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Defying Infinity

In math, three-dimensional space sprawls out to infinity in every direction. With an infinite amount of room, it should be able to hold an infinite number of things inside of it —pearls, peacocks or even planets.

But a recent proof by Olga Frolkina, a mathematician at Moscow State University, shows that one relatively well-known mathematical object can’t be packed an uncountably infinite number of times into an infinite amount of space: the Möbius band, a two-dimensional loop with a half-twist. The result underscores the delicate endeavor of situating surfaces in space, as well as the intuition-challenging nature of infinities.

That’s infinities—plural—as infinities come in different sizes . . .