be the same in any universe, whether quantum or classical; indeed, if you’re a full-gospel, whole-Bible Platonist, you might well argue that those numbers existed even before there was a universe. (At the opposite extreme are the constructivists, who warn that the only numbers you can count on are those you have fingers for.)

Questions like these are easily settled by experiment: Just annihilate the universe, abolish space and time, and try doing a few sums in the void. I would put the proposition to the test right now except that I don’t want to miss this week’s episode of “1 vs. 100.”

Paul Benioff of Argonne National Laboratory has been thinking and writing about these issues for some time. A few weeks ago he posted a preprint on the arXiv titled “Space of quantum theory representations of natural numbers, integers, and rational numbers.” I’ve been struggling to understand that paper, along with a couple of earlier ones on the same theme (here and here). I’m still a long way from mastering this material, but here’s what I’ve made of it so far.

Even if you go along with the Platonist view that the true home of all things mathematical is an ideal realm outside of time and space, we don’t live in that realm. Whenever we want to do something with numbers (or other mathematical entities), we have to represent them somehow in this universe. We chalk them on the blackboard; we twiddle the beads of an abacus; we load patterns of bits into a computer memory; even mental arithmetic requires a mind, which in turn seems to require a body. Thus no one does mathematics without atoms and photons and other toys and tools of the physicist, and so perhaps it’s not utterly crazy to suppose that what we can accomplish in mathematics may depend to some extent on the laws of physics. All the evidence suggests that those laws are inescapably quantum mechanical.

Benioff argues that numbers have to be represented and manipulated in ways that accord with quantum principles; for example, all operations should be reversible, and they should enforce “unitarity,” meaning that the probabilities of all possible outcomes should sum to exactly 1. At the same time, it’s important that numbers behave like numbers: They have to obey familiar axioms of arithmetic, or they’re of no use to mathematics. In the case of the counting numbers (a.k.a. the natural numbers) we have to be able to add and multiply. For the integers (the natural numbers augmented with zero and negative values), subtraction must also be allowed. The rationals introduce division. And for all these kinds of numbers we should be able to determine if two numbers are equal, and put them in order if they are not.

Quantum computing is usually discussed in terms of qubits, or the quantum equivalent of binary digits. Benioff extends the discussion to qukits, the quantum equivalent of base-k digits. Think of a qukit as a black box that holds one of k values, but until you open the lid, you can’t be sure which value. Once you look inside, the uncertainty is resolved, and the box is found to contain a specific value. The question Benioff addresses is: How do you do arithmetic with such unruly numbers? If you can’t even be sure of what numbers you’re working with, how can you add or subtract them, or test them for equality? The answer, ultimately, is that you have to put up with a degree of uncertainty. In the quantum world, the commutative law of addition, a+b = b+a, is not a bedrock principle but a statement whose truth is a matter of probabilities.

Benioff presents a specific implementation of quantum-theoretical numbers, based on a two-dimensional lattice of qukits. Each number occupies its own row of the lattice, with the digits arrayed from left to right. For integers, a designated qukit (actually a qubit, since it has just two states) serves as a sign bit. For rationals, this special qubit also marks the position of the decimal point (or “k-al” point), separating the integer part from the fractional part. On first glance, this lattice of qukits doesn’t seem too different from the hardware of an ordinary computer, but the quantum nature of the qukits brings some peculiarities. For example, the result of every operation has to go into a newly allocated row of the lattice; you can never erase or overwrite an existing value, because quantum operations have to be reversible and cannot destroy information.

Benioff introduces a set of parameters for his numbering system: m and h are the coordinates of a number within the lattice of qukits, k is the base of the qukits, and g is a “gauge-fixing function.” This last item sounds quite arcane, but I think it has a fairly simple explanation. You can imagine a qukit as a vector that can point in any of k directions; the gauge-fixing function defines a reference direction from which all the others are measured.

Here’s a taste of what Benioff has to say about his quantum numbers:

Transformations (k, (m, h), g) → (k’, (m’, h’), g’) in the parameter set induce transformations in the representation space. These consist of unitary translations that move the qukit strings on the lattice, transformations that change states of strings of base-k qukits to states of strings of base-k’ qukits, and unitary gauge transformations for each k….

