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Art with a Mathematical Twist
Euler points out a story about art created through mathematics. The Science News article covers selections from a recent exhibit, where over 40 artists gathered to show their work and the math behind it. The rest of the pieces are also viewable at the exhibit's website.
"Michael Field, a mathematics professor at the University of Houston, finds artistic inspiration in his work on dynamical systems. A mathematical dynamical system is just any rule that determines how a point moves around a plane. Field uses an equation that takes any point on a piece of paper and moves it to a different spot. Field repeats this process over and over again--around 5 billion times--and keeps track of how often each pixel-sized spot in the plane gets landed on. The more often a pixel gets hit, the deeper the shade Field colors it."
When it comes to the relationship between mathematics and the arts, my favourite example is the music of Per Norgard. In 1959 Norgard discovered a way of serializing melody that resulted in endless self-similarity, a type of fractal. He termed it the infinity series, and though the two-tone infinity series had already been discovered by mathematicians, the application of the principle to chromatic and diatonic scales resulted in a series no mathematician had discovered before. The infinity series is a fascinating concept, and in Norgard's works like the Symphony No. 3 it proves immensely beautiful.
Other composers have, of course, made use of mathematical processes. The golden section is often heard in Bartók, for example, though who knows if it was done consciously.
If you're interested in pretty, shiny, mathematical things that you can run on Linux, check out:
- electricsheep: animated fractal flames: http://www.electricsheep.org/ (I highly recommend running this as your screensaver, though it takes a bit for the first sheep to download)
- Jenn: pretty, shiny, blue(?) polytopes, rendered on your computer: http://www.math.cmu.edu/~fho/jenn/
Anyone have any others?
http://mediagoblin.org/
Context Free is a program that generates images from written instructions called a grammar. The program follows the instructions in a few seconds to create images that can contain millions of shapes. The program itself is GPLed and available here.
As you can see from the link below, some of the results from this project are stunning.
Context Free Art gallery.
A few years ago I got the idea to write code that fed massive scene files into POV-Ray. There are probably better tools nowadays but POV-Ray had the virtue of a simple scene description language that I was already familiar with. It's easy to create code to generate it.
I made a heart out of the sextic (huhhuhhuhhuh) polynomial
(2xx+2yy+zz-1)^3 - xxzzz/10 - yyzzz = 0
and had POV-Ray create a bunch of scene files by rotating this thing through 180 degrees to create an animated heart GIF. (This was back in the Dark Ages when the web was full of animated GIFs.) There were probably a thousand other animated hearts out there but this one was mine.
I got the idea to do space filling of the unit sphere with thousands and thousands of small boxes or smaller spheres, playing around with the lighting to see if I could create something vaguely moonlike with inside-out craters. I tried doing this with thousands of hearts but got bitten in the ass by a bug in POV-Ray's polynomial rendering code where it trips over a planar singularity in the heart equation, so every little heart ends up with an unromantic slit running across its equator. There were just too many to fix by hand.
The most interesting image from this technique came from a routine that recursively generated spheres, invoking itself six times per sphere to create smaller spheres on the top, bottom, left, right, front, and back, each of which then does the same thing, to a depth of 5 or 6. You end up with a Sierpinski octahedron.
All this stuff has been done to death by others. I wish I were good at drawing comics.