Paying It Forward

[stoutfellow]

By the time I was in eighth grade, I had already learned such mathematics as the local junior high and high schools were equipped to teach me. I was fortunate; a professor from San Diego State University, by the name of Henry Bray, took me under his wing. For the next five years, he was my mentor, teaching me a sizable chunk of college-level mathematics. I owe him a great deal.

Today was the day of the annual county-wide math contest, in which teams from the local high schools meet in various competitions. As often before, I was a judge in one of the events. Afterwards, I fell to talking with one of the team coaches. He told me of a student, a junior, who had learned what his school had to offer and needed more, and he asked if I would be willing to help.

[Y]ou don't pay back your parents. You can't. The debt you owe them gets collected by your children, who hand it down in turn. It's a sort of entailment. Or if you don't have children of the body, it's left as a debt to your common humanity. Or to your God, if you possess or are possessed by one. (Lois McMaster Bujold, A Civil Campaign)

It's not just parents.

Comments

[pompe.livejournal.com]

I have a math question which puzzles me.

Picture this. You have a rotating disc, like an LP record, rotating counterclockwise. Imagine you put a ruler above the record, and place a pen at the middle point between the edge and the center. Now, you draw the pen against the center as the record rotates, and you get a line. Then I replace the pen at the same spot and draw the line against the edge. Two lines, both affected by the rotation of the disc.

Now I try to do the same thing with a rotating globe. The ruler now looks like the crescent holding a typical globe of the Earth, and I place the pen at the middle point between the "equator" and the "pole", and I rotate the globe counterclockwise. I draw a line on the globe towards the pole, reset and draw another line towards the equator. Then I compare lines on the globe to the disc, and it looks like they link off in different directions. Why is this, mathematically speaking? Or am I missing something?

[stoutfellow.livejournal.com]

I'm not entirely sure what it is you're seeing, but there shouldn't be any major difference.

Think of the globe as being transparent, and the disc as lying in the equatorial plane. Set a light source above the North Pole, high enough that its beams are effectively parallel. Now, the path you describe on the disc should be the shadow, as it were, of the path on the globe; they shouldn't differ in any significant respect.

(In technical terms, the upper hemisphere of the sphere and the disc are "diffeomorphic" - that is, from the point of view of differential geometry, they are essentially the same - and the paths you describe correspond under that diffeomorphism.)

Hmm. Let me do some calculations and get back to you.

[stoutfellow.livejournal.com]

After doing a few calculations, I'm still not seeing what it is that you're describing. Could you make a sketch and e-mail it to me? My address is jparish-at-siue-dot-edu.