This month's topics:
The British physicist J. J. Thomson won the Nobel Prize in 1906 for (among other achievements) his discovery of the electron. Two years before, he posed an essentially mathematical problem, completely natural but very difficult in general: what is the lowest-energy configuration of N classical electrons (repulsion proportional to 1/r2) on a spherical surface? For small N exact solutions are known; some of them (4, 6, 12) correspond to platonic solids. For large N numerical simulations have generated candidates for the best possible arrangement. Recent work of David Wales, Hayley McKay and Eric Altschuler (Physical Review B79 224115) has extended the range of N for which good candidates are known from 400 to over 4000. In a minimum-energy configuration each electron has 6 nearest neighbors, except for disclinations --points at which an electron has 5 ("topological charge" +1) or 7 (charge = -1). Setting Ci as the number of nearest neighbors of the ith electron, the network of nearest-neighbor bonds forms a triangulation of the sphere with V=N vertices, E = (1/2)Σi Ci edges and F = 2E/3 faces. Euler's Theorem V - E + F = 2 yields Σi(6 - Ci) = 12. The sum of the topological charges must be 12. The problem is where to put the disclinations so as to minimize the total energy.
Candidate minimum energy configurations for N = 400, 732, 1632; nearest-neighbor electrons are represented by adjacent polygons. As N increases, larger numbers of heptagons are enlisted to reduce strain. These configurations manifest approximate icosahedral symmetry; for 1632 the symmetry is exact. Images courtesy of David Wales.
One interesting (but disappointing to the platonists among us) discovery reported here is that the most symmetrical configuration is not always the least-energy one. The potential energy is geometrically defined as V = Σi<j |ri - rj|-1, where the ith electron is represented by the unit vector ri in R3. For N=4352 the configuration shown below has V = 9311276, while the configuration with 12 symmetrically placed rosettes (like N=1632 above) occupies a local minimum with V = 9311299.
"Talk of the Town" in the August 3 2009 New Yorker takes us along with the narrator (Nick Paumgarten) on a mathematical walk through midtown Manhattan. The walk was one of those organized by Glen Whitney to publicize his plans for a math museum. "The idea behind the museum ... is that math is ubiquitous, supercool, underappreciated, poorly taught and even more poorly learned." The ubiquitous and supercool part is what Whitney accomplishes on the walks, by pointing out the "math-y bits" of the cityscape. This one started at Lincoln Center, where he explained how "the hyperbolic paraboloids of the grandstand [outside Alice Tully Hall] employed straight lines to form a curved surface." The Philip Johnson clock near Columbus Avenue "led to a disquisition on Pythagoras, octaves, calendars, eclipses, and time." The pentagonal lug nuts on fire hydrants, the timing patterns of the city's stoplights, math was everywhere. The "poorly taught and even more poorly learned" part comes after the end of the tour (in sight of the Columbus Circle unisphere -- Eratosthenes, Christopher Columbus, the measurement of the earth). "In Whitney's view, the standard progression ... --algebra, geometry, trig, pre-calculus, calculus-- is random and baseless, a linear conceit that creates a false sense of increasing difficulty." And the number sense essential to understanding our physical and social world ("The lottery is a tax on the mathematically illiterate.") is developed nowhere.
"Beware the Idolatry of Numbers" is a piece by Ben W. Heineman, Jr., posted August 11, 2009 in The Atlantic online. Heineman examines "how the false idolatry of numbers and systems can lead people, institutions and nations far astray--with catastrophic results." His first example is the Vietnam War, where systems analysis, championed by then Secretary of Defense Robert McNamara, paved the way to disaster: "numbers like body counts, targets hit, enemy forces captured, weapons seized, tons of bombs dropped, and hamlets protected were used to argue that the war was being prosecuted successfully." The second is the recent financial meltdown. Heineman cites Gillian Trett's recent book Fool's Gold as a fine-grained account of how a small group of bankers at J. P. Morgan "(many with degrees in math and computer science)" devised the financial instruments, based on attractive numerical models, that eventually led to the near-collapse of the banking system. As Heineman reminds us, in both cases mathematical models were applied "in the context of complex human society, where some key factors exist that cannot be quantified."
More about Platonic (and Archimedean) solids in the scientific news: Nature's cover story for August 13, 2009 was "Dense packings of the Platonic and Archimedean solids," by Sal Torquato and Yang Jiao (Princeton; Jiao is a graduate student in Mechanical and Aerospace Engineering). Now that the 3-dimensional sphere packing problem has been solved, the question arises of how best to pack many copies of other solids, and the canonical regular solids are obvious candidates. For cubes there is no problem, but for the others, which do not tile space, the solution is not obvious. Torquato and Jiao devised a 2-pronged optimization process (the adaptive shrinking cell or ASC method) in their search for dense packings: they varied the relative positions of solids in a parallelepipidal cell, and allowed the cell shape to vary to accomodate the more efficient arrangements.
One step in the optimization process for achieving a dense tetrahedron packing: the cell geometry is modified, allowing a more efficient placement of the tetrahedra inside. Images from Nature 460 876-879, used with permission.
Torquato and Jiao report, after meticulous numerical experimentation, that for the octahedron, the dodecahedron and the icosahedron, the densest known packings appear to converge on the Bravais lattice packings: the solids, identically oriented, have their centroids at the vertices of a regular lattice in 3-space (i.e. the set of iA + jB + kC, where i, j, k are integers and A, B, C are three given linearly independent vectors). For the tetrahedron the situation is different: it is too far from round for Bravais packing to be efficient. The ASC algorithm leads to an irregular assembly of 72 solids in each cell of a regular lattice, and thereby sets a new world record for density of tetrahedron packing.
Densest known packing for octahedra (a) and tetrahedra (b). The octahedra are all identically oriented, and spaced regularly. For tetrahedra Torquato and Jiao achieve a new density record by packing 72 solids (as shown) in each cell of a regular lattice; the configuration has no smaller-scale regularity. Images from Nature 460 876-879, used with permission.
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