Mandelbrot set

The Mandelbrot set in this visualization

The Mandelbrot set $M$ is the set of complex numbers $c$ for which the sequence below stays bounded. Once $|z_n| > 2$ the sequence runs off to infinity, so $c$ belongs to $M$ exactly when $|z_n| \le 2$ for every $n$.

What the two pictures show

The c-plane, the large picture: every point is a value of $c$, coloured by how soon the sequence from 0 leaves the disc of radius 2. The smooth colouring is for display only. Black means “no escape within the iteration limit $N$” — a numerical verdict, not a proof that $c$ is in the set.

The z-plane, the small screen: for the $c$ under the red reticle, every point is a starting value $z_0$, and black marks the filled Julia set $K_c$, the starting values whose orbits stay bounded (escape radius $\max(2, |c|)$). The yellow dots are the orbit of 0: it lives in the z-plane, not among the parameters. Click a point of the z-plane to start the orbit there instead.

Why it looks like this

The red key shows the regions $L_n = \{c : |z_n| \le 2\}$ one after another, for increasing $n$ chosen to suit the view, ending at $N$: each inside the one before ($L_1$ would be the disc of radius 2). $M$ is the part they all share.

The big cardioid is where $z^2 + c$ has an attracting fixed point: $c = \mu/2 - \mu^2/4$ with $|\mu| < 1$. The disc on its left, $|c + 1| < 1/4$, is where it has an attracting 2-cycle. Each bulb attached directly to the cardioid at the fraction $p/q$ (in lowest terms) holds an attracting cycle of period $q$, and its antenna splits into $q$ spokes at its main junction (counting the one back to the bulb).

Douady and Hubbard proved in 1982 that $M$ is connected. Fatou and Julia had shown that the Julia set of $z^2 + c$ is connected exactly when the orbit of 0 is bounded, so $M$ is also the set of $c$ whose Julia set is in one piece; for $c$ outside $M$ it falls apart into dust.

Easy to misread

• Black is “did not escape within $N$ steps”. Near the boundary, a larger $N$ can turn black pixels into colour.
• Islands that look separate are joined to the main body by filaments too thin to see.
• The small copies are not exact miniatures: each one is distorted, and its surroundings differ.
• The orbit is drawn in the z-plane only; the c-plane holds the parameter, not the orbit.

History

Fatou and Julia built the theory of iterating rational functions in 1918–1920 [1, 2]. Brooks and Matelski produced an early computer image of the set; their paper appeared in 1981 [3]. Mandelbrot published computer images of complex quadratic iteration in 1980 [4], and Douady and Hubbard, who proved the set connected in 1982, named it after him [5].

Related concepts

Julia set · Logistic map · Newton fractal (page planned) · Complex numbers

References

1. G. Julia, “Mémoire sur l’itération des fonctions rationnelles”, Journal de Mathématiques Pures et Appliquées, série 8, 1 (1918) 47–245.
2. P. Fatou, “Sur les équations fonctionnelles”, Bulletin de la Société Mathématique de France 47 (1919) 161–271; 48 (1920) 33–94, 208–314.
3. R. Brooks, J. P. Matelski, “The dynamics of 2-generator subgroups of PSL(2, C)”, in Riemann Surfaces and Related Topics, Annals of Mathematics Studies 97 (1981) 65–71.
4. B. B. Mandelbrot, “Fractal aspects of the iteration of z → λz(1 − z) for complex λ and z”, Annals of the New York Academy of Sciences 357 (1980) 249–259. doi:10.1111/j.1749-6632.1980.tb29690.x
5. A. Douady, J. H. Hubbard, “Itération des polynômes quadratiques complexes”, Comptes Rendus de l’Académie des Sciences, Série I 294 (1982) 123–126.

Further reading: Wikipedia: Mandelbrot set; MacTutor History of Mathematics: Benoit Mandelbrot.