See: our gallery of how the Mandelbrot set burst out of two dimensions
It may look like a piece of virtuoso knitting, but the makers of an image they call the Mandelbulb (see right) claim it is most accurate three-dimensional representation to date of the most famous fractal equation: the Mandelbrot set.
Fractal figures are generated by an 鈥渋terative鈥� procedure: you apply an equation to a number, apply the same equation to the result and repeat that process over and over again. When the results are translated into a geometric shape, they can produce striking 鈥渟elf-similar鈥� images, forms that contain the same shapes at different scales; for instance, some look . The tricky part is finding an equation that produces an interesting image.
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The most famous fractal equation is the 2D , named after the mathematician of Yale University, who coined the name 鈥渇ractals鈥� for the resulting shapes in 1975.
But there are many other types of fractal, both in two and three dimensions. is one of the simplest 3D examples.
Fake fractal
There have been previous attempts at a 3D Mandelbrot image, but they do not display real fractal behaviour, says , an amateur fractal image maker based in Bedford, UK.
Spinning the 2D Mandelbrot fractal like wood on a lathe, raising and lowering certain points, or invoking higher-dimensional mathematics can all produce apparently three-dimensional Mandelbrots. Yet none of these techniques offer the detail and self-similar shapes that White believes represent a true 3D fractal image.
Two years ago, .
The next dimension
鈥淚 was trying to see how the original 2D Mandelbrot worked and translate that to the third dimension,鈥� he explains. 鈥淵ou can use complex maths but you can also look at things geometrically.鈥�
This approach works thanks to the properties of the 鈥渃omplex plane鈥�, a mathematical landscape where ordinary numbers run from 鈥渆ast鈥� to 鈥渨est鈥�, while 鈥渋maginary鈥� numbers, based on the square root of -1, run from 鈥渟outh鈥� to 鈥渘orth鈥�. Multiplying numbers on the complex plane is the same as rotating it, and addition is like shifting the plane in a particular direction.
To create the Mandelbrot set, you just repeat these geometrical actions for every point in the plane. Some will balloon to infinity, escaping the set entirely, while others shrink down to zero. The different colours on a typical image reflect the number of iterations before each point hits zero.
White wondered if performing these same rotations and shifts in a 3D space would capture the essence of the Mandelbrot set without using complex numbers 鈥� real numbers plus imaginary numbers 鈥� which do not apply in three dimensions because they are on only two axes. In November 2007, White published a formula for a shape that came pretty close.
Higher power
The formula published by White gave good results, but still lacked true fractal detail. Collaborating with the members of , a website for fractal admirers, he continued his search. It was another member, , who eventually realised that raising White鈥檚 formula to a higher power 鈥� equivalent to increasing the number of rotations 鈥� would produce what they were looking for.
White鈥檚 search isn鈥檛 over, though. He admits the Mandelbulb is not quite the 鈥渞eal鈥� 3D Mandelbrot. 鈥淭here are still 鈥榳hipped cream鈥� sections, where there isn鈥檛 detail,鈥� he explains. 鈥淚f the real thing does exist 鈥� and I鈥檓 not saying 100聽per cent that it does 鈥� one would expect even more variety than we are currently seeing.鈥�
Part of the problem is that extending the Mandelbrot set to 3D requires many subjective choices that influence the outcome. For example, you could extend a flat plane to 3D by stretching it to form a box, but you could also turn it into a sphere.
鈥淚t鈥檚 an interesting academic exercise to think what you should get,鈥� says , a computer scientist specialising in fractal images at the University of Manchester, UK, 鈥渂ut it all depends on what properties you want to keep in the third dimension.鈥�
The equations White used may get the job done, but the system of algebra used is not applicable to all 3D mathematics. 鈥淭he next stage is mathematical rigour,鈥� says Turner.
See: our gallery of how the Mandelbrot set burst out of two dimensions
