Mandelbrot Explorer
Zoom into the Mandelbrot set
حول هذه الآلة الحاسبة
The Mandelbrot Explorer zooms into the Mandelbrot set — the complex numbers c whose orbit under z to z squared plus c stays bounded. Enter a center point, a zoom factor and an iteration budget: it renders the fractal around your point and classifies that point, proving membership exactly when it lies in the main cardioid or the period-2 bulb and otherwise iterating to find the escape iteration and final modulus. Every view is shareable by URL for complex-dynamics and fractal study.
How to explore the Mandelbrot set
- Enter the center point as a real part and an imaginary part.
- Set the zoom factor — higher values dive deeper into the fractal.
- Raise the iteration budget for more detail near the boundary.
- Read the verdict for the center point and share the view by copying the link.
أمثلة شائعة
- c = 0 → in the main cardioid (period 1); the orbit stays at 0
- c = −1 → in the period-2 bulb; the orbit cycles 0, −1, 0, −1
- c = 2 → escapes at iteration 2 as |z| grows without bound
- Center −0.5, zoom 1 → the classic whole-set view
- Center −0.743 + 0.113i, zoom 2000 → a deep spiral in the boundary
الأسئلة الشائعة
What is the Mandelbrot set?
It is the set of complex numbers c for which the sequence z to z squared plus c, starting from z = 0, never runs off to infinity. Points inside stay bounded; points outside escape.
How does this calculator decide whether a point is in the set?
Two large regions — the main cardioid and the period-2 bulb — have exact membership formulas, so points inside them are proven to belong. Every other point is iterated up to your iteration budget; if the modulus never passes 2 the point is reported as bounded.
What does the escape iteration mean?
It is the first step at which the modulus of z exceeds 2. Once the orbit passes radius 2 it is guaranteed to diverge, so a smaller escape iteration means the point leaves the set faster — that is what the color gradient shows.
Why is deep zoom limited?
The visual samples the plane with ordinary double-precision numbers, which lose accuracy past roughly a trillion-fold zoom. The calculator caps the zoom there so the picture stays faithful.