An interesting result is that the axioms and theorems for each of the three types of numbers [i.e., natural numbers, integers and rationals] are invariant under these transformations. They represent symmetries of the systems. This is the case even though the specific expressions of the axioms and theorems in terms of basic arithmetic relations and operations are different for different representations. This is like the situation in physics where the laws of physics are invariant under Lorentz transformations even though their specific expression in different reference frames may be different.

Another interesting result is that qukits q_k where k is a prime number function as elementary qukits. These are the “elementary particles” as far as quantum representations of numbers are concerned. Qukits where k is not prime can be considered as composites of the prime number q_k.

If you want to delve more deeply into these matters, I must send you to Benioff’s paper. Here I want to return to the broad question of whether this line of inquiry can really lead us to some kind of quantum mathematics, as opposed to an abstract version of quantum computing. Personally, I’m not quite persuaded, although I find the proposition intriguing.

In most of this work the focus is on the representation of numbers, rather than the numbers themselves—a preoccupation that seems more characteristic of computing than of mathematics. Interesting mathematical properties of numbers tend to be independent of their representation; for example, the number seven is a prime whether you write it in decimal notation as 7, in binary as 111 or as the Roman numeral vii. Of course you must choose some representation, and the choice can make a difference in what you can accomplish. Benioff points out that unary notation (in which seven becomes 1111111) is inherently less efficient than other schemes, because the amount of work expended in manipulating a number is proportional to the number itself rather than to the logarithm of the number. For similar reasons Benioff objects to building all of arithmetic on the successor function (n → n+1). But this fretting over efficiency again suggests a more computational than mathematical frame of mind. After all, unary numbers and the successor function are essential building blocks in theories of the foundations of mathematics, such as in the Principia Mathematica of Whitehead and Russell—who were blissfully unconcerned with efficiency.

If you accept that the physical representation of mathematical objects is a fundamental issue in mathematics, then it’s an easy step to the conclusion that any such representation has to be consistent with quantum principles. But is the mathematical imagination truly fettered by the bounds of the physical universe? Mathematicians routinely reason about objects and operations that have no explicit material representation—irrational numbers, for example, or Georg Cantor’s infinite sets. A century ago, in response to another challenge from those who wanted to fence in the scope of mathematics, David Hilbert defiantly proclaimed: “No one shall expel us from the paradise that Cantor has created for us.”

If I remain a tad doubtful about the mathematical status of quantum-theoretical numbers, I do think they offer an interesting perspective on quantum computing. So far, no one has succeeded in building a practical quantum computer with a large number of interacting qubits (or qukits). Technological skeptics contend it will not be done any time soon. Benioff’s work turns this argument on its head. Quantum computers are the only ones we can build, he says, because we live in a world where quantum physics is the law of the land. We may think we have classical computers, but that’s an illusion. We merely have quantum computers that are heavily biased toward specific classical outputs, but they always retain the possibility of delivering a quantum surprise. Usually we regard computing—whether it’s done with a machine or with pencil and paper—as an approximation to a mathematical ideal. If a calculation shows that a+b does not equal b+a, we don’t question the commutative law of addition; we look for a bug or an error. But maybe we need to consider the possibility that it’s the quantum computation that’s the ultimate reality, and the mathematical law is just our convenient and tidy approximation to it.

Doing some laundry last night, I threw a duvet cover and nine pairs of socks into the dryer together. (Household hint: Don’t.) The duvet cover is a giant fabric pouch with a slit along one side; think of a queen-size pita pocket. Initially, all the socks were outside the pouch. When I pulled the load out of the dryer, all but three of the socks were inside the cover.

There’s nothing obvious about the geometry of this big, floppy bag that would suggest it has any special ability to capture socks. It’s not shaped like a fish trap with a funnel opening. In the random tumbling of the dryer, I would have thought that socks would move into or out of the opening with the same probability. But if that’s the case, then finding a 15–3 distribution is quite a fluke. There are 2¹⁸ = 262,144 ways of arranging the socks in two groups, and only 5,220 of them have three or fewer socks in one of the groups. That’s less than 2 percent and a little beyond the 2σ level of unlikelihood. Does Maxwell’s sock demon live in my dryer?

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

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

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

I can cook anything, as long as the recipe starts with “Take it out of the freezer” and ends with “Put it in the microwave.”

