![](/content/sites/plus.maths.org/files/styles/small_square/public/latestnews/sep-dec09/mandelbrot/icon.jpg?itok=1fiEk6OF)
An amateur fractal programmer has discovered a new 3D version of the Mandelbrot set. Daniel White's new creation is based on similar mathematics as the original 2D Mandelbrot set, but its infinite intricacy extends into all three dimensions, revealing fractal worlds of amazing complexity and beauty at every level of magnification.
![A 3D Mandelbrot set](/latestnews/sep-dec09/mandelbrot/mandel3d.jpg)
White's 3D version of the Mandelbrot set. Image © Daniel White.
![The Mandelbrot set](/latestnews/sep-dec09/mandelbrot/mandel.jpg)
The Mandelbrot set (top left) and details you get when repeatedly zooming in. Image courtesy Wolfgang Beyer.
The original 2D Mandelbrot set, named after Benoît Mandelbrot who discovered it in the 1970s, is arguably the most famous fractal of them all. Unlike ordinary 2D shapes, whose outlines typically become smooth when you look at them closely enough even if they look crinkly at first, the Mandelbrot set displays the same level of complexity at every scale. Zooming in on it is like going on a never-ending journey into hidden worlds. This infinite complexity, together with a certain level of self-similarity — you can find copies of the original Mandelbrot set appearing again and again on smaller and smaller scales as you zoom in — is what earns the Mandelbrot set the name fractal.
![A 3D Mandelbrot set](/latestnews/sep-dec09/mandelbrot/mandelsmooth.jpg)
An existing 3D version of Mandelbrot set. There is fractal structure in the horizontal plane, but in the vertical plane the object is smooth.
Several ways of creating 3D versions of the Mandelbrot set already exist, either by modifying the original 2D version, for example spinning it around an axis, or by translating the underlying mathematics into higher dimensions. But in most cases the "fractal-ness" of the resulting images does not extend into all three dimensions. You will see fractal structures when you look across the object along certain directions, but along other directions what you see is disappointingly smooth. White's new Mandelbulb, however, does appear to have genuine fractal features no matter which way you look at it, so in terms of fractal-ness, it's arguably got the biggest claim yet to being a true 3D analogue of the original Mandelbrot set.
The classical Mandelbrot set
So how did White produce his new beast? The classical two-dimensional Mandelbrot set is based on relatively simple mathematical objects: quadratic functions, like, for example,
It now makes sense to consider a quadratic function like
Fixed points, cycles and escaping to infinity
But complex numbers also have a geometric interpretation: the number
This is in stark contrast to what happens for
It turns out that no matter which complex value you choose for
So the secret of the classical two-dimensional Mandelbrot set lies in the fact that complex numbers, which can be added and multiplied, can also be visualised as points on the plane. The addition and multiplication rules mean that it makes sense to look at functions like
The 3D version
![A detail of the 3D Mandelbrot set](/latestnews/sep-dec09/mandelbrot/mandel23D.jpg)
A detail of White's 3D version of the Mandelbrot set. Image © Daniel White.
One way of creating a three-dimensional version of the Mandelbrot set is therefore to invent a new class of numbers, which correspond to points in three-dimensional space, rather than just the plane. Then define some meaningful way of adding and multiplying these numbers, so that it makes sense to consider functions such as
White and Nylander's formula for a 3D analogue of complex numbers
Consider the points
This is exactly what White did, using a 3D analogue of complex numbers, and tinkering around with ways of defining addition and multiplication that are akin to addition and multiplication of complex numbers. The initial results were slightly disappointing in terms of fractal detail. "I scoured everywhere to find signs of the 3D beast," says White, "but nothing turned up. Pretty 3D fractals
were everywhere, but nothing quite as organic and rich as the original 2D Mandelbrot set." Eventually the mathematician Paul Nylander came to White's rescue. He suggested to adjust the "squaring part" in the formula
It's now a task for mathematicians to investigate the exact fractal nature of the beast, and to see whether it is of genuine mathematical interest, rather than just a stunning visual oddity. The pretty pictures that arise in this area of mathematics conceal a variety of mathematical concepts that mathematicians are eager to explore further. Many will argue that there is no single 3D version of the Mandelbrot set. It all depends on what you want to generalise to higher dimensions — particular fractal structures, or other aspects underlying the mathematics of these dynamical systems.
![A detail of the 3D Mandelbrot set](/latestnews/sep-dec09/mandelbrot/honey.jpg)
Another detail of White's 3D version of the Mandelbrot set. Image © Daniel White.
Meanwhile, White's quest continues. Having to move all the way up to the eighth power to get the required intricacy is slightly unsatisfactory, and it suggests that one could do better by modifying the technique. What's more, White's Mandelbulb still contains some "smeared-out" areas that aren't quite as intricate as one might wish for. "As exquisite as the detail is in our discovery, there's good reason to believe that it isn't the real McCoy," he says. "That means the biggest secret is still under wraps, open to anyone who has the inclination, and appreciation for how cool this thing would look. For sure I'll still keep looking."
You can find out more about the 3D Mandelbulb on Daniel White's website.
Comments
David Makin said...
Those interested in more about the Mandelbulb and the search for the "true 3D" Mandelbrot including an almost complete history of the last couple of years search may wish to look here http://www.fractalforums.com/
miner49er said...
What's the explanation for the fracvtal nature of the mandelbrot set? Is it an anomoly in the number system? Is it basically an error?
I have been fascinated by fractals for 20 years but never really thought about _why_ they (mandelbrot/escape-time) exist.
I wonder if discovering why they exist at all, may lead to a 'better' 3D analog?
Paolo Bonzini said...
It is possible to describe this fractal also using quaternions. This is interesting in that it removes the need to define a special, non-standard exponentiation function.
See http://github.com/bonzini/mbulb/raw/master/mbulb.pdf [PDF] (thanks to the people on reddit.com and fractalforums.com for proofreading!)