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
Quick quiz
Test yourself on mandelbrot
10 multiple-choice questions. Pick an answer for each, then submit to see explanations.
Q1.The Mandelbrot set is the set of complex numbers c for which the orbit of z = 0 under which map stays bounded?
Q2.A common escape radius used to decide that an orbit is unbounded is:
Q3.What is the topological dimension of the Mandelbrot set's boundary?
Q4.The large heart-shaped region at the center of the Mandelbrot set is called the:
Q5.Mandelbrot is sometimes called a 'parameter space' picture. What does each pixel represent?
Q6.Smooth coloring near the escape boundary typically uses which quantity?
Q7.Which parameter c lies in the Mandelbrot set?
Q8.The Mandelbrot set is connected. Who proved this?
Q9.Which property of an orbit decides membership in the Mandelbrot set?
Q10.The cardioid is parameterised by which formula?