We often discuss the practical applications of scientific computing and numerical analysis—how to solve complex mathematical systems with some arbitrary precision or how to perform some statistical analysis from a set of real-world data while minimizing error. Sometimes, however, it’s just cool to use these complex ideas to make pretty pictures. Science News recently published an article on mathematical art—that is, pictures generated by computing and analyzing patterns natural in mathematical systems.

The most interesting of the mathematical art samples provided in this article to me was the one based off of dynamical systems. Created by mathematician Michael Field, these involve taking a point and moving it across a piece of paper according to some simple rules and keeping track of the frequency each pixel on the paper is touched. This seemingly random process is repeated for billions of iterations, and eventually some neat patterns come out.

A more common example of using computing to generate art, actually, is in the use of fractals. Most of us math/CS nerds have seen images of things like the Mandelbrot setor a Koch snowflake, and images of these themselves, when rendered well, look pretty awesome. In the article, one of the artists (I think we can safely call them artists) took another well-known fractal, the Sierpinski Triangle, and digitally created a large portrait of Sierpinski himself. Instead of triangles, he used squares that followed the same rules as Sierpinski Triangles (given a square, divide it into 9 squares and remove the middle, and repeat), found the average intensity of these squares overall (stopping the iterations to achieve different degrees of darkness), and had each pixel of a large picture be created by one of these Sierpinski carpets.

All of these pictures show the power of having very accurate computing and visualization systems. Even with very simple rules, mathematics can produce a fascinating array of patterns.

Sources:

http://www.sciencenews.org/articles/20080216/mathtrek.asp

http://en.wikipedia.org/wiki/Fractals