Fractal Brownian Archipelago

Erik Rauch

Introduction | A Continuum of Islands | More Islands

Some surprisingly simple processes can give rise to shapes that resemble natural features, such as coastlines.

Brownian motion is a simple concept. A particle making random jumps traces out a trail which, if one steps back, has structure on all scales - it is a fractal. (Though a true fractal has structure down to infinitesimally small scales, a particle making finite jumps will approach a fractal if you "step back" and look at it from far away.)

Regular Brownian motion forms a special case where the steps taken by the particle are all independent. (Many interactive demonstrations of regular Brownian motion are available.) This is part of a trail traced by a single brownian particle:

Fractional Brownian motion is produced when the steps taken by the particle are correlated in time. One expresses this by saying a parameter, the Hurst exponent H, is different from 1/2. The fractal dimension of the path is equal to 1/H. When H > 1/2, the particle tends to keep moving in the same direction it has been moving, producing smoother trails, as in this example with H=0.7:

Lower H makes the trails more rough.When H < 1/2, the particle's motion is anti-correlated: it will tend to move in a direction different from the one it has been following. This produces trails which are very tangled:



A "Brownian Bridge" is a Brownian trail that returns to its starting point. One can trace the "hull" of this trail, that is, the boundary between points enclosed by the trail and points outside, to produce islands, complete with bays, inlets, peninsulas and isthmuses on all scales. (The hull will in general have a lower dimension than the trail.)

On any of the following pictures, you can click to load a much larger (2000x2000) version where you can explore the coast by scrolling with your browser. Although large, these do not take an inordinately long time to load with a conventional Internet connection.

Examples of Brownian Islands (H=0.76)

Benoit Mandelbrot asked the famous question, "How long is the coast of Britain?" in a paper that pointed out that natural shapes like coastlines have detail on all scales and their length depends on the size of the measuring stick used. Coastlines are fractals (down to, of course, a cutoff where there can be no more detail) and the British coast has a dimension of about 1.31. We can choose a value of H so that the Brownian island that has this dimension - H=0.76. Here are some islands that have the same dimension as Britain's coast.

This island was generated for the cover of Benoit Mandelbrot's latest book (Gaussian Self-Affinity and Fractals ).

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