Complex dynamics · Drawn by a one-line rule
A simple rule, an endlessly detailed edge.
A fractal is a shape that stays complicated no matter how close you look. Zoom into a coastline, a fern or a branching river, and you keep finding the same kind of wiggly detail at every scale. The Mandelbrot set is the most famous fractal of all, and the surprising thing is how little it takes to make it.
Here is the whole recipe. Pick a point on a flat sheet of paper. Follow one simple rule that turns your number into a new number, then feed that answer back in and do it again, and again, forever. For some starting points the numbers stay small and well behaved. For others they blow up and race off to infinity. Colour every point by which of those two things happens, and this shape falls out.
The black part is made of the points that stay calm forever. Everything around it is points that escape, coloured by how long they held on first. All the beauty lives on the border between the two. That edge is where the pattern never settles down. Zoom into it and you find spirals, curls, and little black copies of the whole set, each one surrounded by its own swarms of detail.
Nobody drew that edge by hand. It is not a picture someone designed. It is what one repeated sum looks like when you let it run, and it is endless. You could keep zooming forever and never reach the bottom.
The flat sheet is the complex plane. A complex number \(c = a + b\,i\) is just a point with two coordinates, a horizontal part \(a\) and a vertical part \(b\), where \(i\) is the square root of minus one. You can add and multiply these points with ordinary algebra, and multiplying them turns out to rotate and stretch, which is exactly what gives the set its swirling look.
Fix a point \(c\). Start with \(z = 0\) and apply the rule
\[ z \;\mapsto\; z^2 + c \]
over and over. That gives a sequence: \(0\), then \(c\), then \(c^2 + c\), and so on. Two things can happen. Either the numbers stay bounded, trapped near the origin forever, or they eventually grow without limit and shoot off to infinity. The Mandelbrot set is the collection of every point \(c\) for which the sequence stays bounded. Those are the black points.
There is a clean test for escape. If the size of \(z\) ever gets bigger than 2, the sequence is doomed and will run away. So in practice you iterate a fixed number of times and watch: if \(z\) crosses that radius, you colour \(c\) by how many steps it took, and if it never does, you paint it black. That escape count is the source of the coloured bands, and turning up the iteration count sharpens the fine detail near the boundary.
The edge is a genuine fractal. Its detail does not smooth out as you magnify it; new structure keeps appearing at every scale, and the boundary is so crinkled that it is, in a real sense, more than a one-dimensional line. Scattered along it you find miniature Mandelbrot sets, near-perfect small copies of the whole thing, linked to the big one by fine filaments. The set is also deeply tied to the Julia sets, a related family of fractals, where each point \(c\) in the plane picks out one Julia set, connected in a single piece if \(c\) is inside the Mandelbrot set and shattered into dust if it is outside.
The definition, precisely. For each \(c \in \mathbb{C}\) consider the quadratic map \(f_c(z) = z^2 + c\) and the orbit of the critical point \(0\), that is the sequence \(0,\, f_c(0),\, f_c(f_c(0)), \dots\). The Mandelbrot set is \(M = \{\, c \in \mathbb{C} : \text{the orbit of } 0 \text{ under } f_c \text{ stays bounded} \,\}\). One checks boundedness with an escape radius: if \(|z|\) ever exceeds \(2\) the orbit diverges, since beyond that point \(|z^2 + c| \ge |z|^2 - |c| > |z|\) forces monotone growth. The whole set sits inside the disc of radius \(2\).
Why the critical orbit. Watching only the orbit of \(0\) is not arbitrary. For a rational map the behaviour of its critical points controls the global dynamics, a theme going back to Fatou and Julia. This is what links \(M\) to the Julia sets \(J_c\): the fundamental dichotomy says \(J_c\) is connected exactly when \(c \in M\), and is a totally disconnected Cantor set otherwise. So \(M\) is precisely the connectedness locus of the family \(f_c\).
Connected, but wild. Douady and Hubbard proved that \(M\) is connected, by exhibiting a conformal isomorphism from the complement of \(M\) to the complement of the closed unit disc. The interior is organised into hyperbolic components, cardioids and discs, each carrying an attracting cycle of a fixed period; the main cardioid holds an attracting fixed point, the disc budding off it holds a period-2 cycle, and so on. The boundary is where all the difficulty lives. Shishikura proved that \(\partial M\) has Hausdorff dimension \(2\), as large as a region of the plane, even though it has no area.
Self-similarity and universality. The small copies of \(M\) are not a visual coincidence; they are explained by Douady and Hubbard's theory of polynomial-like maps and renormalisation. The same renormalisation carries a further surprise: the way period doubles along the real slice reproduces the Feigenbaum constant, the same universal number that governs the route to chaos in the logistic map. The great open problem is the MLC conjecture, that \(M\) is locally connected; it is known to imply the density of hyperbolic dynamics in this family, one of the central questions of the subject.
Fractals more broadly. The Mandelbrot set is one landmark in a wider idea Benoit Mandelbrot named in the 1970s: shapes whose detail persists across scales and whose fractal dimension exceeds their topological dimension. Coastlines, turbulence, blood vessels and the distribution of galaxies all show this roughness, and the mathematics of iteration and self-similarity that draws this figure is the same mathematics that describes them.
Related: Chaos Theory & the Butterfly Effect · next: Fourier Transforms · or go back to all topics.