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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 have the photorealism of the Rennaisance, you get all of the math involved in regular life (e.g. the golden ratio). With various less realistic artists (e.g. Pollock, Van Gogh), haven't mathematicians found various deep mathematical patterns in their work? This is what you get when you start out with pure math, and turn it into art, whereas most of art is what you get when you have an intuitive understanding of math (i.e. what looks good) and go with that. All art has math in it.
What I found more interesting than mathematical art was the music produced from differential equations and such.
I really wish I remember more details but a few years ago I saw a presentation by a mathematician in which he had a little program that solved some sort of equations. Grr, I'm going to hate myself now for not remembering. Well, regardless the details, it solved something and assigned the solution values specific notes/chords from a piano, so that whenever a value was obtained, the computer played that note. Thus, the time evolution gave a sequence of notes, and so he recorded this sequence.
He played a few excerpts, I tell you what, it sounded like Mozart or Beethoven. Well, certain parts you could pick up a very forced/electronic feel to it, but other parts glided so beautifully that it sounded like a master pianist was playing.
That was an incredible lecture. Perhaps anyone else knows what I speak of? I'd like to find out what program and equations were used, it was fascinating.
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.
Roman Verostko and others have been doing something he calls algorithmic art for awhile. E.g., put a paintbrush in a pen plotter and then write an algorithm to paint on canvas. Although sometimes I feel like artists like Verostko (who call themselves algorists) are tremendously arrogant sometimes (which I suppose makes them like many other artists), a lot of their stuff seems really beautiful to me. In particular, Verostko's pseudo-calligraphy is just mesmerizing to me-- it looks sort of like a written language, but it's not.
And of course, you can't forget the grandmaster of algorithmic art: Bach. Bach was a master of counterpoint, and the mathematical beauty of some of his works (e.g., The Art of Fugue) is readily apparent. If he indeed did not generate his works in an algorithmic way, well, that's surprising to me. Listen to Glenn Could play Bach, Partitas 1,2, and 3 being my favorite...
I did some mart art work awhile ago, based on Daubechies' scaling functions. Check it out: The Strangers Series.