FAQ

Informally, a shape with detail at every scale, often self-similar. Formally (per Falconer), a set whose Hausdorff dimension strictly exceeds its topological dimension. Mandelbrot coined the term in 1975 from the Latin 'fractus' meaning broken or fractured.

A way to measure how a set fills space at different scales. For a self-similar set with N copies at scale 1/r, the similarity dimension is log(N)/log(r). The more general Hausdorff dimension captures the same idea via covers. Box-counting dimension is the easiest to compute numerically.

The set itself has Hausdorff dimension 2 because it has interior. Its boundary also has Hausdorff dimension 2 (Shishikura, 1991), which is striking: a curve that fills enough space to match the plane around it.

They have fractal-like statistics over a range of scales, not all scales. Coastlines have fractal dimension ~1.2 to 1.3 over scales from kilometers down to ~1 meter, below which there is no further sub-structure. Mandelbrot's 1967 'How long is the coast of Britain?' is the canonical paper.

Brownian motion has trajectories with Hausdorff dimension 2 in the plane (and 2 in any dimension d >= 2). Fractional Brownian motion can have any dimension between 1 and 2. So yes, many random processes produce fractal traces.

Iterated Function System: a finite set of contractive maps. By Hutchinson's theorem, the system has a unique compact attractor that's invariant under the union of the maps. Most classical fractals (Sierpinski, Cantor, Koch, Barnsley fern) are IFS attractors.

A simple algorithm for rendering IFS attractors. Start at any point. Repeatedly: pick one of the IFS maps at random and apply it to your current point, plot the result. After a burn-in, the plotted points densely cover the attractor.

Yes (Douady and Hubbard, 1980s). For every c outside the Mandelbrot set, you can find a continuous path to infinity that stays outside. Equivalently, the Mandelbrot set has connected complement.

Each iteration multiplies the perimeter by 4/3 (goes to infinity) but adds bounded area in a geometric series that converges. Finite area, infinite perimeter is the headline counter-intuition of fractal geometry.

Yes. The Mandelbrot explorer is live now; Julia explorer and IFS playground are coming. All client-side, no server compute.