Imagine a giant hoop passing through south and north poles. If I build a railroad along the equator and pull the hoop one complete
round, the hoop would have swept the entire surface exactly twice. The hoop perimeter is 2?R or 2? assuming R=1.
Imagine yourself holding one point of the hoop during the sweep. You can further imagine all the people (about 7,000,000,000)
holding on the hoop shoulder to shoulder from North pole to the equator. We would cover a quarter hoop. In one round we would sweep
half the sphere area. Each person's travel path distance will be different. The longest path is the equator — 2?R. At the pole the
sweep path is 0.
Q: what's the average travel among all those people? I think answer turns out to be 4R, which is slightly short of 66.666% of the
Q: (To keep this question simpler, we can keep the hoop still.) What's the average distance from the axis? I think it's 2R/?