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Science 6 July 2007:
Vol. 317. no. 5834, p. 39
DOI: 10.1126/science.317.5834.39a
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News Focus

GEOMETRY AND THE IMAGINATION:
Bizarre Pool Shots Spiral to Infinity

Barry Cipra

GEOMETRY AND THE IMAGINATION, 7-11 JUNE, PRINCETON UNIVERSITY, PRINCETON, NEW JERSEY

If a mathematician invites you to play billiards, watch out. You're likely to wind up trying to make shots on a table of some weird, polygonal shape--or even on the outside of such a table.

The notion of "outer billiards" was proposed in the 1950s by Bernhard Neumann and popularized (among mathematicians and mathematically minded physicists) in the 1970s by Jürgen Moser as a stripped-down "toy" model of planetary motion. The setup is simple: An object starting at a point x0 outside some convex figure such as a polygon zips along a straight line just touching the figure to a new point x1 at the same distance from the point of contact (see figure). It then repeats this over and over, thereby orbiting the figure in, say, a clockwise fashion. Neumann asked whether such a trajectory could be unbounded; that is, whether the object could wind up landing progressively farther and farther from the central figure. This is analogous to the question of whether planetary orbits in the solar system are stable. All proven results, however, went the other way. For regular polygons, all trajectories are bounded, and for polygons whose vertices have rational coordinates, trajectories are not only bounded but also periodic: After a finite number of steps, each trajectory winds up back where it started.

Figure 1 Outer limits. Billiard balls aimed around a Penrose kite (blue) will travel outward forever, if you pick the right starting point.

CREDIT: RICK SCHWARTZ

Richard Schwartz of Brown University has given a positive answer to Neumann's question: There is indeed a convex figure with an unbounded trajectory--an infinite number of them, in fact. The example turns out to involve a famous shape, the Penrose kite, which Roger Penrose introduced in the 1970s as one of two pieces (the other is known as the Penrose dart) that produce nonperiodic tilings of the plane with local fivefold symmetry.

Schwartz discovered the unbounded trajectory around the Penrose kite by writing a graphics program for systematically exploring trajectories around kites, which he picked as the simplest figures for which unbounded trajectories could possibly exist. "I think of myself as a good experimenter," he says. "I tried lots of things that didn't work out!"

A key to the discovery was that he computed not only individual trajectories but also entire regions consisting of equivalent trajectories. For the Penrose kite, he found three large, octagonal regions within which trajectories bounce periodically from one region to the other (see figure). Around these regions lies a cloud of smaller regions (color-coded red in figure) with similar trajectory behavior, and around these regions is a larger cloud of yet smaller regions, and so on. The larger and larger clouds of smaller and smaller regions, Schwarz found, converged to a set of points from which the trajectories are unbounded.

Schwartz's initial proof was heavily computational. He has made much of it conceptual, but parts are still computer-assisted. (Schwartz's program, Billiard King, is available at his Web site, www.math.brown.edu/~res.) At the same time, he has found a general class of kites for which, with the help of the computer, he can show unbounded trajectories exist. "The work is very beautiful," says Sergei Tabachnikov, a (mathematical) billiards expert at Pennsylvania State University in State College. "It is an elegant piece of programming and a deep insight into the complicated dynamical phenomena revealed by the experiments." Schwartz, however, admits that the problem is still a puzzlement: "I don't completely understand what's going on."




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