Patterns from chaos — symmetric fractals

Fractal structure shows up in many places in nature: in coastlines, river networks, lung airways, snowflakes, and — closer to what we study — in turbulent flows, where eddies cascade across a continuum of scales and the geometry of the resulting flow patterns is statistically self-similar. A scalar field stirred by turbulence wraps and folds at every scale; the boundary between turbulent and non-turbulent fluid in a jet or a cloud has a fractal dimension; even particle trajectories trace fractal-like paths. Simple iterated maps can mimic some of that visual complexity from very few rules — a useful reminder that intricate spatial structure does not require an intricate generator.

I came across this beautiful example when I was doing my PhD and developing a module on Chaotic Processes together with Prof. Harmen Jonker. I coded this in Fortran 90 and spent many hours changing the parameters to see what emerged. Fast-forward about twenty years and the same idea can be embedded directly in a web page, recalculated in real time as you move the sliders.

The demo iterates a complex map of the form

Here z is a point in the complex plane, m sets the rotational order, and alpha, beta, gamma, and lambda tune the nonlinear map. The picture is not drawn from a closed-form curve. It is built by repeatedly applying the map to many initial points and accumulating where those iterates land.

The reason the images have such clear structure is symmetry. With real coefficients and the conjugate term above, the map is equivariant under the dihedral group D_m: rotations through multiples of 2 pi / m and reflections are respected by the dynamics. The individual orbit can be chaotic, but the set of points it visits inherits the symmetry of the map.

This construction follows the symmetric-chaos maps introduced by Michael Field and Martin Golubitsky. A useful original reference is their paper Symmetric Chaos, Notices of the American Mathematical Society, 42(2), 230-243, 1995; see also their book Symmetry in Chaos: A Search for Pattern in Mathematics, Art, and Nature.