How to use it
The picture is a map of the complex plane. Every pixel stands for one complex number c, and its colour reports what happens when you start at z = 0 and repeat the step z → z² + c. Points whose values stay small for ever get the interior colour: together they form the Mandelbrot set. All other points run away to infinity, and their colour shows how quickly.
Drag to move and scroll or pinch to zoom. Press Orbit to see the first 50 steps of the marked point drawn as a path, with a note saying whether it escapes or settles into a cycle. Press Julia set to open a second picture: the Julia set that belongs to the marked c. The Places buttons jump to well-known spots.
Controls
- Mouse: drag to pan, turn the wheel to zoom around the cursor, double-click to zoom in. With Orbit or Julia set on, the marker follows the pointer; click to fix it, and click the marker to release it.
- Touch: drag to pan, pinch to zoom, double-tap to zoom in, tap to place the marker or drag it.
- Keyboard: focus a picture, then use the arrow keys to pan, + and − to zoom, Home to reset, O and J for Orbit and Julia set, and Shift with the arrows to nudge the marker.
- Panels: formula, iteration limit (automatic or manual), palette, colour offset and density, interior colour, Copy link and Save PNG, optionally at double size.
How it works
The Mandelbrot set is the set of all c for which the orbit of 0 under z → z² + c stays bounded. One theorem makes it computable: once |z| is greater than 2, the orbit is certain to escape to infinity. So the program repeats the step until z gets large or the iteration limit runs out. The number of steps taken, the escape time, picks the colour. To avoid visible bands, the whole number n is replaced by a smooth value, measured when |z| first passes a large radius R = 256:
The big heart-shaped region is the main cardioid, where the orbit settles on one fixed point. The disc on its left, centred on −1 with radius ¼, holds the orbits of period 2. Every smaller bulb around the edge has its own period: 3 for the large bulbs at the top and bottom, then 4, 5 and so on. The program tests the cardioid and the disc with a formula and watches for orbits that repeat, which saves most of the work.
Along the boundary, small warped copies of the whole set turn up at every scale, each wrapped in spirals and filaments. The boundary is so crinkled that its Hausdorff dimension is 2, as Mitsuhiro Shishikura proved in the 1990s. The area of the set is known only from numerical estimates: about 1.5066.
A Julia set uses the same step but keeps c fixed and lets the starting point vary. The two families are tied together: the Julia set is connected exactly when c lies in the Mandelbrot set. Otherwise it falls apart into a dust of separate points.
Things to try
- Turn on Orbit and move from the cardioid into the period-2 disc and then into the top bulb. The readout counts 1, 2 and 3.
- Open the Julia set and drag the marker across the edge of the Mandelbrot set. The Julia set breaks into dust as soon as you leave.
- Visit Seahorse Valley, then zoom into one of the spirals until a mini-Mandelbrot appears.
- Set the iteration limit to 50 and watch fine detail vanish into the interior colour. Then raise it again.
- Open the Deep zoom place and keep zooming until the precision warning appears.
- Switch the formula to Burning Ship and look for the small ship on the long spike on the left.
Frequently asked questions
Why does zooming stop?
The page calculates with double-precision numbers, which hold about 16 significant digits. At a magnification near 1013 neighbouring pixels differ only in the last digit, so they can no longer be told apart. Going deeper needs much slower arbitrary-precision arithmetic.
Why do the interior areas shrink when I raise the iteration limit?
A point is drawn as inside only because it has not escaped yet. Close to the boundary an orbit can need thousands of steps to escape, so a higher limit shows more detail but takes longer to draw.
Who discovered the Mandelbrot set?
Robert Brooks and Peter Matelski printed the first crude pictures in 1978. Benoit Mandelbrot studied the set with far better images at IBM in 1980 and made it famous. The name was given by Adrien Douady, who worked out much of its mathematics with John Hubbard.
What are the other formulas?
The Burning Ship takes the absolute values of the real and imaginary parts of z before squaring. The Tricorn squares the complex conjugate of z. The Multibrot sets replace the square with a third or fourth power.
What do the colours mean?
Only the escape time. Points of one colour take about the same number of steps to get away. The palette repeats in cycles, so the colours carry no meaning beyond that.