Fat chance

GAMBLING, of all our many sins, has finally gone and redeemed itself. Darts, as often a game of chance as of aim and calculated trajectories, has provided theoretical mathematicians -- who casually play ball with tesseracts and other hypothetical objects of more than three dimensions -- a new method of calculating their volumes.

The task is so painstakingly tedious that it slows down to a drunken crawl the fastest of computers. Help came from a quarter that leaves most scientists cold: gambling. The game of chance could be a labour-saving device in assessing the volumes of (mathematically possible) n-dimensional objects.

Take the dartboard: the "mathematical truth" it suggests is complicated but mockingly self-evident: when a given number of darts are thrown at random, the fraction that hits a particular region of the dartboard is equal to the fraction of the board occupied by the region. Take a n-dimensional figure sitting inside an object of known size -- say, a cube with each edge measuring X. The volume of the n-dimensional figure can be approximated by tossing a given number of (mathematical) points randomly into the cube, and then counting the fraction that lands within the region that the figure occupies.

In 1989, Martin Dyer of the University of Leeds in England and Alan Frieze and Ravi Kannan of Carnegie Mellon University in the US devised an ingenious trick to simplify the problem. The basic idea is to place the n-dimensional shape in question within a set of other n-dimensional shapes -- like Russian Babushka dolls, each a little larger than the one inside it, working up to the abovementioned cube. The mathematical "dartboard" for approximating the volume of each shape is the next larger shape; through a process of successive reduction, you arrive at the volume of the smallest shape.

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