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
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“We are conscious of an animal in us, which awakens in proportion as our higher nature slumbers. It is reptile and sensual, and perhaps cannot be wholly expelled; like the worms which, even in life and health, occupy our bodies. Possibly we may withdraw from it, but never change its nature. I fear that it may enjoy a certain health of its own; that we may be well, yet not pure.”
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