The Mandelbrot set is named after mathematician Benoît Mandelbrot, who studied it extensively at IBM in 1980 using early computer graphics to visualize a formula first explored by Pierre Fatou and Gaston Julia decades earlier. For every point c in the complex plane, the set asks a simple question: starting from z = 0 and repeatedly applying z = z² + c, does the sequence of z values stay bounded forever, or does it escape to infinity? Points where z stays bounded are colored black and belong to the set; points that escape are colored based on how quickly they escaped, producing the set's famous swirling, self-similar boundary.
What makes the Mandelbrot set endlessly fascinating is that its boundary contains infinite detail at every scale — zooming into any edge reveals new spirals, filaments, and miniature copies of the whole set, a property called self-similarity. No matter how far you zoom, you never run out of structure to discover. This viewer computes the escape-time algorithm live in your browser: each pixel maps to a point in the complex plane, and the color reflects how many iterations it took to escape a radius of 2 (or black if it never escaped within the iteration limit).
Use the mouse wheel (or pinch on touch devices) to zoom toward your cursor, and drag to pan around. Higher iteration counts reveal finer detail near the boundary at the cost of render time, and the color palette selector lets you recolor the escape-time bands without recomputing the underlying math. Try zooming into the "seahorse valley" near −0.75, 0.1, or the filaments along the negative real axis, to see the fractal detail for yourself.