← Back to Paradoxes

The Mandelbrot Set

zn+1 = zn² + c

One simple formula. Infinite complexity. Zoom forever and discover new patterns— seahorses, spirals, lightning bolts—that have been hiding in mathematics for eternity, waiting to be computed. The most famous fractal ever discovered.

Rendering...
Click to zoom in Max iterations: 100

Iterations

Current Zoom

1×

Color Scheme

Famous Locations

Infinite Complexity from z² + c

The Simple Rule

For each point c in the complex plane: start with z = 0, repeatedly compute z = z² + c. If z stays bounded (|z| < 2), c is IN the set (black). Otherwise, color by how fast it escapes. That's it!

Self-Similarity

Zoom into the boundary and you'll find miniature copies of the whole set. These "mini-Mandelbrots" appear everywhere, connected by intricate spirals and tendrils. Each one is slightly different.

Connected But Infinitely Complex

The Mandelbrot set is connected—a single piece with no isolated points. Yet its boundary is infinitely complex, with dimension ~2. More complex than any curve, but less than a filled area.

Discovered by Computer

Benoit Mandelbrot first visualized this in 1980 at IBM. The beauty was invisible until computers could iterate millions of times. Mathematics had hidden this for thousands of years!