One of the things I occasionally enjoy doing in my spare time, odd though it may sound, is playing with math software to make it create cool visuals. I've had a couple of results I've been particularly happy with, so I thought I'd share them with the world.
The first is an animated graphic of a particular concept in vector calculus. The idea is that you have a curve in space, and at any given point you can define three orthogonal vectors with respect to the curve. The first is tangent to the curve (it points along the curve); the second is the normal vector which points in the direction of greatest curvature; and the third is the binormal which is perpendicular to the first two. In terms of physics concepts, if you think of the space curve as the path along which an object is travelling, the tangent vector is in the direction of its velocity (and tangential acceleration), the normal vector is in the direction of its centripetal acceleration, and the binormal vector is, I suppose, just a convenient normal vector to identify the plane in which the object is travelling at a given instant. In this picture, the green vector is the tangent, blue is the normal, and red is the binormal.
The second animation I have for you is an illustration of what's called a parametric surface. The equation for this surface is rather ugly and complex, but the surface itself is quite beautiful. I have it rotating to give you a complete visual of it. This is an example of math-as-art.
Both of these were done in Maple, which is my personal preferred software for math-art. However, you can do similarly cool things in not only other proprietary software like Mathematica, but also with free (in all senses of the term) software like Maxima, although I don't know the extent to which any free software does animations.
The thing I like most about Maple is that you can talk to it almost entirely in standard math notation, with a few (relatively intuitive) text commands for things like plotting and animating. What I like least about it is that it's a horrendous memory hog and a bit unstable on your standard PC. However, my laptop runs it quite nicely on 64-bit Linux, despite having been entirely incapable of running it under Windows, so it's possible that it may simply have Windows issues.
I rather dislike Mathematica's interface, but there are people who swear by it. As far as Maxima, if you're the sort of person who finds Matlab and command-line Linux easy to deal with, then Maxima is the package for you.
Regardless, however, I do recommend playing with some 3-d graphing-capable software if you're currently a math student (or if you last took math back when slide rules were in vogue); the coolness factor of today's software is really high in the graphics department, and these programs can do some really amazing symbolic math work too.
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