Fractals – geometric shapes that look similar however much you magnify them – have kept mathematically minded people entertained for decades. The Mandelbrot Set, perhaps the most famous fractal of all, has been described as one of the most astonishing objects in the whole of mathematics.
The extraordinary detail of the Mandelbulb, a 3D equivalent of the Mandelbrot Set, as rendered by Visions of Chaos.
But despite the amazing complexity and beauty of this bizarre geometrical figure, it’s firmly rooted in two dimensions. For those of us who can vividly imagine the incredible potential of the Mandelbrot Set in 3D, there’s happy news. For years mathematicians have known of the concept of 3D fractals, and recently, a three-dimensional figure with many of the same qualities as the Mandelbrot Set was discovered.
The Sierpinski Tetrahedron
The Mandelbrot Set is an example of an Escape Time Fractal (see below for more information), but the first fractals to be discovered were somewhat different. Whereas the Mandelbrot Set is the result – unexpected to many – of applying a very simple mathematical operation, the earliest fractals were designed to have fractal properties. The Sierpinski Triangle – sometimes called the Sierpinski Gasket – is a good example. Take a solid (filled-in) equilateral triangle, divide it up into four smaller equilateral triangles and then remove the middle one of those triangles so that you end up with a large equilateral triangle with a smaller triangular hole in it. Now, for each of the remaining three smaller triangles, repeat the operation. Keep on doing this, with ever-smaller triangles, forever. The end result is a fractal figure – however much you magnify it, you’ll see the same detail at ever-smaller scales.
Along with the cube, the octahedron, the dodecahedron and the icosahedron, the tetrahedron is an example of the class of regular polyhedra known as the Platonic Solids. Each one of the tetrahedron’s four faces is an equilateral triangle, so, in this sense, it can be thought of as a three-dimensional extension to the equilateral triangle. As such, it suggests a method of extending the Sierpinski Triangle into the third dimension. The process is much the same as we saw for the Sierpinski Triangle except that instead of splitting the triangle into four half-size triangles and removing the middle one, for each generation the Tetrahedron is split into five half-size tetrahedra and the middle one removed. This is not nearly as easy to do on-screen, but we have reproduced an image of one below.
By extending the Sierpinski Triangle into three dimensions we get the Sierpinski Tetrahedron, shown in red. The blue figure is its inverse.
Mandelbrot Mountain
The Mandelbrot Set is special in a way that the likes of fractals such as the Sierpinski Triangle aren’t. For a start it’s produced from a mathematical function, and a very simple one at that. And second, unlike the Sierpinski Triangle and its ilk (which are exactly the same at all levels of magnification), the Mandelbrot Set shows self-similarity at all levels of magnification. Rarely, if ever, do you see exactly the same things as you zoom in, though you do usually see similar pinwheels, starbursts, irregular shapes that look like lakes and so forth. The fact that you don’t know exactly what you’ll see as you zoom in is a major part of the Mandelbrot Set’s appeal and explains why enthusiasts can spend so much time delving into it in the hope of finding hidden treasure. So can such a complex fractal really be created in 3D?
Before we answer that question, let’s just say that it’s quite possible to use three dimensions to display the standard two-dimensional Mandelbrot Set. Here we’re going to use a free fractal package called Fractal Explorer together with the Fractal Landscape Library. You can find it here: http://www.eclectasy.com/Fractal-Explorer/index.html Start up this package and select ‘New Fractal’ from the File menu. The familiar Mandelbrot (the default fractal) will appear. If you want to see it better then select a larger size in the Size menu (though when we come to turn it into a fractal landscape, the size it can handle is limited). If you haven’t delved into the Mandelbrot Set before, we suggest you do so now before proceeding.
To zoom in on any portion, drag a box round it and then double-click inside that box. Note that you’ll only get an interesting result if you zoom in where there appears to be lots of action – if you zoom in on the central black area then you’ll see nothing but blackness. Now that you’ve satisfied your curiosity, get the complete (not zoomed-in) Mandelbrot Set on-screen. The black irregular shape in the centre is the actual Mandelbrot Set, and the coloured area that surrounds it shows contours that indicate how close points are to being within it. This might be hard to appreciate with the default colour scheme (the colours were chosen to blend into each other) but if you choose a different group of colours then the contours will become more visible.
To do this select ‘Colour control’ from the Fractal menu and then, on the Palette dialog box, ensure that the Palette tab is selected. Next, bring up the Palette browser by clicking on the appropriate black triangle. Choose the Examples folder and then select ‘Polinom3.frs’ by double-clicking on it. Contours are generally associated with altitude, so this suggests one possible means of rendering the normally two-dimensional Mandelbrot Set in three dimensions.
When rendered as a 3D landscape the normally flat Mandelbrot Set takes on the appearance of a bizarre mountain range.
To see the Mandelbrot Set reproduced as a 3D figure, in which the information in the contours is translated to height above sea level, simply click on the 3D icon on the tool bar. Having looked at the entire Mandelbrot Set in 3D –which resembles a strangely symmetrical mountain range with a flat plateau – try doing the same with zoomed-in portions. There are also lots of options in Fractal Landscapes to change your viewing angle and how the landscape is rendered.
