The result: a program that many people enjoyed. In only 20 hours, 46 users had voted it up. It has been 4 days since I wrote the program, and it now has at least 97 up-votes.
To create the fire, I used a particle system of a lot of transparent ellipses, a good bit of math, and Perlin noise. I added the wind yesterday, using Perlin noise to make transitions smoother. The program doesn't interact with the user; it's only supposed to look cool.