Roses

I'm still hobbling around. The toe wasn't bothering me much this morning, but spending an hour and a half in front of my calculus class left me limping badly, and I went home as soon as possible after that. (Apparently the same psycho-physiological mechanism that keeps fatigue at bay when I'm lecturing also cuts my sensitivity to minor pains. Or perhaps I should say it postpones it...)

Anyway, the class went pretty well. (A few students were visibly dozing, but I don't have the heart to do to them what Dr. Fan did to me...) We're covering polar coordinates, and I was discussing "roses" with them. The graph, in polar coordinates, of an equation like r = cos (n theta), where n is a whole number, looks like a flower with multiple petals. (Actually, they look more like daisies to me, but "rose" is the traditional name.) I pointed out the difference between the case n even (there are 2n petals, pointing in all directions) and n odd (there are n petals, pointing in only half of the possible directions), and one of the students asked what it would look like if n were a fraction - say, 1/2. I had to admit that I didn't know, but decided to go ahead and try to freehand it. It was rather interesting, although not much like a rose - or a daisy. (After I got back to my office, I checked it on Mathematica, and my freehand was pretty accurate. Yay me.)

It always tickles me when a student asks a creative question, especially one I don't have an immediate answer for.