Go ahead and scoff, but I’m proud of my culinary accomplishments. Furthermore, I submit that the art of microwaving frozen foods is not without intellectual challenge. Inferior technique could leave some bits of your burrito still frozen while other parts are overcooked. The underlying cause of this well-known problem has lately been explored in depth and detail through computer simulations done by Motohiko Tanaka and Motoyasu Sato of the National Institute for Fusion Science in Toki, Japan. They present their results in an arXiv preprint.

The first mystery of microwave cookery is that it works at all. To heat a substance, you have to agitate the molecules, augmenting their random motions. Radiation at visible or infrared wavelengths does a good job of this: A visible or infrared photon is absorbed by a single molecule, which then goes bouncing off in some unpredictable direction. But microwaves are orders of magnitude too large and slow and weak to stimulate individual molecules. (In this respect the prefix micro is misleading; the wavelengths range from about a centimeter up to 10 or 20 centimeters.) A microwave’s energy is spread out over many millions of molecules—a situation that doesn’t seem like a very promising way to get them all moving in different directions.

One clue to how microwaves induce heating is what you’re not supposed to put in the microwave oven: aluminum foil. Metals have an abundance of free electrons, which are accelerated by the electric field of the microwaves. This field changes polarity a few billion times per second, and so the electrons slosh back and forth through the foil at that frequency. This motion of the electrons does not in itself constitute heating because it is not random; on the contrary, it is a highly organized oscillation—an alternating electric current. But the oscillating electrons collide with imperfections in the metal, and soon the orderly movement degrades into heat.

So much for tinfoil; most of us eat little in the way of metallic food. And nonmetals do not have an abundance of electrons (or other charged particles) free to move under the influence of an alternating electric field. What foods have is water. Although the H₂O molecule is electrically neutral, it has positive and negative poles, where opposite charges congregate (most of the positive charge on the hydrogen atoms, most of the negative charge on the oxygen). When this dipole structure is immersed in an electric field, it takes up a preferred orientation, antiparallel to the applied potential. Since the microwave field changes direction at a frequency of a few gigahertz, the water molecules must continually flip or spin to keep pace. As with the sloshing of the electrons in tinfoil, this rotation of the water molecules can’t be considered heat because it’s too orderly; all the molecules are twirling in synchrony. But in liquid water the molecules are tightly packed, and as they spin they bump into one another like dancers on a crowded floor. These collisions randomize the motion, and so the temperature of the water rises.

Tanaka and Sato study this process in a simulated volume of water measuring about 40 Ångstroms on a side and containing about 2,700 water molecules. The molecules are attracted to one another through electrical forces, but there’s also a “hard core” repulsion that prevents them from getting too close. In the simulation, all of the forces acting on each molecule are recalculated every femtosecond (10^–15 second), after which the positions and orientations of the molecules are updated. The system is allowed to come to equilibrium at a specified temperature during an initialization phase that lasts for a simulated time of 100 picoseconds (100 × 10^–12 second), and then the microwave field is turned on for 500 picoseconds. Tanaka and Sato use an unrealistically intense field—about a million volts per centimeter—in order to speed up the process. (Although the simulated time is only about half a nanosecond, the actual running time on a cluster of four-processor Pentium workstations is 48 hours.)

In liquid water at 300 Kelvins (roughly room temperature), Tanaka and Sato find that microwave heating is quite efficient: The colliding, spinning molecules raise the temperature to about 350 Kelvins. But here’s the problem: In ice, unlike liquid water, Tanaka and Sato see almost no heating. The reason is that water molecules in an ice crystal are immobilized by strong electrostatic bonds, and microwaves have too little energy to break them free. In the oscillating microwave field, the ice molecules wobble back and forth a bit, but they cannot twirl, and thus they cannot collide. Tanaka and Sato don’t explicitly discuss the culinary implications of their work, but the inference is obvious: It’s because of this icy recalcitrance to microwaves that nuking a frozen burrito takes as much skill as baking a perfect soufflé or whipping up a sauce Bearnaise.

