Brownian tree

A Brownian tree example

A Brownian tree, whose name is derived from Robert Brown via Brownian motion, is a form of computer art that was briefly popular in the 1990s, when home computers started to have sufficient power to simulate Brownian motion. Brownian trees are mathematical models of dendritic structures associated with the physical process known as diffusion-limited aggregation.

A Brownian tree is built with these steps: first, a "seed" is placed somewhere on the screen. Then, a particle is placed in a random position of the screen, and moved randomly until it bumps against the seed. The particle is left there, and another particle is placed in a random position and moved until it bumps against the seed or any previous particle, and so on.

A circular Brownian tree

Growing Brownian tree

A Brownian tree that looks like a snowflake

The resulting tree can have many different shapes, depending on principally three factors:

• the seed position
• the initial particle position (anywhere on the screen, from a circle surrounding the seed, from the top of the screen, etc.)
• the moving algorithm (usually random, but for example a particle can be deleted if it goes too far from the seed, etc.)

Particle color can change between iterations, giving interesting effects.

At the time of their popularity (helped by a Scientific American article in the Computer Recreations section, December 1988), a common computer took hours, and even days, to generate a small tree. Today's computers can generate trees with tens of thousands of particles in minutes or seconds.

These trees can also be grown easily in an electrodeposition cell, and are the direct result of diffusion-limited aggregation.

A Brownian tree grown from a copper sulfate solution in an electrodeposition cell

Fractals
Characteristics	Fractal dimensions Assouad Box-counting Correlation Hausdorff Packing Topological Recursion Self-similarity
Iterated function system	Barnsley fern Cantor set Dragon curve Koch snowflake Menger sponge Sierpinski carpet Sierpinski triangle Space-filling curve T-square
Strange attractor	Multifractal system
L-system	Space-filling curve
Escape-time fractals	Burning Ship fractal Julia set Filled Lyapunov fractal Mandelbrot set Nova fractal
Random fractals	Brownian motion Brownian tree Diffusion-limited aggregation Fractal landscape Lévy flight Percolation theory Self-avoiding walk
People	Georg Cantor Felix Hausdorff Gaston Julia Helge von Koch Paul Lévy Aleksandr Lyapunov Benoit Mandelbrot Lewis Fry Richardson Wacław Sierpiński
Other	"How Long Is the Coast of Britain?" Coastline paradox List of fractals by Hausdorff dimension The Beauty of Fractals (1986 book)