Julia Set Explorer: Drag c and Watch the Fractal Change
A free Julia set explorer: drag a point on a map of the Mandelbrot set and watch its Julia set change live, type any constant c, walk c around the cardioid, zoom and cycle the colours, save pictures and share links. Runs in your browser.
Drag to move · scroll, pinch or double-click to zoom · click a point to inspect it. In the Mandelbrot picture each point is a number c; the colour tells how fast the orbit of z → z² + c runs away. Black points never do.
Drag to move · scroll, pinch or double-click to zoom. In a Julia set the constant c is fixed and each point is a starting value z for z → z² + c; the colour tells how fast it runs away and black points never do. Choose c on the map in the corner or below.
The Julia constant c
Every c makes a different Julia set. Click or drag on the small map in the corner of the picture to choose it, type it here, or let it walk around the edge of the Mandelbrot set.
Type a number c = a + bi and see what its orbit does: it either escapes to infinity after some number of steps (outside the set) or stays bounded (inside). For the fractal chosen in the Explorer.
The orbit
| step n | zn real | zn imaginary | |zn| |
|---|
Julia set explorer
A Julia set is a fractal made by one simple rule: pick a complex number c, then for every point z of the plane repeat z → z² + c. Points whose orbit stays bounded are in the set (drawn black here); the rest are coloured by how fast they escape. Change c a little and the whole shape changes: this explorer lets you drag c across a map of the Mandelbrot set and watch the Julia set transform live.
How to use it
- Choose c on the map. The small Mandelbrot picture in the corner is a map of all the constants. Click or drag on it and the Julia set of that point appears. Make the map bigger or smaller with + and −, or close it with × (tick “c map” to bring it back).
- Type c exactly, or pick a famous one from the list: Douady's rabbit, the dendrite, San Marco, a Siegel disk, Fatou dust, the spiral galaxy.
- Walk c around the Mandelbrot set. Press the walk button and c travels along the edge of the main cardioid; you see the Julia set grow bulbs, pinch off and shatter into dust as it goes.
- Zoom and explore. Drag, scroll or pinch to zoom; the orbit of the point under the pointer is drawn so you can see why it escapes or not. Save a picture or copy a link to the exact view.
The link with the Mandelbrot set
The Mandelbrot set is the set of constants c whose Julia set is connected (one piece). Choose a c inside the Mandelbrot set and you get a connected shape (a rabbit, a dendrite, a Siegel disk); choose one outside and the Julia set falls apart into infinitely many separate specks called Cantor dust. The Mandelbrot set is a kind of index of all Julia sets: its bulbs tell you how many “arms” the Julia set has and its shape near c predicts the Julia set's shape. See it in the Mandelbrot Explorer, where hovering over any point previews its Julia set.
Things people ask
- What is the difference between the Mandelbrot set and a Julia set? In the Mandelbrot set c changes with the pixel and the start value is always 0. In a Julia set c is fixed and the start value changes with the pixel.
- How many Julia sets are there? One for every complex number c: infinitely many, each different.
- Why is the picture sometimes mostly black or mostly dust? Inside the Mandelbrot set the Julia set encloses a black filled region; outside, it is dust that is nearly invisible, so the picture is just colours.
- Who was Julia? Gaston Julia, a French mathematician who studied these sets around 1918, decades before computers could draw them.
Everything runs on your device; nothing is uploaded. Related: Mandelbrot Explorer · Buddhabrot Explorer · 2D graphing calculator.