It sounds like a poorly thought through Pokémon name from 1995. But in fact the Mandelbulb is a beautiful three-dimensional mathematical object that generalises the two-dimensional Mandelbrot set, which has become well known as an example of a fractal.

The Mandelbrot set (black) with additional coloring (blue to white).

The Mandelbrot set is generated by a very simple idea: we take a point in the 2d plane, and use that as the input to a rule which generates a new point. We then use the new point as the input to the same rule, to generate a third point, then a fourth point, and so on. If we start with the point $z_0 = 0$ (the origin) we generate the sequence $z_1, z_2, z_3,\dots$ from it. This simple but powerful idea, iteration, is at the heart of chaos theory and fractal images.

The rule that generates the Mandolbrot set is given by the following formula, where $c$, $z_n$ and $z_{n+1}$ are complex numbers:

$z_{n+1} = z_n^2 + c$

If we think of a point in the plane defined by its magnitude (or distance from the origin) and the angle it makes with the horizontal, then this rule simply says: “to get the next point in the sequence you square the magnitude, double the angle, and then shift the point by a constant amount $c$.”

There are two possibilities when we start iterating points in this way: for some values of $c$ the points get further and further away from where they started, and head off to infinity (we say that the sequence diverges) but for other values of $c$ they never stray further away than some limit (in which case we say that the sequence is bounded.) To make the Mandelbrot set we color a point $c$ black if the sequence for that value of $c$ is bounded, and leave it white if the sequence diverges.

The boundary of the Mandelbrot set is infinitely complicated: no matter how far you zoom into it, you never stop seeing more and more detail. This never-ending detail qualifies it to be a fractal. We can make prettier pictures by coloring the points that diverge according to how long they take to diverge: in the image above the points that diverge quickly are shown in blue, whereas points that diverge only after many iterations (i.e. they’re nearly in the Mandelbrot set) are shown in white.

The challenge of the Mandelbulb is to extend this concept into three dimensions, to allow fully 3d renders that incorporate dramatic lighting and shadows. As we’ll see this isn’t as easy as it sounds at first.

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