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🫧 Mandelbrot / Multibrot Set (2D)

The real 2D Mandelbrot set: z_n+1 = z_n^n + c, iterated per pixel with a true escape-time algorithm and smooth continuous coloring. Pan, zoom and slide the power n to see the Multibrot variants that the 3D Mandelbulb generalizes into spherical coordinates.

Mathematics2DModerate60 FPS📱 Mobile-adapted⇄ 3D version

About this simulation

This is the real, standard 2D Mandelbrot set: for every pixel c in the complex plane, z starts at 0 and is repeatedly updated by z → zⁿ + c. If |z| ever exceeds a bailout radius the point has "escaped" and is colored by how quickly that happened, using smooth continuous coloring; points that survive the full iteration budget are considered inside the set and painted near-black. With n=2 this is the classic Mandelbrot set; sliding the power exponent produces the related family of Multibrot sets. The 3D Mandelbulb takes this same power-n idea and applies it to spherical coordinates in three dimensions — this page is its 2D parent.

What it shows

A true escape-time fractal computed per pixel with real complex-number iteration, not an approximation — z_{n+1} = z_n^n + c tested against a large bailout radius, with a smooth normalized escape count driving the color.

How to use

Drag to pan, scroll or pinch to zoom into the boundary. Power n reshapes the set (n=2 is the classic Mandelbrot, higher n gives Multibrot variants with more lobes). Max iterations trades fine boundary detail for speed. Reset view returns to the default framing.

Relation to the Mandelbulb

The 3D Mandelbulb has no true complex-multiplication analogue, so it reuses this same power-n exponent idea in spherical coordinates (r→r^n, θ→nθ, φ→nφ). This page is the 2D original that generalization is built from.

⚙ Under the hood

The real 2D Mandelbrot set: z is iterated as z^n + c per pixel with a true escape-time algorithm and smooth continuous coloring. Pan and zoom into the fractal boundary and slide the power exponent to explore the Multibrot family -- the flat 2D fractal that the 3D Mandelbulb generalizes into spherical coordinates.

fractalmandelbrotmultibrotcomplex numbersescape timemath

2D · HTML5 Canvas 2D · 60 FPS target · runs fully client-side, no install