Julia set

Parametric equations

For a fixed complex constant $c$, substitute each starting point $z_0$ into $f(z)=z^2+c$ again and again. Some of those orbits stay bounded forever. Others escape to infinity. The Julia set $J_c$ is the boundary between these two kinds of starting points. When $|c|\le 2$ (as on this page), $|z_n|>2$ forces the orbit to diverge. How fast it diverges is drawn as color bands, and the interior that does not diverge stays black. Changing $c$ can take the boundary from one connected branching shape to a cloud of separate points (Cantor dust).

Interaction

• Parameter drift: while this is on, $c$ moves slowly near the boundary of the Mandelbrot set, and the shape of the fractal boundary keeps changing
• Manual c: turn drift off, then set $c$ with Real part Re(c) and Imaginary part Im(c). The boundary is recomputed
• Maximum iterations: raising the iteration cap refines the boundary, but the calculation is slower

What to notice

• For a fixed $c$, the Julia set is the boundary between starting points that stay bounded and starting points that escape to infinity.
• With $|c|\le 2$, once $|z_n|>2$ the iteration diverges, and how fast it diverges sets the color bands outside.
• As $c$ changes, the set can pass from a connected branching shape to a disconnected cloud of fractal points.

Related

• Euler’s formula, rotating
• Sierpinski triangle
• Logistic map bifurcation diagram

Further reading

• Julia set (Wikipedia)