The Mandelbulb
If you consider our foray into the Mandelbrot Mountain to be cheating and still yearn for a genuine 3D Mandelbrot Set, fear not. After years of searching, experimenters have come up with something that is a very close 3D analogue. Called the Mandelbulb, it shares much of its two-dimensional counterpart’s appeal and exhibits fractal detail in all three dimensions.
Trying it out for yourself is far better than reading someone else’s description, so start up Visions of Chaos, which you’ll find at http://softology.com.au/voc.htm. Select ‘Hypercomplex Fractals | Mandelbulb’ from the Mode menu and then select ‘Generate’ from the Image menu. The Mandelbulb Options dialog box will appear, complete with a staggering range of parameters and tickboxes to choose from. For now we’ll accept all the default values, so just click ‘Render’ to get your first glimpse of the Mandelbulb. This default figure offers an amazing degree of variety in its fractal depths, and the Mandelbulb Options window lets you experiment with lots of other similar 3D fractals if and when you get bored of this one. For now, though, we suggest you stick with the default Mandelbulb and try your hand at zooming in. Unlike 2D fractals, which are normally zoomed into by dragging a box round an area of interest, zooming into a 3D figure is much harder to define using mouse actions – so the controls are all provided by parameters in the Mandelbulb Options window.
Once you’ve discovered an interesting area of the Mandelbulb, it’s worthwhile playing around with colour schemes to get the best effect.
There are two ways of navigating a 3D object. If you know an area you want to look at – perhaps because someone has suggested it – then you can enter values for the various parameters in the top left of the dialog box. Otherwise, your best option is to progressively zoom in and pan around the Mandelbulb using the Camera Controls at the top right. Note that in addition to the various buttons such as ‘Forward’ and ‘Backward’, you can choose how much to move when moving or tilting the camera by entering values for ‘Move distance’ and ‘Rotation/tilt degrees’. If you use the camera controls, the Preview window updates automatically; if you change a parameter then you’ll need to click on ‘Preview’ to update it. When you’re happy with the preview, click on ‘Render’ to see your creation in in all its three-dimensional glory.
Escape Time Fractals
The 2D Sierpinski Triangle and the 3D Sierpinski Tetrahedron can be thought of as ‘synthetic fractals’ in the sense that the rules that are used to create them were designed to produce a fractal result. The 2D Mandelbrot Set and the 3D Mandelbulb are quite different. They are both examples of a class of mathematical shapes known as Escape Time Fractals, and to the non-expert it comes as quite a surprise that the algorithms used can give rise to literally infinite detail.
To cut a long story short, an Escape Time Fractal is generated by testing points (on the complex plane, in the case of 2D fractals) to see whether or not they’re a part of the fractal figure. Testing a point involves carrying out an operation (in the case of the Mandelbrot Set, squaring it and adding a constant) repeatedly. In some cases the value soon becomes very large and heads off to infinity. This point is said to have escaped, and represents a point that isn’t in the fractal figure. Other points give a small result, however many times the operation is repeated, and these are the points contained within the fractal figure.
See the Mandelbulb
So you’ve seen an image of the three-dimensional Mandelbulb, but, fascinating as it might be to zoom in on its amazing complexity, it still looks flat on-screen. Thanks to Vision of Chaos’ anaglyph feature, you can see it in stereoscopic 3D. On the Mandelbulb Options dialog box, select the ‘Anaglyph’ tickbox near the bottom left corner. Then, depending on which type of 3D glasses you’re using, select either ‘Anaglyph Red Blue’ or ‘Anaglyph Red Cyan’ from the dropdown menu. You could also try ‘Anaglyph Colour’ or ‘Anaglyph Half Colour’, both of which will work with red/cyan glasses, but depending on the colour palette you’ve chosen, the 3D effect might not be as good. You can also increase the value of the ‘Degrees between eyes’ tickbox.
Having introduced you to Visions of Chaos and the Mandelbulb, we’ve given you every opportunity to waste untold hours in search of the ultimate fractal image. However, if you’d prefer to save some time then you can travel straight to some already-known beauty spots in this strange alternative world.
The secret is to use some of the canned scenes that are shipped with Visions of Chaos. In the Mandelbulb Options window, simply click on ‘Load’ and then select one of the Mandelbulb Parameter files. If you are interested in the original Mandelbulb as opposed to its many close relatives, make sure you load one of the files with a name that starts ‘mandelbulb sin power 8’, or ‘anaglyph mandelbulb sin power 8’ if you want to view it in stereoscopic 3D. Alternatively, you could use one of the canned scenes as a starting point to zoom in yet further on this otherworldly object. Whichever way you choose, though, bear in mind that rendering the Mandelbulb involves some serious number crunching. Jason Rampe, creator of Visions of Chaos, estimated that because of the need to first calculate in 3D and then raytrace the Mandelbulb, it requires many hundreds of times more calculations per pixel than the Mandelbrot Set does. Obviously some patience (and a faster PC) is called for.