Higher Dimensions
The connection with the Euler characteristic χ suggests the correct generalisation: the 2n-sphere has no non-vanishing vector field for n ≥ 1. The difference in even and odd dimension is that the Betti numbers of the m-sphere are 0 except in dimensions 0 and m. Therefore their alternating sum χ is 2 for m even, and 0 for m odd.
Read more about this topic: Hairy Ball Theorem
Famous quotes containing the words higher and/or dimensions:
“The higher processes are all processes of simplification. The novelist must learn to write, and then he must unlearn it; just as the modern painter learns to draw, and then learns when utterly to disregard his accomplishment, when to subordinate it to a higher and truer effect.”
—Willa Cather (1873–1947)
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—J.L. (John Langshaw)