Mathematicians have long wondered how “shapes of constant width” behave in higher dimensions. A surprisingly simple construction has given them an answer.
In 1986, after the space shuttle Challenger exploded 73 seconds into its flight, the eminent physicist Richard Feynman was called in to find out what had gone wrong. He later demonstrated that the “O-ring” seals, which were meant to join sections of the shuttle’s solid rocket boosters, had failed due to cold temperatures, with catastrophic results. But he also discovered more than a few other missteps.Among them was the way NASA had calculated the O-rings’ shape. During preflight testing, the agency’s engineers had repeatedly measured the width of the seals to verify that they had not become distorted. They reasoned that if an O-ring had been slightly squashed — had become, say, an oval, instead of maintaining its circular shape — then it would no longer have the same diameter all the way around.These measurements, Feynman later wrote, were useless. Even if the engineers had taken an infinite number of measurements and found the diameter to be exactly the same each time, there are many “bodies of constant width,” as these shapes are called. Only one is a circle.Arguably the best known noncircular body of constant width is the Reuleaux triangle, which you can construct by taking the central region of overlap in a three-circle Venn diagram. For a given width in two dimensions, a Reuleaux triangle is the constant-width shape with the smallest possible area. A circle has the largest.In three dimensions, the largest body of constant width is a ball. In higher dimensions, it’s simply a higher-dimensional ball — the shape swept out if you hold a needle at a point and let it rotate freely in every direction.