Interestingly, microwaves also lose much of their sizzle when water is superheated to 400 Kelvins. Under these conditions, the water molecules are easily set to spinning, but the bonds between them are so feeble that the rotation is not converted into random, thermal motion. I am tempted to see a certain philosophical significance in this curious behavior. There’s been much written about the specialness of the water molecule—most notably the geometric quirk that makes solid ice lighter than liquid water. If it were otherwise—if rivers and lakes and oceans could freeze from the bottom up—life would have had a hard time getting established on planet Earth. Now we know that modern slacker civilization also depends on a peculiarity of the water molecule. If we didn’t have this glorious interval of susceptibility to microwaves in a narrow window of temperatures near 300 Kelvins, I’d have to survive on Poptarts in the toaster oven.

It’s no secret that the way to win fame and fortune in physics is to invent a better refrigerator. Michael Faraday and James Prescott Joule and J. J. Thomson (Lord Kelvin) were all thinkers or tinkerers in refrigeration; the Kelvinator brand alludes to the last of those pioneers. Einstein and Leo Szilard held dozens of patents on refrigerator designs. And workers in cryogenics have won at least 10 Nobel prizes, starting with Heike Kammerlingh Onnes in 1913.

Here’s the latest cool idea on how to chill out: Let the hot atoms in a fluid escape through nanopores that block the lower-energy atoms. William J. Mullin of the University of Massachusetts in Amherst and Neal Kalechofsky of Oxford Instruments America Inc. suggests three ways this might work. The abstract of their recent paper:

We consider the possibility of adding a stage to a dilution refrigerator to provide additional cooling by “filtering out” hot atoms. Three methods are considered: 1) Effusion, where holes having diameters larger than a mean-free path allow atoms to pass through easily; 2) Particle waveguide-like motion using very narrow channels that greatly restrict the quantum states of the atoms in a channel. 3) Wall-limited diffusion through channels, in which the wall scattering is disordered so that local density equilibrium is established in a channel. We assume that channel dimensions are smaller than the mean-free path for atom-atom interactions. The particle waveguide and the wall-limited diffusion methods using channels on order of the deBroglie wavelength give cooling. Recent advances in nano-filters give this method some hope of being practical.

The plan sounds a little like Maxwell’s demon without the demon. The nanopores don’t have to be opened and shut for individual atoms; they discriminate naturally between various energy states of the atoms. Or at least that’s what Mullin’s and Kalechofsky’s calculations suggest; the crucial experiment has yet to be performed.

arXiv link: Theory of cooling by flow through narrow pores.

Update 2006-05-16: Oops. Lord Kelvin = William Thompson. Lord Kelvin ≠ J. J. Thompson.

Scientists are selfless seekers after truth, unswayed by worldly emoluments, immune to the tawdry enticements of fame, indifferent to prizes and honors. Thus I can’t quite imagine why anyone would bother ranking a collection of scientific papers by applying the algorithm that Google uses to decide which Web pages deserve the most prominent display. Nonetheless, if you’re an author of anything published in Physical Review or its various offshoots over the past century or so, P. Chen, H. Xie, S. Maslov and S. Redner have calculated the Google PageRank of your publications. The database they analyzed consists of 353,268 papers published in various American Physical Society journals between 1893 and 2003.

The best-known measures of “impact” for scholarly publications are based on counting citations; a paper’s impact rises when other papers refer to it. The PageRank algorithm applies this principle recursively. If paper A is cited by paper B, that fact will be weighted more heavily if paper B itself gets many citations. Chen et al. adapted this method of evaluation to the 3,110,839 citations within their network of Physical Review articles. They found that most of the papers with a high PageRank score are well-known works that also get high marks under other kinds of citation analysis. But there were some surprises—papers they call scientific gems. An example is a 1980 article by H. Rosenstock and C. Marquardt titled “Cluster formation in two dimensional random walks: Application to photolysis of silver halides.” Only three other articles in the database cite Rosenstock and Marquardt, and thus it remains an obscure publication from the point of view of most citation analyses; but it rises to 85th place in the PageRank standings because one of those three citing papers is itself a highly rated work. (The latter paper, by T. Witten and L. Sander, is “Diffusion-Limited Aggregation, a Kinetic Critical Phenomenon,” with 680 citations as of June 2003. Chen et al. note: “The Witten and Sander article has only 10 references; thus a substantial fraction of its fame is exported to [Rosenstock and Marquardt] by the Google PageRank algorithm.)

Unfortunately, the database used by Chen, Xie, Maslov and Redner is not publically available, and so there is no convenient way for physicists to check their own PageRank standings. But, then again, physicists are so egoless, I’m sure they wouldn’t bother anyway.

