Mandelbrot explorer · Julia set generator
Julia Portal
The Mandelbrot set is a map of every Julia set: each point c on it stands for the shape you get by repeating "square it, add c". Tap any point to open its Julia set, drag or pinch to fly deeper, and save the one you like full screen.
- c
- In the set
- Julia set
- Orbit of 0
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The link reopens this exact point and view.
How it works
One rule, two pictures
1. Square it, add c
Everything on this page comes from one rule: take a number z, square it, add c. The numbers are complex numbers, points on a flat plane, where squaring doubles the angle and squares the distance from the center. Repeat the rule and a point either stays close forever or flies off to infinity. Once it gets farther than 2 from the center, it never comes back.
2. The Mandelbrot set: change c, start at 0
For the big map, every pixel is a different c, and each one starts at z = 0. The black region is every c whose numbers stay close. The colors outside show how many steps the others took to escape, which is where the glowing edges come from. Its main body is a cardioid, the same heart shape the 2 times table draws on a circle.
3. A Julia set: fix c, change the start
For the Julia set, c stays fixed at the point you tapped and every pixel is a different starting point. The black region is every start that stays close. If your c is inside the Mandelbrot set, the Julia set is one connected piece. If c is outside, it shatters into dust. That is why the Mandelbrot set is called a map of Julia sets.
4. The cycles behind the bulbs
Inside the main cardioid, the orbit of 0 settles on a single point. In the big disc to its left, it settles into a cycle of 2, and in the bulbs on top and bottom a cycle of 3, which gives the rabbit its three-way junctions. The panel under the Julia set runs the orbit of 0 for your c and reports which cycle it finds.
Real vs legend
What is math and what is story
| Claim | Status |
|---|---|
| If c is in the Mandelbrot set its Julia set is one connected piece; if not, it is dust. | Real |
| The main body of the Mandelbrot set is a cardioid, the shape of the 2 times table on a circle. | Real |
| The edge of the Mandelbrot set is so crinkled it is as complex as a flat area (its fractal dimension is 2, proved by Shishikura in 1998). | Real |
| Gaston Julia studied these sets in 1918, decades before computers could draw them. | Real |
| Zooming in never runs out of new detail. | Real |
| The Mandelbrot set is the fingerprint of God or a map of consciousness. | Legend |
| Ancient temples or crop circles show the Mandelbrot set. | No source |
Questions
Mandelbrot and Julia FAQ
What is the Mandelbrot set?
Pick a number c on the plane. Start at 0, square it and add c, then square the answer and add c again, over and over. If the numbers stay within a distance of 2 from the center forever, c is in the Mandelbrot set. The black shape is every c that stays; the colors show how fast the others escape.
What is a Julia set?
Keep c fixed and change the starting point instead. The filled Julia set is every starting point that stays bounded when you repeat square and add c. Each c has its own Julia set, so there are infinitely many. Gaston Julia studied them in 1918, long before computers could draw them.
How are the Mandelbrot set and Julia sets connected?
The Mandelbrot set is a map of all Julia sets. If c is inside the Mandelbrot set, its Julia set is one connected piece. If c is outside, its Julia set breaks into dust: infinitely many separate points. Near the edge of the Mandelbrot set, zooming in often shows shapes that look like the Julia set of that same c.
Why is it called the Douady rabbit?
At c ≈ −0.1226 + 0.7449i the Julia set looks like a rabbit with a body and two ears, and every ear has its own smaller body and ears, without end. It is named after the mathematician Adrien Douady. That c sits in the bulb where the orbit of 0 settles into a cycle of 3.
How far can I zoom?
About 600 billion times, until the computer runs out of decimal places. The set itself has detail at every scale, so the only limit is the arithmetic, not the shape.
Can I use a Julia set as a wallpaper?
Yes. Open it full screen and press Save to keep it at your screen's size, or get the 4K wallpaper at 2160 × 3840. The share link reopens the exact same point and view.
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