Complex dynamics
Mandelbrot set
The set of complex numbers c for which the iteration z ← z² + c, starting from z = 0, stays bounded. Named after Benoît Mandelbrot, who computed the first images in 1980 while at IBM. The most famous shape in mathematics outside of Euclid.
At a glance
| First imaged | 1980 (Mandelbrot, IBM) |
|---|---|
| Domain | Complex plane |
| Iteration | zn+1 = zn2 + c, z0 = 0 |
| Bounded region | Contained in the disk |c| ≤ 2 |
| Connected? | Yes (Douady & Hubbard, 1980s) |
| Area | ~1.5065918849 (numerical estimate) |
| Boundary dimension | 2 (Shishikura, 1991) |
Key features
- Main cardioid: the heart-shaped region in the center, parameterized by
c = (eiθ − e2iθ/2)/2. - Period bulbs: circular regions attached to the cardioid, each containing parameters of a fixed period. The largest is the period-2 disk at
c = -1. - Mini-Mandelbrots: scaled copies of the full set embedded inside the boundary at every zoom level. The largest is in the “needle” near
c = -1.75. - Seahorse / elephant valleys: classic regions for deep-zoom tourism, near
c ≈ -0.75 + 0.1iandc ≈ 0.275. - Misiurewicz points: parameters where 0 maps to a repelling periodic point; the local structure mirrors a Julia set.
Dimension
Mitsuhiro Shishikura proved in 1991 that the boundary of the Mandelbrot set has Hausdorff dimension exactly 2. The interior has dimension 2 (filled regions); the exterior has dimension 2 (the complement). What makes the result striking is that despite being a curve, the boundary fills enough space to be 2-dimensional, the same as the plane itself.
Try it
The Mandelbrot explorer renders the set live in your browser, with click-to-zoom, drag-to-pan, and a max-iterations slider.
References
- Mandelbrot, B., The Fractal Geometry of Nature, W. H. Freeman, 1982.
- Douady, A. and Hubbard, J. H., “On the dynamics of polynomial-like mappings,” Ann. Sci. ENS, 1985.
- Shishikura, M., “The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets,” Ann. of Math., 1998.
- Peitgen, H.-O. & Saupe, D. (eds.), The Science of Fractal Images, Springer, 1988.
- Julia sets · Mandelbrot explorer