### Amphigorical Math

This is a section of Giovanni Maria Cassini’s globe, a very beautiful object, which you can view in great detail at David Rumsey’s map site. You can see that Cassini knew something I was just guessing at, how to find the radius of the curve of map gores. Dr. Math tried to explain it to me, but he’s a mathematician and I’m a dilletante and we have trouble communicating. Every time I read the term “radians,” a wee pixie comes in through the cat door and blows out my pilot light. The eyes glaze over and the frontal lobes grow cold and I wake up eight hours later, surrounded by empty 40 ounce malt liquor bottles and professional wrestling, blaring on the television.

This is not the royal road to geometry. This is not even the on-ramp.

I kept thinking it was a spherical triangle thing. It probably is, but since I can’t manage to get the concept into my head, I tried doing it with good old Pythagoras and a piece of paper. (Well, actually, I was sketching madly on my tablet pc, amusing the other bus passengers no end.) It is not beautiful, but I think it works. Feel free to disagree; I won’t argue.

(For our foreign readers, I will mention that a pixie is a small, semi-mythical being, very like a leprechaun, but with a post-punk sensibility.)

Hey, speaking of Pythagoras, are any of our South Asian readers familiar with the medieval mathematician, Bhaskara II? Came across a reference to his having done a proof of the Pythagorean theorem by folding paper, but I can’t find any details on this.

Well, you know I must comment!

That Dr. Math thing looks confusing to me too. Your analysis looks close enough, although I don’t know why you can assume the bottom leg of your triangle will be r-1.

Here’s another way you might be able to do what you want: Each section of the Cassini globe is an approximation of a “lune.” A lune is the section of the surface of a sphere that looks like an orange wedge. That is, it goes from pole to pole and is basically a wedge of the sphere. Lunes are determined by the angle they make at each pole. So a 90 degree (pi/2 radians — oops, sorry. Didn’t mean to cause convulsions) lune would cover 1/4 of the whole sphere.

Yeah. Anyway, a nice thing is that the area formula for a lune is very nice. A lune whose angle is theta (in radians — oops) drawn on a sphere with radius r will have area:

lune area = 2 * theta * r^2.

Right. So if you can measure the area of one of your Cassini sections (in square inches) and measure the angle of the lune (the angle at the top of the section) AND convert this angle to radians (ooops! radians = (degree angle) * (pi/180) ) then you can use this lune area formula to calculate what the radius of the sphere must be.

I thought I saw a proof of the Pythagorean Thm via folding before. But I never heard of this Bhaskara II fellow before. Do let me know if you learn or know more!

January 19th, 2007 at 6:49 pm