.GIF Animations
These are some animations that I made just for fun.
Diffusion-Limited Aggregation
During a lecture, my real analysis professor told us a story that graduate students should not be introduced to diffusion-limted aggregation (DLA). The reasoning is that the mathematics of DLA are so beautiful, yet strikingly difficult to study, that students might waste their graduate career studying it.
The DLA process is described as follows. We start with a single particle fixed in place. At each iteration, a new particle wanders into the space according to a random walk, and wherever the new particle touches the fixed particle it sticks. Again, new particles will wander in and stick to the cluster. If you repeat this thousands of times, you get very beautiful pictures like the one on the right.
The more red the point, the more recently it attached to the DLA cluster. The animation shows the evolution of the DLA cluster as hundreds of points are added each frame.
Buddhabrot
The Mandelbrot set is one of the most famous fractals in mathematics and it has a seemingly endless number of interesting properties. Since the Mandelbrot set is obtained by studying the dynamical system f(z)=z² + c, we might ask what else can be learn from this system. The Buddhabrot is a probability distribution that encodes how probable a point is to be visited by this dynamical system, for specific values of the parameter c. If you attempt to visualize these probabilities, you will generate an image that's somewhere between the Mandelbrot set and a picture of the Buddha.
For the figure on the left, I sample points that are visited by the system f(z)=z² + c. Every 1,000 samples, I plot the resulting Buddhabrot image. As I get more samples, you can see the image more clearly. People more clever than I have played with the Buddhabrot extensively and generated some much cooler images, which you can see in the following links:
Rotations and Oscillations
A very easy and fun way to make images that move nicely is to pick some parameters that you use when drawing pictures of an object, and to vary those parameters according to a sine wave. I do this with a number of parameters in these 3D graphics, varying the height, amplitude or spatial orientation in this way, and the results tend to produce some animations that are fun to watch.
Stony Brook University's mathematical mascot, the umbilic torus.
