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
Famous quotes containing the words fields and/or spheres:
“Ah happy hills! ah pleasing shade!
Ah fields beloved in vain!
Where once my careless childhood stray’d,”
—Thomas Gray (1716–1771)
“It launch’d forth filament, filament, filament, out of itself,
Ever unreeling them, ever tirelessly speeding them.
And you O my soul where you stand,
Surrounded, detached, in measureless oceans of space,
Ceaselessly musing, venturing, throwing, seeking the spheres to connect them,
Till the bridge you will need be form’d, till the ductile anchor hold,
Till the gossamer thread you fling catch somewhere, O, my soul.”
—Walt Whitman (1819–1892)