What Is It?
The Mandelbulb is a 3D fractal built by extending the Mandelbrot set's iteration into three dimensions using spherical coordinates. Where the 2D Mandelbrot squares complex numbers, the Mandelbulb raises 3D points to a power n by converting to spherical form (r, θ, φ), multiplying the angles by n, and raising the radius to rn.
The result looks nothing like a simple extrusion of the 2D set. At power 8 - the value that produced the original discovery - the Mandelbulb forms bulbous, organic shapes covered in surface textures that repeat at every scale.
How It Was Found
Daniel White started the search on the Fractal Forums around 2007, trying different ways to extend complex multiplication into 3D. His "triplex algebra" using spherical coordinates was the key idea. Paul Nylander suggested trying power 8, and in 2009 the first renders of the Mandelbulb appeared.
The community had been looking for a "true 3D Mandelbrot" for years. Previous attempts - quaternion Julia sets, hypercomplex numbers - produced interesting shapes but didn't have the same boundary structure as the 2D set. The Mandelbulb isn't a perfect analogue either (there's no rigorous proof it's connected, for instance), but it's the closest anyone has found.
What the Power Parameter Does
Changing the power n completely reshapes the fractal:
- Power 2: A smooth, simple shape - barely recognizable as a fractal
- Power 8: The classic Mandelbulb. Rounded bulbs covered in fine detail
- Power 12+: More lobes appear, surfaces get sharper, and the whole thing starts looking like a sea urchin
The Parameter Morphing animation sweeps through power values so you can watch these transitions happen in real time.
Exploring It
Sweeps through different power values, showing the fractal reshape itself continuously.
Slow camera rotation around the fractal - good for seeing the full 3D structure.
Moves the camera close to the surface, skimming over the detail at close range.
Full manual control. Mouse or touch to orbit, zoom, and fly through the geometry.
How It's Rendered
The Mandelbulb is rendered using ray marching with distance estimation - the same approach used for all our 3D fractals. A fragment shader casts a ray for each pixel, steps forward using the distance estimator, and applies lighting when it hits the surface. The distance estimator tracks a scalar running derivative alongside the main iteration, which keeps the math cheap enough for real-time frame rates on a GPU.