Strike a billiard ball on a frictionless table with no pockets so that it never stops bouncing off the table walls. If you returned years later, what would you find? Would the ball have settled into some repeating orbit, like a planet circling the sun, or would it be continually tracing new paths in a ceaseless exploration of its felt-covered plane?
These kinds of questions occurred to mathematical minds centuries ago, in relation to the long-term trajectories of real objects in outer space, and for nearly that long they’ve seemed impossible to determine exactly. What will a bouncing ball be up to a billion years from now? It’s as hard to answer as it sounds.
More recently, though, mathematicians have achieved a succession of stunning breakthroughs. One of the latest results, yet to be published, describes a new category of what are known as “optimal” billiard tables — shapes whose particular angles make it possible to understand every billiard path that could occur within them. The newfound shapes are among a handful of optimal billiard tables ever discovered, and part of an even more select group of quadrilaterals with that property.
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