In mathematics, the discussion of **vector fields on spheres** was a classical problem of differential topology, beginning with the hairy ball theorem, and early work on the classification of division algebras.

Specifically, the question is how many linearly independent vector fields can be constructed on a sphere in *N*-dimensional Euclidean space. A definitive answer was made in 1962 by Frank Adams. It was already known, by direct construction using Clifford algebras, that there were at least ρ(*N*) such fields (see definition below). Adams applied homotopy theory to prove that no more independent vector fields could be found.

Read more about Vector Fields On Spheres: Technical Details, Radon–Hurwitz Numbers

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**Vector Fields On Spheres**- Radon–Hurwitz Numbers

... These numbers occur also in other, related areas ... In matrix theory, the Radon–Hurwitz number counts the maximum size of a linear subspace of the real n×n matrices, for which each non-zero matrix is a similarity transformation, i.e ...

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—Willa Cather (1873–1947)