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This month's topics:

The math beneath the Interplanetary Superhighway

 The Genesis mission parked in a halo orbit while it studied the solar wind, then looped around L2 for a leisurely and inexpensive trip back to Earth.Image by Shane Ross (USC), used with permission.

The cover story for the April 16 2005 Science News was Erica Klarreich's "Navigating Celestial Currents," with subtitle: "Math leads spacecraft on joy rides through the solar system." The spacecraft in question was NASA's Genesis. [The joy ride ended in what NASA terms a "hard landing" in the Utah desert. Fortunately much useful scientific information survived. More on the NASA website.] Klarreich's piece is about the way the mathematical analysis of the Solar System Gravitational Dynamical System (the sum of the gravitational fields of the all the objects in the system) led to the discovery of extremely fuel-efficient orbits. Edward Belbruno, now at Princeton, pioneered this approach twenty years ago, and scored its first great success in 1991 when he rescued the Japanese Hiten spacecraft, stranded in Earth orbit without enough fuel, it seemed, to reach the moon. Belbruno showed how to exploit the chaotic nature of the SSGDS to calculate a long-duration, low-cost trajectory which would lead the spacecraft to its destination.

 At L1, the sum of Earth's gravity and the centrifugal force exactly balance the gravitational attraction of the Sun. At points on the (black) halo orbit the vector sum would pull a mass towards L1; this can be balanced centrifugally by motion of the mass along the orbit.

Chaotic here does not mean disorderly, but refers to the enormous change in behavior that can be produced by a tiny change in initial conditions near an unstable critical point of the system. The Hiten rescue used the unstable critical points in the Earth-Moon system; there are three of them, known as the Lagrange points L1, L2 and L3. The Genesis mission used the similarly defined and labeled points in the Earth-Sun system: with notation from the diagram at right, at the L1 point the Sun's gravity (red) is exactly balanced by the sum (blue) of the Earth's gravity and the centrifugal force produced by the yearly rotation of the Sun-L1-Earth axis. (At L2 and L3 that centrifugal force balances the sum of the two gravities). The axis is crossed at L1 by a two-dimensional surface at each point of which the (green) resultant of the red and blue vectors is tangent to that surface. A mass on that surface will fall towards L1 unless it is orbiting rapidly enough, staying in the surface, to balance that attraction by centrifugal force. These are the "halo orbits" shown in Shane Ross's illustration above. But if the mass strays ever so slightly away from the surface, it will spiral either away from the Earth or away from the Sun. A tiny bit of fuel can send it on its way. We can think of "freeways" linking halo orbits around the three unstable equilibrium points. Want an inexpensive trip to Jupiter? Time your trajectory so that you're there when one of the Sun-Jupiter freeways crosses a Sun-Earth freeway; then a little nudge from your thrusters will do the trick. Klarreich's article is available online.

Peter Lax in the New York Times

Peter Lax won the Abel Prize this year. On that occasion, he was interviewed by Claudia Dreifus of the Times; the interview appeared in the Science section on March 29, 2005, with a photograph showing Lax in his NYU office in front of a blackboard bearing the prominent and talismanic chalk inscription: δ = log 4/log 3. Dreifus leads Lax through his early days: Budapest and Stuyvesant High School ("I didn't take any math courses at Stuyvesant. I knew more math than most of the teachers"). Lax was drafted in 1944 at the age of 18, and ended up at Los Alamos. "I arrived six weeks before the A-bomb test. ... Looking back, there were two issues: should we have dropped the A-bomb and should we have built a hydrogen bomb? Today the revisionist historians say that Japan was already beaten ... I disagree. ... I also think that Teller was right about the hydrogen bomb because the Russians were sure to develop it. And if they had been in possession of it, and the West not, they would have gone into Western Europe. What would have held them back? Teller was certainly wrong in the 1980's about Star Wars. ... The system doesn't work. It's a phantasmagoria." Dreifus asks what Von Neumann would think about the ubiquity of computers today. "I think he'd be surprised. ... But remember, he died in 1957 and did not live to see transistors replace vacuum tubes." Did he know John Nash? "I did, and had enormous respect for him. He solved three very difficult problems and then he turned to the Riemann hypothesis. ... By comparison, Fermat's is nothing." Does he believe high school and college math are poorly taught? "... In mathematics, nothing takes the place of real knowledge of the subject and enthusiasm for it."

The math of craquelure

 Craquelure in ceramics results from the differential shrinking of coats of glaze. The characteristic pattern is different from other naturally occurring tilings, which usually involve hexagons.

"Four Sided Domains in Hierarchical Space Dividing Patterns" is the title of an item published on February 9, 2005 in Physical Review Letters, and picked up in the "Research Highlights" section of the February 24 2005 Nature. The authors, Steffen Bohn, Stephane Douady and Yves Coudert (Rockefeller University and ENS, Paris) begin with the observation that, in the tilings formed by the cracks in ceramic glazes, the average number of sides of a tile is four. This seems unnatural, at first glance: Generically the edges of a tiling meet three by three. Euler's charcteristic for a convex domain gives Vertices - Edges + Faces = 1 or V - E + F = 1. Since every edge joins two vertices, generically 2E = 3V; Euler's equation then gives 3V - 3E + 3F = 3 and so 3F - E = 3. When the number of faces is large we can write 3F = E and since each edge is shared by two faces, this means that the faces must be, on average, six-edged. How the six edges become four sides in crackle finishes is clear from the picture. The authors explain the general mechanism at play: they define a hierarchical space-dividing pattern as one formed by "the successive divisions of domains and the absence of any further reorganization," and they show that "the average of four sides is the signature of this hierarchy." Another example is the organization of veins and sub-veins in the framework of a leaf (earlier work of theirs in this direction was referred to in the cover illustration of Science for February 6, 2004). Finally, they remark that the street network in a city where "growth resulted from self organization" is also of this type, and exhibit as evidence part of a 1760 map of Paris. Article available online.

Proof checking by computer assistants

Anyone who has ever been hoodwinked by a false proof of an intricate statement will be grateful to know that computers have been trained to take over the job of checking arguments. This is explained by Dana Mackenzie in the March 4 2004 Science, in an article with the title "What in the Name of Euclid Is Going On Here?" Mackenzie evokes the following problematic "scenario that has repeated itself, with variations, several times in recent years: A high-profile problem is solved with an extraordinarily long and difficult megaproof, sometimes relying heavily on computer calculation and often leaving a miasma of doubt behind it." The remedy is now at hand: software packages ("proof assistants") which "go through every step of a carefully written argument and check that it follows from the axioms of mathematics." The best-known examples are Coq, HOL and Isabelle. Recently Coq was used by Georges Gonthier to check the proof of the Four-Color Theorem, the archetype of Mackenzie's scenario. It passed; Gonthier's paper is available online. Isabelle was put through its paces by Jeremy Avigad to check the proof of the Prime Number Theorem. HOL-light has been used by Thomas Hales to check the Jordan Curve Theorem, a warm-up perhaps for a verification of his work on the Kepler Conjecture (see this column for May 2004). Mackenzie muses on the philosophical implications of these new develpments. "Ever since Euclid, mathematical proofs have served a dual purpose: certifying that a statement is true, and explaining why it is true. Now these two epistemological functions may be divorced. In the future, the computer assistant may take care of the certification and leave the mathematician to look for an explanation that humans can understand."

Tony Phillips
Stony Brook University
tony at math.sunysb.edu