Mandelbrot
The Mandelbrot set is a famous fractal pattern that looks incredibly detailed and beautiful no matter how closely you zoom in. It’s created by repeating a very simple process, yet the result is an endless swirl of shapes that never stops revealing new detail.
Mathematically, the Mandelbrot set is defined as the set of complex numbers for which a specific sequence stays bounded. This means the values don’t escape to infinity as you keep iterating the formula. This property makes it a central object in complex dynamics and a key example in chaos theory, with applications ranging from computer-generated art to modeling natural patterns.
Mandelbrot Explorer
How does it work?
This visual shows the Mandelbrot Set, a famous fractal.
It's drawn on the complex plane, where numbers have a real part and an imaginary part (like $c = a + bi$). Each pixel on the canvas represents a different complex number, $c$.
To decide the color of a pixel, we repeat a simple calculation:
$z_{n+1} = z_n^2 + c$
We always start with $z_0 = 0$. We then see what happens to $z$ as we repeat the calculation.
If the number $z$ gets very large (its distance from the origin is > 2), we say it "escapes". If it stays small forever, it's "trapped".
Points inside the set (black) are the ones that are trapped. Points outside are colored based on how quickly they escape.