Invaders from The Fourth Dimension

So no updates for a couple days, but haven't been doing a whole lot of anything new until today.

Today I started rewriting my programs... again. Except this time, they're being implemented to work in four-dimensional space. How cool is that? It's not actually as simple as it sounds, though. Namely because, whereas in three dimensions every surface has only one straight line that is perpendicular to it (called the normal vector), in four dimensions those same surfaces have two lines that are perpendicular. Crazy, huh?

General Relativity

Today I started reading about this titular subject, and with a massive scientific irony, is possibly one of the easiest concepts to understand in physics, modern or classical. Ever wonder why the Earth pulls us down? Newton says because two masses attract each other... but why? I mean, we know why two magnets attract each other: they're constantly sending photons back and forth "telling" the other to move closer (or further apart). But there is no such chatter in Newtonian gravity. So how do we know which way is down?

Well many people know about "great circles". If you're looking at a map, the path of the plane traveling in a straight line appears an arc of a circle. That's because even though the plane is traveling in a straight line in three dimensions, the projection of the path of the plane into two dimensions is curved. The idea behind general relativity is that objects are always traveling in a straight line in four dimensions (the "spacetime continuum"), but when you project that line into three dimensions (aka, the universe that we observe) you get a curve!

Now you're probably wondering how mass is involved. Well, recall what I've been doing for the past couple weeks: calculating curvature. A straight line has no curvature. A slow tilt has small curvature. A sharp turn has high curvature. Well, with gravity and general relativity, mass is the curvature. In free space, nothing around you, you go in a straight line. Near a planet, you curve into it. Near a black hole, you curve into it faster.

Week in Review: May 7-May 11

Okay, so I've missed a few days. Here's what went down. I started calculating the other kind of curvature, which is basically multiplying the same two values that were previously added to find the first kind. And I had to do that in two ways. The first way was proving to be horribly inaccurate (machine error popping up), but the second way (which calculates dead-on) is only valid for 3 dimensions. Once we go to four when only the first way will work. Quite the dilemma.

Symmetric, bitch

Today I rewrote my programs entirely from scratch... again. Except this time there was a useful point. Consider the shapes I was working with. A sphere, torus, parabloid... those are all round, aren't they? You can spin them all you want around the z-axis and it won't change how they look. Any surface like this is called rotationally symmetric and mathematically that means its curvature is dependent on only one variable instead of two.

So now my algorithms calculate the exact same stuff they did before, except they do it in a tiny fraction of the time it used to take, and the error shrinks much faster as well. So woot!

Numerical Techniques: Part Two

So today I expanded on the programs I made yesterday. Now it can handle a sphere, an ellipsoid, or torus, and ultimately finds what is called the "intrinsic curvature" of the object in terms of a particular distortion (currently involving just increasing the radius, essentially, but can be modified for any kind of distorting). I have no idea what this curvature actually represents, but apparently my programs calculate it right! That's always good.

Numerical Techniques

Today I began implementing some simple numerical differentiation and integration routines, applying them to some techniques found in differential geometry.

In the span of nearly three hours, I wrote a series of five programs. These programs must be run sequentially, as each uses the data of the previous program (except, of course, for the first which generates the data).

Five programs and three hours and I can successfully calculate the area of a sphere with radius 1.

Okay, so maybe I'm demeaning this process a little. My initial application was a unit sphere. But the first program can be modified to generate data for any kind of surface, and the remaining programs can find the area of said surface with little-to-no modification.

Differential Geometry

So today I read up on this titular subject, and came to the conclusion that it is the possibly cannibis-induced lovechild of set theory and vector calculus. For those unfamiliar with the subjects, they are two very core fields in mathematics. I know many people who are good at one and only one (myself included); I know of none adept at both.

The following words have defined mathematical meanings that I was completely unaware of before today:

• Patch
• Umbilical
• Monkey saddle
• Maximal
• Compact
• Pullback
• Pushout
• Categorical dual
  
• Bundle
• Fibre product
• Functor

And no, I am not exaggerating in any way. Each and every one of these terms is related in some manner to differential geometry, I have discovered them all in my reading.

Update: I reworded things a little, as previously this entry sounded far more harshly than I intended.