Roses

[stoutfellow]

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.

Comments

[gryphons-lair.livejournal.com]

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.

Hmm... interesting.

I wonder if the term "compass rose" for the thingy under the needle, is related in any way?

[stoutfellow.livejournal.com]

It's... conceivable. My first impulse is to doubt it, though. I'm not sure when polar coordinates were developed, but they can't predate Descartes - early 17th century - and I think the term "compass rose" is older than that. I'll look into it when I get to the office tomorrow - I have a book or two that might say something.

[ndrosen.livejournal.com]

I have an impression that Archimedes did some work with polar coordinates, but if so -- and I am neither a mathematician nor a historian of mathematics -- it must have been just a little, using words rather than a decent symbolism, and not leaving much of a path for lesser minds to follow.

[stoutfellow.livejournal.com]

You may well be right; one of the other curves I mentioned yesterday was the spiral of Archimedes. (I've got his collected works around here somewhere; I'll check there too.) But - as with logarithms and integral calculus - nothing came of it.

[stoutfellow.livejournal.com]

I just took a look at Archimedes' treatise on spirals, and yes, something like polar coordinates is implicit in his definitions there:

Definitions.[...]
2. Let the extremity of the straight line which remains fixed while the straight line revolves be called the origin of the spiral.
3. And let the position of the line from which the straight line began to revolve be called the initial line in the revolution.

This is from the Heath translation; he admits taking a few liberties here, but these are essentially the notions of pole and polar axis, as used in polar coordinates today. The ideas are only implicit, but they're there.

[stoutfellow.livejournal.com]

From what I've been able to find out, polar coordinates as such were first used in the early 18th century. I haven't been able to attach a date to the term "compass rose" yet, but I'm sure it's older than that.

[filkferengi]

It always tickles me when professors are enthusiastic; passion [for a subject] is contagious. Hey--a virus it's ok to spread! :)