This month's topics:
Math and the art of mattress flipping
Mattress flipping is one of those household chores that is bothersome because you never know if you are doing it right. Mattress manufacturers recommend periodic flipping for even wear: the four possible combinations of head and foot, top and bottom should receive equal exposure. Ideally there would be a maneuver you can execute each time you flip your mattress such that after four repetitions all four combinations will have been used. Brian Hayes calls such a maneuver a "golden rule" in his treatise on the subject in the September-October 2005 issue of American Scientist, and he gives us the bad news: no such golden rule exists.
Mathematically speaking, there are four ways to rotate a mattress so that it ends up aligned with the bed. Hayes uses the symbols I for the Identity rotation (wait until next week) and R, P, Y for the nautical terms Roll, Pitch and Yaw. Image courtesy Brian Hayes.
His argument runs as follows: no matter how creatively you manipulate your mattress, once it's back on the bed you will have performed one of the four operations I, R, P, Y shown in the figure. Each of these operations has the property that if you repeat it, you end up where you started. So you will have missed two of the configurations.
Hayes goes on to define the mathematical concept of group and to give it content by comparing mattress flipping with another chore: "rotating" (interchanging) the tires on an automobile so that each tire is used, and undergoes wear, in the four different positions. Here there is a "golden rule:" repeating Q (counterclockwise substitution around the outside of the car) four times brings you back to where you started, and each tire will have seen all four positions.
The multiplication tables for mattress flipping (left) and counter-clockwise tire rotation (right). For example, a P (flipping end over end) followed by an R (flipping right over left) has the same effect as a Y (planar rotation by 180 degrees). Each of these tables defines a group with four elements, but the two groups are intrinsically different. Images courtesy Brian Hayes.
The article, available online, ends with some fancier material: the complete group of permutations of four objects, and the group of rotations of a cubical mattress.Virus geometry
A virus is essentially genetic material in a box. The box, or capsid, is assembled from specialized proteins called capsomers. Watson and Crick had observed in 1956, on topological grounds, that viral capsids could be expected to show the regularities of platonic solids. In fact, icosahedral-type symmetry is the most prevalent.
Satellite Tobacco Mosaic Virus (diameter = 168Å), Type 1 Poliovirus (304Å) and Simian Virus 40 (488Å) have different sizes and capsid structures, but all exhibit icosahedral symmetry. Images from Virus Particle Explorer (VIPER), a website for virus capsid structures and their computational analysis.
Recent progress in understanding this bias towards icosahedra was reviewed ("Armor-plated Puzzle") by Peter Weiss in the September 3 2005 Science News. Weiss first describes research by the UCLA team of Roya Zandi, David Reguera, Robijn Bruinsma, William M. Gelbart and Joseph Rudnick (PNAS 101, 15556-15560). This team used Monte-Carlo simulations to find locally energy-minimizing configurations of "pentamers" and "hexamers." As Weiss explains it, "They developed a computer model that treated capsomers as malleable disks. ... Then, by having the computer repeatedly shuffle those disks into arbitrary arrangements on a spherical surface, they simulated the formation of millions of hypothetical capsids. ... To explore all possible ratios of pentamers and hexamers, the researchers also programmed into the process random switching of disks between the two types." The lowest energies occurred with arrays of 12 pentamers, surrounded by 0, 20, 30 and 50 hexamers respectively. These corresponded exactly to the prediction, made in 1962 by Donald Caspar and Aaron Klug, of capsids made of 12 pentamers, or 12 pentameters with 20(T-1) hexamers, where T is one of the series 3, 7, 13, 19, ... of numbers of the form h2 + hk + k2; h and k are integers with (h,k)=1.
The Satellite Tobacco Mosaic Virus (12 pentamers), and the Poliovirus (12 pentamers plus 30 hexamers) fall into this classification, but the Simian Virus 40 does not: every one of its 72 capsomers is a pentamer. Weiss explains how Reidun Twarok (York University) read about this problem and saw how her previous work on quasi-crystals could be applied. "The technique employs some mind-bending concepts, such as a six-dimensional lattice based on a hypercube or other building block. Twarock considered lines and planes projecting from such a lattice onto a three-dimensional sphere representing a viral capsid. ... Exploring the expanded portfolio of possible capsid structures that her tiling method had revealed, Twarock found a tile arrangement for a capsid comprising 72 pentamers and no heptamers." This turned out to be exactly the SV-40 structure pictured above. Her work appeared in the Journal of Theoretical Biology last year (226, 477 - 482), with a more general classification of possible capsid structures available online.Dirac: algebraist or geometer?
The September 15 2005 Nature has an Essay by Graham Farmelo with the title "Dirac's hidden geometry." Farmelo mentions a 1963 interview with Thomas Kuhn in which Dirac describes himself as having a fundamentally geometrical approach. And he quotes a memoir in which Dirac "divided mathematicians cleanly into algebraists and geometers" and said of himself that his preference was "strongly on the side of geometry, and has always remained so." How then can his papers have been so algebraic, with nary a diagram offering "any solace to readers trying to visualize what was represented by the blizzard of abstractions" while Dirac was "steam hammering his way through the mathematics"? Here is Farmelo's conclusion: "Perhaps even Dirac did not fully understand the connection between his private geometry and his public algebra? As Ludwig Wittgenstein insisted at the end of his Tractatus Logico-Philosophicus, 'Whereof one cannot speak, thereof one must remain silent.'" [Good advice. This essay is based on a misunderstanding of what Dirac meant by geometry (and of what geometry has come to mean in modern mathematics, to a great extent because of Dirac). The monopole that sometimes bears his name epitomizes his revolutionary understanding of physical phenomena as having a profoundly geometric meaning. Let's remember the date of this discovery: 1931. The language required to speak of it geometrically had not been invented. -TP]
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