Julia Set Explorer
Render Julia sets for any complex parameter
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.
How to use the Julia Set Explorer calculator
- Enter the values for your problem into the input fields.
- Read the result — it updates instantly as you type.
- Check the formula and the visual explanation to follow how the answer was found.
- Copy the page URL to share the exact calculation.
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)
Frequently asked questions
How does the Julia Set Explorer calculator work?
Enter your values and the calculator applies the exact mathematical method for this problem, showing the result together with the formula it used. Everything is computed with high-precision arithmetic, so the answer you see is not limited by ordinary floating-point rounding.
When would I use the Julia Set Explorer calculator?
It is useful for homework and exam preparation, for checking work done by hand, and whenever a step in a larger problem needs this calculation done quickly and reliably.
How accurate are the results?
Calculations run at 30 significant digits of precision internally. Displayed values are rounded for readability, but the underlying result is far more precise than a typical hand or pocket-calculator computation.
Can I share a specific calculation with someone else?
Yes. Every value you enter updates the URL, so copying the address bar and sharing the link reopens the exact same calculation for whoever clicks it.