Hairy Ball Theorem...

If f is a continuous function that assigns a vector in R3 to every point p on a sphere, and for all p the vector f(p) is a tangent direction to the sphere at p, then there is at least one p such that f(p) = 0. In other words, it is not possible to comb the hair on a ball smooth, not without a bald spot.

Nuts, ain't it?

As AC/DC used to say,

some balls are held for charity and some for fancy dress, but when they're held for pleasure, they're the balls that I like best...

Note that donuts CAN be combed smooth in a fascinating variety of ways, thru the hole, or along the edge, or in a spiral.

Might take some work, tho...

Hours of endless fun with nary but a hairy donut (link not safe for male readers prone to fainting spells or penis envy) and a comb, proving that the male fascination with holes is as powerful and inevitable as, say, nuclear fusion.