A different(ial) integral

Today I am simply posting an entry from the blog which I mistakenly visited instead of this one, the short-lived and now defunct 'Marinating in the Blood of the Proletariat', whose URL is for some reason integral.blogspot.com. It is one of three posts from late fall 2001.

Why am I doing this? You can marinate on that y'self for a while, as OutKast once said. Maybe it is a kind of "Pierre Menard, Author of the Quixote" thing - the reclaiming, reunderstanding, reproducing, of a text, or something. Comments are encouraged.

Here follows the post:

Alrighty, then. Well, here it is Friday already; another week rushing by in the blink of an eye (trite alert). I need to find a way to make time slow down. My life is speeding by me and I'm on the couch eating popcorn. Well, figurative popcorn. Although I do eat popcorn occasionally. And I occasionally sit on the couch. So I am not being entirely dishonest. And by dishonest I mean not telling the truth in its entirety. And by that I mean that I like peas.

I would suggest that maybe I just don't do enough, but I am constantly busy. Maybe too busy? There has to be a line to draw somewhere. I'm just afraid that I'll close my eyes and when I open them again I'll be old and have never gotten to do what I want / need to do this time around. I'm not sure I even know what that is, yet. I'm pretty sure that this whole doctor thing is a good idea. I really care about people and I'm glad that my career will help people live better lives and be happier in general. I just... wanted to be Indiana Jones, that's all. I want some adventure. And I want to kill nazis.

Kevin's stuck at the computer lab again tonight finishing up his Open GL project. I was trying to get ahold of Dave so that I would have something to do other than hang out on somethingawful all night (I spend so much time there). Jeff wanted to do something, but I know that that would be a bad idea, and plus seeing him and talking to him still makes me really uncomfortable. I'm not sure we'll ever be friends again and I can't say I'd mind. Every time we talk he brings up my "needing to leave Kevin." It sucks. Kevin is the best thing that has ever happened to me, and I don't mean that in the cliche sense. Jeff has always been jealous, and it's getting really annoying. He's going to Peru for Christmas break, so I don't have to worry about him for at least a month. I can't wait until next year when he doesn't live right fucking next door to me. Maybe he'll leave the country. Maybe he'll leave the planet.

I need to find some friends who are female.

Weird geometry

The area of a hyperbolic triangle, call it ABC, strangely enough doesn't depend on the lengths of the sides. It is totally determined by the angle measures. So for any two hyperbolic triangles ABC and DEF, if angles A=D, B=E, and C=F, then ABC and DEF are congruent. Weird!

The area of a hyperbolic triangle is given by

Area = π-A-B-C

where A, B, C are the measures of the named angles in radians. (In the special case of a triply-asymptotic or ideal triangle, where all three vertices lie on the boundary "at infinity", the area is simply pi. Weirder still! The area of a TRIANGLE is the ratio of the circumference of a circle to its diameter? (The picture shows two ideal triangles in the Poincare upper half plane model

Perhaps weirdest yet? In spherical geometry, the area of a triangle ABC is r²(A+B+C-π). r² here is the square of the radius of the sphere. Usually one chooses the unit sphere, so the term disappears. But what happens if you use i, the imaginary unit, instead? Well, you get

Area = i²(A+B+C-π)
= -1(A+B+C-π)
= π-A-B-C

which is the formula for the area of a hyperbolic triangle. Consequently, hyperbolic geometry "is just" spherical geometry on a sphere of radius i.


:O