Fractals

How it works

For every point c of the complex plane, the calculation z → z² + c is repeated, starting from z = 0. Once |z| exceeds 2, the sequence is sure to escape to infinity and the point is coloured by how quickly it escaped. Points that have not escaped after the chosen number of iterations are dark: either they belong to the set, or the calculation simply did not run long enough. A Julia set uses the same formula, but c is fixed and the starting z changes.

zₙ₊₁ = zₙ² + ca point escapes once |z| > 2

Try it yourself

1. Click “Seahorse valley” and zoom with the mouse wheel. The shapes on the boundary repeat in smaller and smaller variations.
2. When zoomed in deeply, set “Detail / iterations” to 100 and then to 2000. Some dark areas break apart: they were dark only because there were too few iterations.
3. Change “Exponent of z” to 3. The formula z³ + c gives two main lobes instead of one; in general, exponent d gives d − 1 lobes.
4. Switch “Fractal family” to Julia and choose “Julia spiral”. A small change of the parameter c transforms the shape smoothly.

Model limitations

The computer uses finite number precision and a limited number of iterations, so zoom is limited to 10¹²×. A dark point is therefore not proof that it belongs to the set.

Related experiments

• Galton board
• Double pendulum
• Logic circuits