Mandelbrot Explorer
Zoom into the Mandelbrot set to the limit of double precision, or switch to Julia sets. Five palettes, named landmarks, and a save button.
Click to zoom in · Shift-click to zoom out · Drag to pan · Scroll to zoom at the pointer
What you are actually looking at
- Take a point, square it, add the point, and repeat. If the result stays bounded forever the point is in the set and is drawn black. If it runs away, the colour says how many steps it took.
- Escaping past a distance of 2 is the whole test. Once a value gets that far it is guaranteed to run away, so there is nothing to gain by iterating further. That one fact is what makes the picture computable at all.
- The black shape is not smooth anywhere. Zoom in as far as the arithmetic allows and the boundary keeps producing new detail — and, buried in it, complete small copies of the whole set.
- Deeper views need more iterations, so the count rises automatically as you zoom. Without that a deep view collapses into a flat blob.
- A Julia set is connected exactly when its constant lies inside the Mandelbrot set, and scattered dust when it does not. That is what the Mandelbrot set really is: a map of which Julia sets hold together.
- Zooming stops at about 10¹³×, where ordinary double-precision arithmetic runs out and the image would go blocky. It stops rather than pretending.
How to use
- Click to zoom in, shift-click to zoom out, drag to pan.
- Use Go to for named places worth seeing.
- Switch to Julia for the related family of shapes.
Frequently asked questions
What is actually being calculated?
Take a point, square it, add the point, repeat. If the result stays bounded forever the point is in the set and is drawn black; if it runs away, the colour records how many steps it took. Escaping past a distance of 2 is the exact test — once a value gets that far it can never come back — and that single fact is what makes the picture computable at all.
Why does it stop zooming?
At around ten trillion times, ordinary double-precision arithmetic runs out of digits and the image would break into flat blocks. It stops there rather than pretending to go deeper. Going further needs arbitrary-precision arithmetic, which is far slower.
Does the detail ever run out?
No. The boundary is not smooth at any magnification — every zoom produces new structure, and buried in it are complete small copies of the whole set. There are infinitely many of those.
What is a Julia set?
The same rule with the added constant held fixed instead of varying with the point. Each constant gives a different picture. The connection is exact and rather beautiful: a Julia set is connected — one joined-up piece — precisely when its constant lies inside the Mandelbrot set, and is scattered dust when it does not.
Why does the iteration count keep going up?
Because deeper views need it. Points near the boundary take longer and longer to declare themselves, so a fixed count turns a deep zoom into a flat blob. The count rises with depth automatically and is shown so you can see it happening.
Where should I zoom first?
The Go to list has the classic places — the seahorse and elephant valleys either side of the main body, a triple spiral, and a miniature copy of the whole set sitting on the spike at about −1.75.
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