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 |
Über diesen Rechner
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.
Häufige Beispiele
- 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)