Diffusion Limited Aggregation
(DLA) in 3D

Written by Paul Bourke
January 2004

See also: Constrained Diffusion Limited Aggregation in 3 Dimensions
Computers and Graphics, Volume 30, Issue 4 , August 2006, Pages 646-649

Buy software for Mac OSX

The following are examples of growth in three dimensions based upon the same principles as diffusion limited aggregation, normally performed in two dimensions. The basic principle involves a particle that moves around randomly until it gets close to part of the existing structure, at which point it sticks and becomes part of the structure. The process is started with a seeding structure, normally a single point, and continues until the structure reaches a desired size.

Click on the images here for higher resolution versions


Exhibited at Museum of Design, Zurich. einfach komplex – Bildbäume und Baumbilder in der Wissenschaft. Simply Complex -- Picture-Trees and Tree Pictures in Science. 30 April 2005 -- 4 September 2005



The renderings here are created using POV-Ray, each particle is actually represented as a cone and a piecewise Bezier curve is used to smooth continuous chains of particles. As particles are added the thickness of the branch leading to that particle in thickened.


A further extension is to constrain the DLA growth to some "vessel". In the following the constraint takes the form a box (left) or cylinder (right) with one end open.


Grown over a sphere (planetarium), left: looking from inside the planetarium, right: from the outside of the dome.


DLA grown within a 3D scanned model of the human head (plus antlers), described by a STL file. Part of modelling exercise for Philippe Vranjes.



The software presented here is made available for Mac OS-X, for Mac OS-X X-Windows must be installed and running. DLA structures can be grown in 3D, constraint surfaces applied, and resulting structures saved in PovRay format for high quality rendering.

Screen dump from dla3dviewer

Resulting render from PovRay (Unmodified from export)

The following downloads are fully functional except saving of the geometry is disabled and size of growth is limited is limited to 4000 branches. The software is provided for a number of versions of Mac OSX, please contact the author if you experience problems. This allows you to check the software, the unrestricted version is available for a modest shareware fee.

Demo versions



Buy full version

> dla3dviewer -h
Usage: dla3dviewer [command line options]
Command line options
       -h    this text
       -f    full screen
       -s    active (quad buffer) stereo
      -ss    dual screen stereo
Mouse controls
     left    rotate camera
   middle    roll camera
    right    menus
Function keys: see menus
Keyboard commands
        q    quit
        h    reset camera
      [,]    roll camera
      <,>    more camera forward, backward
      +,-    zoom camera in, out
        w    window dump to TGA file

Screen dump from dla3dviewer

Resulting render from PovRay

Software Notes

  • You need to install XQuartz from here: https://www.xquartz.org/

  • The software expects a three button mouse, note that under Mac OS-X the middle and left button can be activated by using the option and control keys. The left mouse button rotates the camera about the model, the middle button rolls the camera, and the right button brings up a menu system.

  • Useful keys include the '<' and '>' keys for moving the camera towards to away from the model. The camera is always pointing towards the origin which is where the initial seed particle is placed.

  • Using the menus the distribution of the introduced particles can be varied, the default is to introduce new particles on the surface of a sphere. When "extras" are turned on the last few thousand positions of introduced particles are shown. Particles are introduced just past the maximum range of the current model.

Settings dialog

  • Random seed: Each seed creates a unique structure, alternatively the same structure will result from the same seed (all other conditions being equal).

  • Bias towards origin: On each time-step the current particle moves in a random direction, the bias moves the particle towards the origin, generally by a very small amount.

  • Branch length: This sets the scale of the model and is the length of an added branch as well as defining how close a particle needs to get to the existing DLA structure in order to "stick" to it.

  • Stickiness: This is the probability of a particle sticking to the existing DLA structure once it is close enough. In general this control determines the density and fuzziness of the final structure. More information.

  • Particle rotation: On each time step this rotates the current particle about the z axis by (usually) a very small angle. Adds a swirling effect to the DLA and can vary the fractal dimension.

  • Branch thickness: This is used when the DLA structure is drawn with solid branches, instead of lines and points. This is also used in most of the export formats. The minimum value will be the thickness of the end of the branches.

  • Intersection range is used to avoid self intersections.

  • Multiple roots can be created, the range is used by the built in root seed distributions distribution.

    The constraint surface can be scaled, translated, and rotated.

Growth inside a cup


Physical realisations of various growth stages by David Sutton.

Multiple independent seeds

Each growth acts as a constraint on others


Further examples by David Sutton

Project by Michael Eden, building DLA structure within Architectural elements.

Constrained Diffusion Limited Aggregation in 3 Dimensions

Paul Bourke
Poster: Graphite (ACM Siggraph), Dunedin November/December 2005


Diffusion Limited Aggregation (DLA) has usually been studied in 2 dimensions as a model of fractal growth processes such as branching, lightning, snowflakes, mineral deposits, and coral. Here, the basic principles are extended into 3 dimensions and used to create believable models of root systems. A straightforward approach is introduced for constraining the growth of the 3 dimensional DLA by a surface or containing it within a vessel.

PDF version: dla3d.pdf