Julia Set Explorer

Render Julia sets for any complex parameter

f(z) = z² + c | c = -0.8 + 0.156i

Parameter c

-0.8 + 0.156i

|c|

0.8151

Escape radius R

1.532

Connectivity

Connected — a single connected piece

The critical orbit (starting at z₀ = 0) stays bounded, so c lies in the Mandelbrot set and the Julia set is connected.

Fixed points of z ↦ z² + c

z	\|λ\| = \|2z\|	Behavior
1.5275 − 0.0759i	3.0588	Repelling
-0.5275 + 0.0759i	1.0659	Repelling

About this calculator

The Julia Set Explorer renders the Julia set of f(z) = z² + c for any complex parameter c. It reports whether the set is connected (the Fatou–Julia dichotomy: connected exactly when c is in the Mandelbrot set), finds the two fixed points z = (1 ± √(1 − 4c)) / 2 with their multiplier |λ| = |2z| and stability, computes the escape radius R = (1 + √(1 + 4|c|)) / 2, and tests whether a seed z₀ belongs to the filled Julia set by escape time.

Common examples

c = 0 → Julia set is the unit circle (connected)
c = −1 → the 'basilica' (connected; period-2)
c = −0.8 + 0.156i → a connected Julia set
c = 0.5 + 0.5i → outside the Mandelbrot set, a disconnected dust
c = −0.123 + 0.745i → the Douady rabbit (connected)