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What Is the Mandelbulb?

A 9th Power Mandelbulb, a 3D Fractal of the Mandelbrot Set

A typical 9th power Mandelbulb

The Mandelbulb is a three dimensional manifestation of the Mandelbrot set. It is an infinitely complex, naturally occurring fractal object.

The two dimensional Mandelbrot set, discovered in the early 20th century, is named for the mathematician Benoit Mandelbrot—the ‘Father of Fractal Geometry’—who studied and popularized it.

The Mandelbrot Set is the most widely recognized mathematical fractal form. Along with fractal geometry, it entered popular culture in the mid 1980s. About thirty years passed between the first digital rendering of the flat Mandelbrot set in the late 1970s and the first rendering of its three dimensional equivalent, the Mandelbulb, in 2009.


The Mandelbrot set. Image credit: Wikimedia

The Mandelbulb was discovered by Daniel White and Paul Nylander, and developed collaboratively in the Fractal Forums community. These individuals set out to find a three-dimensional equivalent of the Mandelbrot set, and they found what they were looking for.

Using a spherical coordinate system, and some ingenious math, White and Nylander projected the Mandelbrot set into three dimensions, creating the Mandelbulb. In 3D-space, we see a more fully realized rendering of the Mandelbrot set. While the flat set exhibits infinite complexity, the Mandelbulb reveals that complexity in a fuller magnitude.

The Mandelbulb is an archetypal fractal form, embodying principles of deterministic chaos. The Mandelbrot set is the “most complex object on the complex number plane” but arises from a simple formula, commonly expressed as Z = Z² + C

The familiar shape of the Mandelbrot set is apparent when the Mandelbulb is shown in cross section.

The familiar shape of the Mandelbrot set is apparent when the Mandelbulb is shown in cross section.

The Mandelbulb, the Mandelbox, and Hybrid Fractals

The Mandelbulb is a 3D Fractal, also called a “Mandelmorph.” It is not the only Mandelmorph, or the only ‘pure’ expression of the Mandelbrot set… Keep Reading

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