Diffusion limited aggregation is a mathematical model simulating the growth of various physical structures. "Particles" are dropped one by one and diffuse until they hit a nucleation site, where they get stuck, extending the nucleation site. 

Here are pictures obtained when the diffusion is Brownian and the nucleation site is a single point. "Brownian" diffusion means here that the particles move step by step, and at each time can randomly move one unit up, down, left or right. The shade of grey depicts the time where the particle has stopped diffusing, with the lighter greys corresponding to later times.

Here are pictures where the particles are dropped at the top of the picture, and can diffuse only toward the bottom, but not up. In the first picture, there are five allowed moves: left, bottom-left, bottom, bottom-right, right. In the second picture, the left and right moves are forbidden. The original nucleation site is a horizontal segment at the bottom of the picture.

In this picture, particles are dropped at the center and the nucleation site is at the edges of the picture. The growth process stops once the structure reached the center of the picture, where the particles are dropped.

Finally, here is a series of pictures, where the original nucleation site is as before on the edges of the picture, and the particles are dropped uniformly at random on the picture.