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The Schwarz-Christoffel formula (top) gives a conformal map from an arbitrary polygon to the unit disc. The generalization published in March 2007 by Darren Crowdy (below) applies to polygonally bounded domains of arbitrary topology. Image after Crowdy.
The Riemann mapping theorem guarantees a conformal map between any proper simply-connected planar domain and the open unit disc. In general, the map is constructed as the limit of an infinite process; but in case the domain is a polygon, an explicit, finite formula was found in the 1860s by Schwarz and Christoffel. An item by Adrian Cho, published March 6, 2008 on Science's "Science Now" website, covers the publication a year ago (Math. Proc. Camb. Phil. Soc. 142 (2007) 319) of a generalization of the Schwarz-Christoffel formula to multiply-connected polygonal domains. The author was Darren Crowdy (Imperial College London); Cho quotes him: "If you give me any polygon with any number of polygonal holes, I can map it to a circle with the same number of circular holes." Crowdy's discovery "has been creating a buzz this week with coverage in several newspapers in the United Kingdom." For example, "140 Year-Old Schwarz-Christoffel Math Problem Solved" on scientificblogging.com. The point of Cho's piece, however, is not the mathematics but a priority controversy. Thomas DeLillo and Alan Elcrat (Wichita State), together with John Pfaltzgraff (Chapel Hill) published "Schwarz-Christoffel Mapping of Multiply Connected Domains" in the Journal d'Analyse (94 (2004) 17-47), and claim their share of the glory. According to Cho, "The Americans' formula ... involves the multiplication of an infinite number of terms, which goes haywire if the holes are too close together." Crowdy asserts that his method, which "replaces that product with an obscure beast known as Schottky-Klein prime function" (in Cho's words) is more reliable. Pfalzgraff is "very skeptical." Cho ends on a conciliatory note by quoting Michael Siegel (NJIT Newark) "It's a breakthrough, and all these people contributed." Cho's title: "Mathematicians Debate the Hole Truth."
50 generations of midge population in Lake Myvatn. The solid line represents obsevations, the dashed line output from the mathematical model with nine tuned parameters. Image courtesy of Anthony Ives.
"Mathematics Explains Mysterious Midge Behavior" is the title of an article by Kenneth Chang in the March 7 2008 New York Times. At Myrvatn ("Midge Lake") in northern Iceland, during mating season, the air can be thick with male midges (Tanytarsus gracilentus), billions of them. Chang quotes Anthony Ives (Wisconsin) "It's like a fog, a brown dense fog that just rises around the lake." And yet in other years, at the same time, there are almost none. Ives was the lead author on a report in Nature (March 6 2008) that gave an explanation for this boom-and-bust behavior in which, as Chang describes it, "the density of midges can rise or fall by a factor of a million within a few years." In the Nature report ("High-amplitude fluctuations and alternative dynamical states of midges in Lake Myvatn"), Ives and his co-authors characterize the midge ecology as one "driven by consumer-resource interactions, with midges being the consumers and algae/detritus the resources" and they set up a system of three coupled non-linear difference equations, one each for midges, algae and detritus, to model it. The dynamics of this system include a stable state as well as a stable high-amplitude cycle; small variations in parameters can drive the system from one of those attractors to the other.
Alternative stable states of the midge-algae-detritus model. This image shows the tangent plane to the manifold containing the cyclic component of the dynamics around the stationary point. The blue pentagon shows the unstable period 5 cycle that makes up part of the boundary between domains of attraction to the inner invariant closed set and the outer stable cycle. The outer stable cycle is out of this picture. Image courtesy of Anthony Ives.
Eric Altschuler and Antonio Pérez-Garrido published an article in Physical Review last year (E 76 016705 (2007)) in which they described "a four-dimensional polytope, new to our knowledge, with a high degree of symmetry in terms of the lengths of the sides." They found the configuration "by looking at the ... problem of finding the minimum energy configuration of 80 charges on the surface of the hypersphere S3 in four dimensions" with the energy function Σ(1/rij) where rij is the distance between the i-th and j-th points, and the sum is taken over all pairs of distinct points. (They remark that they cannot prove this is actually a global minimum, but add that "even good local minima can be interesting or important configurations.") The other N for which they found symmetric configurations are 5, 8, 24 and 120; corresponding to the 4-simplex, the dual of the 4-cube, the 24-cell and the 600-cell. The authors give a method for visualizing their 80-vertex polytope in terms of the Hopf map S3 --> S2. They triangulate S2 with 16 equal equilateral triangles: 4 abutting the North Pole, 4 the South, and a band of 8 around the Equator. This polyhedron has 10 vertices. Each of these vertices corresponds to a circle of the Hopf fibration, along which they describe explicitly how to place 8 of the polytope's vertices.
Two from a sequence of 20 projections of the 80-vertex polytope from 4-space into the plane. Each of the 10 Hopf-fibration circles has a different color, and appears as an octagon linking its 8 polytope vertices. Entire sequence, each projection composed with an additional rotation by 30o about a fixed plane in 4-space, available here. Images courtesy of Eric Altschuler.
Another description of the 80-vertex polytope was published by Johannes Roth later in the same journal (E 76 047702 (2007)).
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