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Mathematicians Discover New Shapes to Solve Decades-Old Geometry Problem | Quanta Magazine

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There are many “bodies of constant width,” as these shapes are called, which only one is a circle.

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.

In 1988 , Princeton graduate student Oded Schramm asked a simple-sounding question: Can you construct a constant-width body in any dimension that is exponentially smaller than the ball?.

Ukrainian mathematicians have solved the problem of finding the smallest possible body of constant width in all dimensions greater than 2 .

Their work provides a surprisingly simple algorithm for building an n-dimensional shape of constant-width shape whose volume is at most 0.9n times that of the ball.

The group briefly used their construction to investigate one promising candidate in three dimensions.

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