For a fixed complex number c, the Julia set is the boundary between starting points whose orbits under z → z² + c stay bounded and those that escape to infinity. Each c gives a different set; the Mandelbrot set is the catalogue of which c values give connected ones.

Why does the set sometimes fall apart into dust?

When c leaves the Mandelbrot set, its Julia set becomes totally disconnected — a Cantor dust. The circle of radius 0.7885 passes in and out of the Mandelbrot set, so the animation crosses that threshold twice per cycle.

Will this slow my computer down?

It is tuned to run smoothly on an ordinary laptop. If it feels heavy, raise the Pixel size or lower the speed in the settings; every control takes effect immediately.

Is the Julia Set Morph screensaver free?

Yes. It runs free in your browser with nothing to download or install, and it keeps working offline once the page has loaded.

How do I exit the Julia Set Morph screensaver?

Press the ESC key at any time to leave fullscreen and return to the page.

When did Julia and Fatou do this work?

Fatou published in 1917 and Julia in 1918, decades before computer graphics. Mandelbrot's computer visualisation of the Mandelbrot set came in 1980.

Why does colour change smoothly across the set?

Escape times are whole numbers, which would show bands. The renderer blends between counts, giving continuous shading.

Julia Set Morph Screensaver

Julia Set Morph is a free, in-browser fractal screensaver. Every Julia set is the map of which starting points escape to infinity under z² + c for one fixed c; this saver recomputes that map every frame while c slides along a slow orbit, so the fractal morphs continuously from spirals to dendrites to dust and back. Nothing to download, and ESC exits at any time.

How the Julia Set Morph screensaver works

For every pixel, the saver takes its position as a complex number z and repeats z → z² + c. If z ever grows past a radius of 4 it is counted as escaped and the pixel is coloured by how many steps that took; if it survives the full count it is part of the set and painted with the inside colour.

The Mandelbrot set is the same recipe with c varying by pixel. Here c is one value for the whole frame, and that value moves: around the circle of radius 0.7885 in the classic mode, along the edge of the Mandelbrot cardioid for parabolic, dendrite-like sets, or on a slow wander through the interesting region.

Escape counts are made smooth — a fractional correction from how far past the radius the point landed — so the colour bands blend instead of stepping. The frame is rendered at a reduced resolution into an offscreen buffer and scaled up, which is what keeps a full recompute every frame affordable at fullscreen.

Because c never repeats its exact path, the animation does not loop. Pixel size trades detail for smoothness: 1 is full resolution and the heaviest, 5 is fast and painterly.

Julia sets and their changing parameter

Pierre Fatou and Gaston Julia, French mathematicians, began the study of complex dynamics in the early twentieth century; Fatou published in 1917 and Julia his well-known memoir in 1918. The sets that carry Julia's name were studied through proof and algebra long before the pictures were drawn by computer.

The Mandelbrot set is the set of all values of c for which the Julia set of z² + c is connected. For c outside it, the Julia set is a disconnected cloud of points, sometimes called Fatou dust. Benoit Mandelbrot, working at IBM's Thomas J. Watson Research Center, first visualised the set in 1980. Moving c through the plane therefore passes through every kind of shape, and the sliding motion is a way of walking across that map.

The finest detail lies along the boundary, where tiny differences decide whether an iteration escapes after ten steps or never. Because there is a limit on the number of iterations, the rendered edge is an approximation, and zooming deeper requires more computation.

A worked example

At the defaults — Circle path, Spectrum palette, Speed 4, Pixel size 3 — the set starts as a dense whorl of spirals, unravels into a branching dendrite as c passes the top of its circle, dissolves into scattered dust near the bottom, and gathers back into spirals, one full cycle every minute or so. Switch the path to Edge of the main cardioid and every frame is a set that is exactly on the boundary between connected and disconnected.

Settings & tips

Pixel size 1 or 2 for a crisp still-image look; 4 or 5 for fluid motion on older hardware.
The Ice palette on the cardioid path looks like frost; Fire on the circle looks like solar prominences.
Slow the speed to 1 to watch a single spiral arm bud and split.

Frequently asked questions

What is a Julia set?: For a fixed complex number c, the Julia set is the boundary between starting points whose orbits under z → z² + c stay bounded and those that escape to infinity. Each c gives a different set; the Mandelbrot set is the catalogue of which c values give connected ones.
Why does the set sometimes fall apart into dust?: When c leaves the Mandelbrot set, its Julia set becomes totally disconnected — a Cantor dust. The circle of radius 0.7885 passes in and out of the Mandelbrot set, so the animation crosses that threshold twice per cycle.
Will this slow my computer down?: It is tuned to run smoothly on an ordinary laptop. If it feels heavy, raise the Pixel size or lower the speed in the settings; every control takes effect immediately.
Is the Julia Set Morph screensaver free?: Yes. It runs free in your browser with nothing to download or install, and it keeps working offline once the page has loaded.
How do I exit the Julia Set Morph screensaver?: Press the ESC key at any time to leave fullscreen and return to the page.
How is the Mandelbrot set related to Julia sets?: The Mandelbrot set is the set of values c for which the Julia set of z² + c is connected. For c outside it, the Julia set is dust. It works as a map of how Julia sets behave.
When did Julia and Fatou do this work?: Fatou published in 1917 and Julia in 1918, decades before computer graphics. Mandelbrot's computer visualisation of the Mandelbrot set came in 1980.
Why does colour change smoothly across the set?: Escape times are whole numbers, which would show bands. The renderer blends between counts, giving continuous shading.

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