Diffusion Limted Aggregation Diffusion-limited aggregation (DLA) is a relatively simple process that creates natural looking forms. Clicking on any of the following images will open a movie of the DLA process. All images and movies were created with Visions Of Chaos.

2D Diffusion-Limited Aggregation For the two dimensional examples, consider a grid of pixels with one or more pixels active (usually this is a single dot in the middle of the image). A location at the edge of the grid is chosen at random. From this location a new particle randomly moves around the grid. If it comes into contact with the starting simgle particle it sticks to it. If the moving particle goes off the grid then it is disregarded and a new randomly moving particle is injected. This process is repeated as long as required. Result of a simple 2D DLA. The moving particles will stick to any existing particle in all 8 directions around it (Moore Neighborhood). If this is changed to only stick to horizontal and vertical particles (von Neumann Neighborhood) the result becomes as follows. The algorithm can be changed to take into account the number of times the moving particles need to hit an extisting particle before sticking. Result when having each particle hit the existing structure 5 times before sticking (Moore Neighborhood). The structure becomes more "clumped" together. Result when having each particle hit the existing structure 10 times before sticking (Moore Neighborhood). Result when having each particle hit the existing structure 10 times before sticking (von Neumann Neighborhood). The algorithm can also be changed to give the moving particles a probability of sticking to an existing particle. Result when having a 50% probability of each moving particle stick to the existing structure (Moore Neighborhood). Result when having a 1% probability of each moving particle stick to the existing structure (Moore Neighborhood). To get an idea of how many times the moving particles visit the grid locations, the grid can be shaded. The following image shades the grid using a spectrum ROYGBIV color scheme (red shows least visited grid locations - violet shows the most visited grid locations). 3 hits per stick. Moore Neighborhood.

3D Diffusion-Limited Aggregation DLA also works in three dimensions. In 3D there is still a single particle to begin. Particles randomly move in three dimensional space and stick any existing structure. The next image is updated 1000 particles per frame. The final total is 950,000 particles making up the main structure. Moving particles had a 10% chance of sticking. In this example the moving particles randomly start on the surface of a sphere surrounding the structure so the growth is more even and less biased towards the axiis as it grows (although in this example a section of growth did expand faster than the rest of the structure). In the following example the display is updated 250 particles per frame. The final total is 945,000 particles making up the main structure. Note that in this example the random particles started from a cube around the initial particle, so the final result is biased towards the X, Y and Z axiis. The next image is updated 1000 particles per frame. The final total is 2,118,000 particles making up the main structure. Moving particles had a 10% chance of sticking so the result is a more dense structure. This example the random particles started from a cube around the initial particle, so the final result is also biased towards the X, Y and Z axiis.

Links related to DLA Andy Lomas' Images Of Aggregation. Wikipedia's page on Diffusion Limited Aggregation. Paul Bourke has images and descriptions of both 2D DLA and 3D DLA. Golan Levin's Dendron.