Mandelbrot Zoom

What is Mandelbrot Zoom?

Mandelbrot Zoom lets you explore the Mandelbrot set, the most famous fractal: a shape made by one tiny formula, z → z² + c, whose edge holds endless spirals, seahorses, lightning and tiny copies of the whole set, however far you zoom. Drag to move, scroll or pinch to zoom, and press Zoom In to glide deeper on its own.

The Mandelbrot set, Seahorse Valley, a double spiral and a view zoomed 10^18 times, each in a different colour palette, made with Mandelbrot Zoom

Ordinary fractal viewers stop at a zoom of about 10^13, where computer numbers run out of digits and the picture turns to blocks. This one keeps the centre of the view to 60 digits and draws each pixel as a small difference from one precisely computed point, so you can zoom down to 10^30, a billion billion times further, in real time on your graphics card.

Features

• Deep zoom to 10^30, with automatic detail that adds steps as you go deeper.
• Famous places: Seahorse Valley, Elephant Valley, a spiral galaxy, a double spiral, a copy of the set, dendrites, a Julia island and a deep spiral.
• Colour palettes with smooth shading, adjustable bands and colour cycling.
• Julia sets: see the Julia set of any point of the Mandelbrot set.
• Wallpapers up to 4K and phone sizes, drawn with smoothed edges.
• Zoom videos: record a video zooming from the whole set into the view you found.
• Exact coordinates to copy, go to, or share as a link.

How to use Mandelbrot Zoom?

1. Drag to move around, and scroll, pinch or double-click to zoom in (Shift and double-click to zoom out).
2. Press Zoom In to dive into the centre automatically, at the speed you choose.
3. Pick a famous place to start from, and a palette you like.
4. Save a picture or a wallpaper, record a zoom video to where you are, or copy the link to share the view.

Frequently asked questions

What is the Mandelbrot set?

It is the set of numbers c on the complex plane for which the sequence z → z² + c, starting from z = 0, never grows without limit. The points inside are drawn black; the colours outside show how quickly the sequence escapes. Benoit Mandelbrot made the first pictures of it in 1980, and it became the symbol of fractals and chaos theory.

Does the Mandelbrot set go on forever?

Its edge does: it has infinite detail, and zooming never reaches a smooth line. What you find changes as you go, and small copies of the whole set appear at every depth, each surrounded by its own patterns. The only limit is computing power; here the zoom goes to 10^30.

What is the difference between the Mandelbrot set and a Julia set?

Both use z → z² + c. The Mandelbrot set changes c from point to point and always starts at z = 0; a Julia set keeps one c fixed and starts at each point. Each point of the Mandelbrot set has its own Julia set, and they look alike: a Julia set from inside the set is connected, one from outside falls apart into dust.

Why do deep zooms look blocky in other viewers?

Graphics cards work with numbers of about 7 digits, and normal computer numbers have about 16. Past that depth, neighbouring pixels get the same coordinates. This tool works out one point's path with 60 digits, then follows only each pixel's tiny difference from it (a technique called perturbation), which keeps the picture sharp to 10^30.

Can I use the pictures?

Yes. Everything is drawn in your browser, and the wallpapers and videos you make are yours to use. For more simulations, try the Double Pendulum Simulator, the Galton Board Simulator and the Game of Life Simulator.