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 the moral tone, the more suspect the speaker.”
—Mason Cooley (b. 1927)
“I was surprised by Joe’s asking me how far it was to the Moosehorn. He was pretty well acquainted with this stream, but he had noticed that I was curious about distances, and had several maps. He and Indians generally, with whom I have talked, are not able to describe dimensions or distances in our measures with any accuracy. He could tell, perhaps, at what time we should arrive, but not how far it was.”
—Henry David Thoreau (1817–1862)