See: Finding Scientific Gems with Google, on the arXiv.

Every morning I go fishing in the arXiv. Or at least that’s the way I’ve been thinking about this daily ritual: I cast my net over the waters and look to see what strange and wonderful creatures I’ve brought up from the deeps. Today it hit me that I have the metaphor backwards. I’m the fish, and what’s going on here is that the arXiv dangles lures in front of me to see if I’ll take the bait. Some days I’m just not biting. Today, however, I was snapping at one hook after another. I’ll be the first to admit that I sometimes choose a brightly colored bit of fluff in preference to a nutritious worm. (Although I did pass on today’s proof that P ≠ NP—or was it the converse?)

physics/0603229
Title: Laws of Graph Evolution: Densification and Shrinking Diameters
Authors: Jure Leskovec, Jon Kleinberg, Christos Faloutsos

How do real graphs evolve over time? What are “normal” growth patterns in social, technological, and information networks? Many studies have discovered patterns in static graphs, identifying properties in a single snapshot of a large network, or in a very small number of snapshots; these include heavy tails for in- and out-degree distributions, communities, small-world phenomena, and others. However, given the lack of information about network evolution over long periods, it has been hard to convert these findings into statements about trends over time. Here we study a wide range of real graphs, and we observe some surprising phenomena. First, most of these graphs densify over time, with the number of edges growing super-linearly in the number of nodes. Second, the average distance between nodes often shrinks over time, in contrast to the conventional wisdom that such distance parameters should increase slowly as a function of the number of nodes (like O(log n) or O(log (log n))).

cond-mat/0603718
Title: Statistical Mechanics of Community Detection
Authors: Joerg Reichardt, Stefan Bornholdt

Starting from a general ansatz, we show how community detection can be interpreted as finding the ground state of an infinite range spin glass…. The community structure of the network is interpreted as the spin configuration that minimizes the energy of the spin glass with the spin states being the community indices. We elucidate the properties of the ground state configuration to give a concise definition of communities as cohesive subgroups in networks that is adaptive to the specific class of network under study. Further we show, how hierarchies and overlap in the community structure can be detected….

physics/0603215
Title: Computer simulation of language competition by physicists
Authors: Christian Schulze, Dietrich Stauffer

… About every ten days a human language dies out, and in Brazil already more than half of the indigenous languages have vanished as a result of the European conquest. On the other hand, Latin has split in the last two millennia into several languages, from Portuguese to Romanian…. Thus similar to biology, also languages can become extinct or speciate into several daughter languages.

In contrast to biology, humans do not eat humans of other languages as regular food, and thus one does not have a complex ecosystem of predators eating prey as in biology. Instead, languages are meant for communication, and thus there is a tendency of only one language dominating in one region, like German in Germany etc. Will globalisation lead to all of us speaking one language in the distant future? For physics research, that situation has already arrived many years ago….

Thus in the history mankind we may have had first a rise, and later a decay, in the number of different languages spoken. In Papua New Guinea there are now 10³ languages, each spoken by about 10³ people; can this situation survive if television and mobile phones become more widespread there?

While we cannot answer these questions, we can at least simulate such ”survival of the fittest” among languages, in a way similar but not identical to biology….

physics/0510151
Title: Trainspotting: Extraction and Analysis of Traffic and Topologies of Transportation Networks
Authors: Maciej Kurant, Patrick Thiran

The knowledge of real-life traffic pattern is crucial for good understanding and analysis of transportation systems. This data is quite rare. In this paper we propose an algorithm for extracting both the real physical topology and the network of traffic flows from timetables of public mass transportation systems.

math.SP/0603630
Title: Sharp bounds for eigenvalues of triangles
Authors: B. Siudeja

The purpose of this paper is to prove the following theorem.

