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Basic mathematical research pursues questions that are often quite far from actual practical application; but they contribute, in a largely invisible way, to the broader research ecosystem that eventually does generate such application. For instance, consider the problem of packing spheres into space as efficiently as possible - a question first proposed by Kepler in 1611. At a practical level, the solution to this problem has been "known" to greengrocers for centuries - one should stack the spheres in a hexagonal close packing. But mathematicians spent decades to work out how to establish the optimality of this packing, culminating in a formally verified proof in 2012.

Mathematicians also explored variants of this sphere packing problem in other geometries than three-dimensional Euclidean space: for instance in higher dimensions, with the famous recent breakthroughs of Viazovska in 8 and 24 dimensions, or in more discrete geometries over finite fields. Such curiosity-driven questions appear to lack immediate application - nobody had a need to pack eight-dimensional oranges together, for instance. (2/3)