Julia Set Explorer

How the Julia Set Explorer works

Julia set explorer: how it works

Every pixel is a starting value z. The page repeats z = z² + c with the same c for every pixel and counts how many steps it takes for z to get further than 2 from the origin. Points that never escape within the iteration limit are drawn black; the rest are coloured by how fast they escaped, smoothed so the bands blend into each other.

The constant. Each c gives a different set, and the shape depends on where c sits in the Mandelbrot set. A c inside the main cardioid gives a connected blob with a wavy edge; one inside a bulb gives a set with the same number of arms as the bulb's period, which is why the Rabbit (period 3) has three lobes at every fork; a c on the boundary gives a dendrite, a branching line with no interior; and a c outside the Mandelbrot set entirely gives dust, a scatter of disconnected points. Random from seed picks a c on a ring of radius about 0.75 around the centre of the Mandelbrot set, which lands close to the boundary often enough to be interesting; press Regenerate for another.

Zooming. A click recentres on that point and magnifies by 2.5; shift-click zooms out. Unlike the Mandelbrot set, a Julia set looks the same at every scale near any point of its boundary, so zooming is less a search for new shapes than a way to frame the pattern you already have.

Iteration limit. Points near the boundary take longer to escape. Too low a limit and the boundary fattens into a black blur; raise it and the filaments thin out. 300 is plenty for the full view; go higher when you zoom in. Every pixel may run the full loop, so 4K at 2,000 iterations takes a few seconds.

Palette and colour cycle. Colour cycle sets how many escape steps make one trip through the palette. Regenerate shifts the palette's starting point without changing the view (unless the constant is Random from seed, in which case it also picks a new c).

Export. Save PNG recomputes the current view at the chosen size.

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