Theorem 1.1. Let T be a triangle in a plane of area A and perimeter L. Then the first eigenvalue λT of the Dirichlet Laplacian on T satisfies

The constants 9 and 16 are optimal.

math-ph/0603065
Title: Quasicrystals: algebraic, combinatorial and geometrical aspects
Authors: Edita Pelantová, Zuzana Masáková

math-ph/0603068
Title: A Spinorial Formulation of the Maximum Clique Problem of a Graph
Authors: Marco Budinich, Paolo Budinich

In this paper we propose a new representation of the maximum clique problem in complex space. After a brief review of this famous NP-complete problem, we show how the adjacency matrix of a graph can be expressed as the square of a symmetric complex matrix. The vectors forming this matrix have zero length and Cartan has shown that this geometry can be treated elegantly with spinors…. We finish with a formulation of the maximum clique problem in this formalism and show that each graph uniquely identifies a spinor whose properties surely deserve deeper studies.

Still more on reversible and zero-energy computing (see earlier bit-player posts here and here, and the American Scientist column):

M. Maissam Barkeshli of the University of California at Berkeley has a preprint titled “Dissipationless Information Erasure and Landauer’s Principle.” (The paper was first submitted to the arXiv last April, but I missed it then, and noticed it today only because it has just been updated.) Barkeshli’s “dissipationless erasure” does not challenge the basic premise that a reversible computer could operate (in principle) without energy loss, whereas an irreversible computer must dissipate at least some energy. He argues, however, that the energy-dissipating step in an irreversible machine need not necessarily be the erasure of a bit of information, as Rolf Landauer first suggested in 1961. Barkeshli describes a hypothetical computer technology in which erasing is free but writing a new value takes energy. The energy cost for the entire cycle remains the same.

When I was a kid, there were no toys I treasured more than magnets. I had dozens of them: horseshoes, bars, a couple of powerful alnico cylinders salvaged from old loudspeakers. The invisible but very palpable forces acting between these objects—pushing like poles apart, drawing unlike together—were signs to me that mystery still exists in the universe. And yet magnetism also taught me that mysteries can be unraveled and understood. In some book I borrowed from the public library (maybe it was a biography of William Gilbert?) I read an explanation of magnetism that seemed to make sense. Imagine an array of many thousands of magnetic compass needles, all packed tightly together, the book suggested. Each such needle will tug on its neighbors as the magnetic fields interact, and so the needles throughout the array will all tend to line up in parallel. This is how a permanent magnet (or ferromagnet) works, the book said; the imagined tiny compass needles represent individual atoms in magnetic materials such as iron.

There are two problems with this story. First, it fails to explain where the atomic magnetic fields come from. Are we to imagine inside each atom an array of even tinier compass needles? Second, the explanation is just plain wrong. The atomic-scale magnetic dipoles in a ferromagnet do not line up because of magnetic interactions like those between compass needles. The actual interatomic forces are quite different; they are short-range, quantum-mechanical interactions that have no direct counterpart in the world of macroscopic objects.

I have known the truth about ferromagnets for some time, but the debunking of the many-tiny-compass-needles story leaves another question still murky. If an array of compass needles is not a good model of a ferromagnet, what does happen when you bring a bunch of magnetic compasses close together? A recent paper by six authors in Japan, India and the U.S. answers this question in the most direct way possible—through experiments with real compasses. They used small, spherical, liquid-filled compasses meant for mounting on a car windshield.

Before reading on, you might try to guess the outcome of their experiments.

When the compasses are arranged in a line—a one-dimensional array—the needles do tend to line up head-to-tail, all parallel to the line. Even with as few as two compasses, this interaction is strong enough to overcome the influence of the earth’s magnetic field.

But a two-dimensional, square array behaves differently, and not at all like a ferromagnet. In fact, the square lattice of compasses is an antiferromagnet, with nearby elements pointing in opposite directions. Even more surprising, the antiferromagnetic lattice is twisted 45 degrees with respect to the underlying lattice of compass needles. If the edges of the compass array are parallel to the east-west and north-south axes, then the compass needles all point along the diagonals. Sets of four adjacent compasses form a sort of loop, with needles pointing northeast, southeast, southwest, northwest. The paper explains this curious structure as a superposition of two simpler antiferromagnetic states: In one of these states, alternating rows of the latttice are oriented east and west; in the other, alternating columns are directed north and south.

The paper, Ferroics: magnetic-compass lattice and optical phonon dispersions of dipolar crystals, is available from the arXiv and has been submitted to The American Journal of Physics. The authors are Takeshi Nishimatsu, Umesh V. Waghmare, Yoshiyuki Kawazoe, Benjamin Burton, Kazutaka Nagao and Yoshihiko Saito.