archived as http://www.stealthskater.com/Documents/Fractals_03.doc more of this topic at http://www.stealthskater.com/Science.htm#Fractals note: because important web-sites are frequently "here today but gone tomorrow", the following was archived from http://www.newscientist.com/article/dn18171-the-mandelbulb-first-true-3d- image-of-famous-fractal.html on November 18, 2008. This is NOT an attempt to divert readers from the aforementioned web-site. Indeed, the reader should only read this back-up copy if the updated original cannot be found at the original author's site. The "Mandelbulb": first 'true' 3D image of famous fractal by Jacob Aron New Scientist, November 18, 2009 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 3-dimensional representation to date of the most famous fractal equation: the Mandelbrot set. Fractal figures are generated by an "iterative" 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 "self-similar" images -- forms that contain the same shapes at different scales. For instance, some look uncannily like a snowflake. The tricky part is finding an equation that produces an interesting image. The most famous fractal equation is the 2-D Mandelbrot set -- named after the mathematician Benoît Mandelbrot of Yale University who coined the name "fractals" for the resulting shapes in 1975. But there are many other types of fractal both in 2 and 3 dimensions. The "Menger sponge" is one of the simplest 3-D examples. Fake fractal There have been previous attempts at a 3-D Mandelbrot image. But they do not display real fractal behavior says Daniel White, an amateur fractal image maker based in Bedford, UK. Spinning the 2-D Mandelbrot fractal like wood on a lathe, raising and lowering certain points, or invoking higher-dimensional mathematics can all produce apparently 3-dimensional Mandelbrots. Yet none of these techniques offer the detail and self-similar shapes that White believes represent a true 3-D fractal image. 2 years ago, he decided to find a "true" 3-D version of the Mandelbrot. 1 The next dimension "I was trying to see how the original 2-D Mandelbrot worked and translate that to the third dimension," he explains. "You can use complex maths. But you can also look at things geometrically." This approach works thanks to the properties of the "complex plane" -- a mathematical landscape where ordinary numbers run from "East" to "West" while "imaginary" numbers (i.e., based on the square root of -1) run from "South" to "North". 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. T he different colors 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 3-D space would capture the essence of the Mandelbrot set without using complex numbers (i.e., real numbers plus imaginary numbers) 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 Fractal Forums (a website for fractal admirers), he continued his search. It was another member -- Paul Nylander -- who eventually realized that raising White's formula to a higher power (i.e., equivalent to increasing the number of rotations) would produce what they were looking for. White's search isn't over, though. He admits the Mandelbulb is not quite the "real" 3-D Mandelbrot. "There are still 'whipped cream' sections where there isn't detail," he explains. "If the real thing does exist – and I'm not saying 100 percent that it does – one would expect even more variety than we are currently seeing." Part of the problem is that extending the Mandelbrot set to 3-D requires many subjective choices that influence the outcome. For example, you could extend a flat plane to 3-D by stretching it to form a box, but you could also turn it into a sphere. "It's an interesting academic exercise to think what you should get," says Martin Turner, a computer scientist specializing in fractal images at the University of Manchester, UK. "But it all depends on what properties you want to keep in the third dimension." The equations White used may get the job done. But the system of algebra used is not applicable to all 3-D mathematics. "The next stage is mathematical rigor," says Turner. 2 2-D fractal Mathematicians have studied self-similar shapes like this one since the turn of the last century. But it was Benoît Mandelbrot who named them "fractals" in 1975. Fractals are more than pretty pictures, though, as the mathematics behind them can explain natural phenomena from the length of coastlines to the structure of the Universe. There are many types of fractals, both in 2 and 3 dimensions. But the most famous is the 2-D Mandelbrot set shown here. Past attempts to extend it to the third dimension have met with difficulty. (image: Wolfgang Beyer) Spun One method for generating a 3-D Mandelbrot image is to spin the classic 2-D Mandelbrot around a central axis like a block of wood in a lathe. You end up with a 3-D shape. But it's not much different to the original. There are a few ways to produce this shape but none of them add any extra detail. (image: Paul Bourke) Raised The traditional Mandelbrot image is produced through an iterative process with different colors indicating the number of iterations at that point. Another method of making a 3-D Mandelbrot (shown here) keeps the 2-D version but swaps the different colors for changes in height. With some parts higher and some lower, the effect is similar to the hills and valleys of a relief map. But it's still essentially 2-dimensional. (image: Aexion) 3 Whipped This tasty-looking 3-D Mandelbrot is actually a projection from 4 dimensions just as a square is a 2-D projection of a cube. It is generated using a 4-D version of complex numbers called quaternions. But despite that complexity, its smooth appearance means there is no interesting fractal detail when you zoom in. Even higher-dimensional maths can be used to create 3-D Mandelbrots but with no better results. (image: Paul Bourke) Mandelbulb Daniel White's "Mandelbulb" takes a different approach. He took the geometrical properties of the "complex plane" where multiplication becomes rotation and addition becomes movement of the plane in a particular direction and applied them to a 3- dimensional space. (image: Daniel White) Mandelbulb close-up It took 2 years to perfect the formula which contains incredible detail at even 1000x magnification. (image: Daniel White) 4 Mandelbulb close-up 2 Here is a close-up of a spiral section of the Mandelbulb juxtaposed with a 2-D Mandelbrot image. (image: Daniel White) Mandelbulb close-up 3 A close-up of the Mandelbulb "spine" and a similar 2-D Mandelbrot. (image: Daniel White) 5 Mandelbulb close-up 4 The whole Mandelbulb again from a different viewpoint. (image: Daniel White) Readers' Comments 1. Gilding The Lilly? by Tom / Wed Nov 18, 2009 13:48:34 GMT Interesting. But adding another dimension to an infinitely complex fractal will only reveal things that were there anyway if you had looked closely enough. 2. A Little Help by Geraint / Wed Nov 18, 2009 13:53:41 GMT Now admittedly I've scanned over this briefly before I go to work so likely I've skipped an explanation somewhere. But as interesting as this is, what are the practical applications of mapping fractal equations in this way? It mentioned that the patterns are reminiscent of certain elements in Nature. Is this related to things like weather predictions and such like? I'd be grateful if someone could reply to a layman here. Mathematics has never been my most intuitive subject! :) 3. A Little Help by Tony Coleby / Wed Nov 18, 2009 14:45:41 GMT Hey, I'm a layman too. But one obvious real world application -- at least once computers are fast enough to do these kind of equations on the fly (2 years?!) -- is the generation of realistic objects in computer games and simulations which keep their complexity at infinite scales. Apart from that, it's just fascinating and is -- I believe -- a work of Art! 4. re: "A Little Help" by soylent / Wed Nov 18, 2009 15:50:22 GMT "Hey, I'm a layman too. But one obvious real world application -- at least once computers are fast enough to do these kind of equations on the fly (2 years?!) …"" 6 Not two years, not 20 years. If you can extend Moore's law (and the algorithm is embarrassingly parallelizable), that's only a factor 1000 performance improvement. That doesn't even get you close! "is the generation of realistic objects in computer games and simulations which keep their complexity at infinite scales." The market for games centered on abstract art is pretty negligible.