The definitive first fractal
A gigantic, precision-aware explorer for the most famous fractal ever discovered. Zoom from the iconic whole set to filaments billions of times smaller, raise the iteration budget into the thousands, and inspect the orbits that create the image.
Live fractal laboratory
Mandelbrot remains the flagship deep zoom. Switch between complex dynamics, root-finding basins, recursive curves, deletion sets, circle packings, and three-dimensional constructions.
zₙ₊₁ = zₙ² + cHow this world is made
The flagship fractal: an infinite boundary of bulbs, filaments, spirals, and miniature copies organized around one connected black set.
Every pixel supplies c. Starting from z = 0, the renderer repeatedly squares z and adds c, then records whether the orbit escapes.
What to look forZoom around the black boundary for seahorse tails, elephant trunks, islands, and copies of the whole set.
Guided descent
These coordinates are curated starting points, not static pictures. Load one and continue the expedition yourself. Higher-numbered landmarks also raise the iteration budget for slow-escaping detail.
Mandelbrot field guide
The Mandelbrot set is the collection of complex numbers c for which repeated application of z ← z² + c, starting from z = 0, remains bounded. The black interior represents points that did not escape; exterior color records how their orbits departed.
A number a + bi represented as one point on a two-dimensional plane.
One repetition of z ← z² + c, using the last result as the next input.
The sequence of z values generated for one chosen point c.
How many iterations pass before an orbit grows beyond the escape radius.
The infinitely intricate frontier between escaping and bounded points.
Small regions echo the whole set without being perfectly identical copies.
How this renderer works
Convert its screen position into a complex number c.
Repeat z ← z² + c up to the current iteration budget.
If |z| exceeds the radius, its orbit is guaranteed to diverge.
Escape speed becomes a smooth color; unescaped points remain black.
Why deep zoom eventually stops
The mathematical boundary has detail at every scale, but ordinary JavaScript numbers carry about 15–16 significant decimal digits. Near 10¹⁴× magnification, neighboring pixels can collapse onto the same number and the image begins to lose trustworthy detail.
Going farther requires arbitrary-precision arithmetic or perturbation methods. This laboratory shows its remaining precision live so the limit is visible rather than hidden.
Fatou begins studying iterative rational functions.
Julia publishes foundational work on iteration in the complex plane.
Benoît Mandelbrot renders and popularizes the connectedness set now bearing his name.
Deep-zoom methods reveal scales far beyond direct floating-point calculation.
Beyond the flagship
These are not static illustrations. Every specimen is generated live in your browser from its own dynamical or recursive rule. Choose one here to load it in the laboratory above.