New TOC Aggregator! And, a Puzzle.
In other news... recently a friend named William Torres (ordinarily more musical than mathematical) asked me an intriguing question:
Can you rotate a rigid, hollow sphere in space, holding its center fixed (but possibly varying the axis of rotation), in such a way that every point moves the same nonzero total amount?
Note that we're not referring to a point's net displacement; in fact, the net motion of the sphere will always correspond to a rotation about some axis (you shouldn't necessarily find this obvious... see wiki), hence two points will always have zero net displacement. What we're interested in is equalizing the path lengths of points' trajectories--how much 'exercise' they get. Of course, the amount should be finite.
I don't know the answer to William's question, but this doesn't mean it's hard. As a warm-up, see if you can prove the following (which I think I can show): It is possible for each point on Earth (understood as a sphere) for each point to have the same nonzero average wind speed over a 24-hour period (where wind is understood as a continuous vector field on the sphere, with vectors tangent to the sphere's surface, and varying continuously over time).
This contrasts with Brouwer's classical 'Hairy Ball Theorem', which states that at any given time, the wind speed will be zero at two or more points on Earth.