Mandelbrot Explorer
Explore the Mandelbrot set with adjustable iteration depth and colouring, zoom into named features, and see why detail never runs out and where floating point finally breaks.
Last reviewed by the Radiatus Cloud team
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The rule is one line and the boundary is infinitely complex
Take a complex number c, start at zero, and repeatedly replace z with z squared plus c. If the result stays bounded forever, c is in the set. That is the entire definition, and from it comes a boundary with detail at every magnification: zoom anywhere on it and new structure appears, forever, without repeating. The interior is a simple black region and the exterior escapes quickly; everything interesting happens on the line between them, which has a fractal dimension of two despite being a boundary curve.
The colours show escape speed, not membership
Points outside the set are coloured by how many iterations they take to exceed a radius of two, because once the magnitude passes two it provably escapes. That count is what produces the bands, and it means the picture is really a map of escape time rather than of the set itself. Raising the iteration limit changes the picture: points that looked inside at 100 iterations reveal themselves as outside at 1000, so the black region shrinks as you look harder, and at any finite limit some of the black is an artefact.
Double precision runs out around a magnification of ten to the fifteen
Each zoom level needs more significant digits to distinguish neighbouring pixels. A double holds about sixteen decimal digits, so past roughly ten to the fifteen the pixel coordinates stop being distinguishable and the image degrades into blocks. Deep zoom software uses arbitrary precision and perturbation theory to go further; without that the limit is arithmetic rather than artistic, and it arrives sooner than most people expect.
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Frequently Asked Questions
What defines the set?
A complex number c is in the set if repeatedly applying z to z squared plus c, starting from zero, stays bounded forever. Everything visible follows from that one rule.
What do the colours mean?
How many iterations a point took to escape a radius of two. Points inside the set never escape and are drawn black, so the picture is a map of escape time rather than of membership.
Why does the black region shrink when I raise the iteration limit?
Because points that had not yet escaped at the lower limit do escape at the higher one. At any finite limit some of the black is an artefact of not having looked long enough.
How deep can I zoom?
To about ten to the fifteen with double precision, after which the pixel coordinates are no longer distinguishable and the image breaks into blocks. Going deeper requires arbitrary precision arithmetic.
Why radius two?
Because once the magnitude of z exceeds two it provably grows without bound, so no further iteration is needed. It is a proved threshold rather than a chosen one.
Privacy & Security
Everything runs in your browser; nothing is uploaded.
How to Use
Choose a location and zoom to explore the set.
Disclaimer: This tool is provided "as is" without warranty of any kind. Results are for educational and utility purposes.