Fractal set pattern generator

Infinite detail on a small window

A fractal set is the boundary between points that fly off to infinity and points that stay put when a simple formula is applied again and again. The Mandelbrot set, drawn by Benoit Mandelbrot in 1980, is the best known. Julia sets, named after Gaston Julia, come from the same formula with a fixed constant. The Burning Ship, found in 1992, folds the numbers before each step, and the Newton fractal shows which root of a polynomial each starting point is drawn to.

Close to the edge of such a set the detail never runs out, which is why a small window can hold spirals, filaments and tiny copies of the whole.

How the generator builds them

For every point in the window the generator counts how many steps it takes to escape, and cuts that count into bands, so the picture comes out as flat areas of color you can use as vector art. The number of bands per step sets how fast the colors cycle.

A fractal does not repeat by itself, so the window is mirrored at the edges of the tile: each side is a reflection of its neighbor, and the picture joins up without a seam. Zoom and center let you choose any part of the set to put in the window.

What you can adjust

• Set — Mandelbrot, Julia, Burning Ship or Newton.
• Zoom — the half-width of the window; smaller values zoom in.
• Center — where in the plane the window sits.
• Bands — color bands per iteration, from broad rings to fine stripes.
• Angle — the Julia constant, moved around its circle.
• Exponent and roots — the power in the formula, or the number of roots for Newton.
• Detail — the iteration limit, higher for deep zooms.
• Colors — a ground and up to three bands; any palette.

Export and use

Download the seamless tile as PNG or SVG, or copy a ready CSS background. The patterns you create are free to use, personally or commercially.