The Hairy Ball Theorem revisited – a newer, shorter, proof

Once proved, mathematical theorems* tend to stay proved. Nevertheless, they can sometimes still be improved – say, by making them shorter. Take, for example proofs for The Hairy Ball Theorem.

Mathematician Henri Poincaré first drew attention to the Hairy Ball Theorem in 1885 with his treatise ‘Sur les courbes définies par les équations différentielles (III)’ Journal de mathématiques pures et appliquées, 4e série, tome 1 (1885), p. 167-244. It took roughly 27 years, however, for the first formal proof, which was revealed by L. E. J. Brouwer  (a.k.a. ‘Bertus’) in ‘Über Abbildung von Mannigfaltigkeiten’ December 1912, Volume 71, Issue 4, pp 598–59.

Now a new (shorter) proof has been provided by by professor Eugene Curtin of Texas State University, US.  See: ‘Another Short Proof of the Hairy Ball Theorem‘ in The American Mathematical Monthly, 125:5, 462-463,

“We show how the assumption of the existence of a continuous unit tangent vector field on the sphere leads to an explicit formula for a homotopy between curves of winding number 1 and − 1 about the origin, thus proving the hairy ball theorem by contradiction.”

Notes:

[1] Another short proof had earlier been provided in 2016 by Peter McGrath (Brown University, US) See: An extremely short proof of the hairy ball theorem. Amer. Math. Monthly, 123(5): 502–503.

* [2] There is some discussion about the correct plural for the word ‘Theorem’, if { ‘Theorem’ > 1 }

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