What is the Mandelbrot set?

The Mandelbrot set is the set of all complex numbers c for which the sequence z₂ = z² + c (starting at z₀ = 0) never diverges to infinity. When you plot these numbers in the complex plane - real part on the x-axis, imaginary part on the y-axis - the boundary between points that stay bounded and those that escape is infinitely complex: zoom in as far as you like and you will always find new structure, never a smooth curve.

Named after mathematician Benoît Mandelbrot, who popularized it in 1980, the set is one of the most recognized fractal images in mathematics. Its characteristic cardioid and circular bulbs bulge into an infinitely ornate filament boundary that has become a symbol of the surprising beauty hidden in simple equations.

How to navigate

Scroll or pinch to zoom in and out.
Click and drag to pan across the plane.
Click to center on any point you find interesting.
Switch between the Mandelbrot and Julia set view using the controls.
Adjust the max iterations slider for more detail at high zoom levels (at the cost of speed).

Points of interest

Location	Coordinates	What you’ll see
Main cardioid	c ≈ −0.12 + 0.74i	The large heart-shaped region; all points here produce a stable fixed orbit.
Period-2 bulb	c ≈ −1.0	The large circle on the left; orbits alternate between two values.
Seahorse Valley	c ≈ −0.75 + 0.1i (zoom ×500)	Intricate spiral filaments resembling seahorse tails.
Elephant Valley	c ≈ 0.275 + 0.009i (zoom ×200)	Elephant-trunk-like spirals pointing inward.
Mini-brot	c ≈ −1.755 + 0.0i (zoom ×10⁷)	A miniature copy of the entire Mandelbrot set, buried deep in the filaments.

Julia sets

For every complex number c, there is a corresponding Julia set: the set of starting points z₀ whose orbits under z -> z² + c remain bounded. The Mandelbrot set acts as a map of Julia sets: if c is inside the Mandelbrot set, its Julia set is connected; if c is outside, the Julia set is a disconnected cloud of points called Cantor dust. Pick any point in the Mandelbrot view and its Julia set will show you the local fractal character of that region.

Color mapping

Points inside the Mandelbrot set (which never escape) are typically colored black. Points outside are colored by the escape time - the number of iterations before the sequence’s magnitude exceeds 2 (the bailout radius). Fast-escaping points are assigned one color; slow-escaping points another; smooth coloring algorithms interpolate between these to eliminate the visible banding and produce the smooth gradient color maps you see.

Other classic fractals

Sierpiński triangle: formed by repeatedly removing the central triangle from each remaining triangle. Has a fractal dimension of approximately 1.585.
Koch snowflake: constructed by adding a triangular bump to each side of an equilateral triangle repeatedly. Infinite perimeter, finite area. Dimension ≈ 1.2619.
Burning Ship fractal: a variation of the Mandelbrot set using absolute values in the iteration, producing a distinctive ship-like shape with dramatic cliff structures.
Newton fractal: generated by applying Newton's root-finding method to a complex polynomial - the boundaries between basins of attraction produce intricate fractal patterns.

Fractal dimension

Unlike ordinary geometric shapes, fractals have non-integer (fractal) dimensions. A line has dimension 1; a filled square has dimension 2. The boundary of the Mandelbrot set has a Hausdorff dimension of exactly 2 (despite being a curve), while the Koch snowflake boundary has dimension ≈ 1.2619. This captures the intuitive idea that a fractal curve is "more" than a line but "less" than a plane - its complexity lies between integer dimensions.