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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.
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/
The images described in the summary (which are not really representative of most of the stuff in the gallery, just Fields's stuff) are generally known as iterated function systems, and perhaps belong to the subset known as fractal flames. The description is fairly accurate, but the images he has made are rather unimpressive compared to ones I've seen (and made myself). Probably the best known example of a fractal flame program is Electric Sheep. However, another good program for making fractal flames is called Apophysis (regretfully, it's Windows only, but does work fairly well under Wine). I've been working with Apophysis for about 3 years now, and trust me, there's a lot of more artistic stuff out there that uses fractal flames. Even some of the stuff on Wikimedia Commons is better than his stuff.
Coincidentally, my captcha was "artful".
It's true that mathematical proportions and structures have been found in artwork for centuries, but what's different about these things is the role of the algorithm and raw computational power in producing this artwork. These are works that could not have been done before the availability of computers. The artist directs and controls the mathematics, using them like other artists use different kinds of paints, brushes, and canvas. But the computer does the mind-numbingly tedious work of billions of computations to render it on-screen. This is not all that different from artists using 3D sculpting and rendering tools; it's just a different set of algorithms.
Others have pointed out Electric Sheep and Apophysis; these focus on one particular type of non-linear iterated function system, the "fractal flame". There are many other fractal rendering tools out there, some free, some not. Wikipedia has a list if you're interested. This is a medium that has been in constant change for twenty years and doesn't look like it's ready to settle down any time soon.
People are never as simple as their stereotypes. This applies equally to Christians, Muslims, and Emacs-lovers.
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.
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.
>> Anyone have any others?
Perhaps the king of all environments (at least in my mind) is Processing. It is a Java based environment created by Ben Fry and Casey Reas. It's open source, has a huge active community, and plenty of 3rd party libraries for exploring things like computer vision, audio, physics, ray tracing, AI, etc.
There are a ton of really talented people doing cool things in Processing. Too many to list here, check out the Exhibition page for things to play around with.
OS X, Linux, Tivo, Amiga, my fascination with cult-like technologies would intrigue any psychiatrist.
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.
Don't say that! next thing you know somebody is going to sue Pirate Bay for linking to pi. If that was to happen maybe we can determine how many digits are within "fair use". As far as I know, nobody has uploaded the whole thing yet.
What?