Opening Pandora's Box For the Second Time

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The first thing I tried was multiplying phi and theta by two, resulting in the shape you see above. It's nice, but not exactly what I'd call a 3D Mandelbrot (zooming in doesn't show true 3D fractal detail).

This one is the same as to the left, except offsets have been added to the multiplication bit (0.5*pi to theta and 1*pi to phi), to make it appear almost 3D Mandelbrot-esque. Also see Thomas Ludwig's globally illuminated render, and this one from Krzysztof Marczak.

Same as the first, except this time we try only multiplying angle phi by two, but not theta.



Created by Dr. Kazushi Ahara and Dr. Yoshiaki. This looks great, but zooming in will not reveal the variety of style that the Mandelbrot has.

On October 13th, 2006, Marco Vernaglione put out the question and challenge to the world with this memorable document

Is this merely a fool's quest?

Mandelbulb (order 8)

Resulting renders

But it wasn't until I incorporated proper shadowing that the subtleties of this incredible object became apparent. For the renders below and exploration afterwards, I'll be concentrating mostly on the 8th power, since that seems to be around the 'sweet spot' for overall detail and beauty, but lower powers can also produce stunning results too.

ur story starts with a guy named Rudy Rucker , an American mathematician, computer scientist and science fiction author (and in fact one of the founders of the cyberpunk science-fiction movement). Around 20 years ago, along with other approaches, he first imagined the concept behind the potential 3D Mandelbulb (barring a small mistake in the formula, which nevertheless still can produce very interesting results - see later), and also wrote a short story about the 3D Mandelbrot in 1987 entitled " As Above, So Below " (also see his blog entry and notebook ). Back then of course, the hardware was barely up to the task of rendering the 2D Mandelbrot, let alone the 3D version - which would require billions of calculations to see the results, making research in the area a painstaking process to say the least.So the idea slumbered for 20 years until around 2007. I then independently pictured the same concept and published the formula for the first time in November 2007 at the fractalforums.com web site. The basic idea is that instead of rotating around a circle (complex multiplication), as in the normal 2D Mandelbrot, we rotate around phi and theta in 3 dimensional spherical coordinates ( see here for details). In theory, this could theoretically produce our amazing 3D Mandelbrot. But here's the somewhat disappointing result of the formula (click any of the pictures for a larger view):Although the second one looks somewhat impressive, and has the appearance of a 3D Mandelbulb very roughly, we would expect the real deal to have a level of detail far exceeding it. Perhaps we should expect an 'apple core' shape with spheres surrounding the perimeter, and further spheres surrounding those, similar to the way that circles surround circles in the 2D Mandelbrot.Zooming in reveals some interesting detail, but I didn't find a great deal beyond the usual chaos. Had I known at the time where to look however, I would've already started to see some very curious detail emerge. The best shot I could find was this view in the YZ plane from the middle picture (found just before this article was published actually):A glimpse of this detail can be traced back to as early as November 20, 2007, with this preliminary cross-section render created by Thomas Ludwig - the first person to render and witness the (Power 2) Mandelbulb in 3D (other than my early crude height map images like this one ). The red box in that linked picture is the area in the large picture shown above if you zoomed in!Full size shown here . For other 'hot spots', try here , and this one from the inside I went to great lengths to explore the concept, including the utilization of various spherical coordinate systems and adjusting the rotation of each point's 'orbit' after every half-turn of phi or theta. But it didn't work. Something was missing. I scoured everywhere to find signs of the 3D beast, but nothing turned up. Pretty 3D fractals were everywhere, but nothing quite as organic and rich as the original 2D Mandelbrot. The closest turned out to be Dr. Kazushi Ahara and Dr. Yoshiaki's excellent Quasi-fuchsian fractal (see right), but it turned out that even that doesn't have the variety of the Mandelbrot after zooming in.Some said it couldn't be done - that there wasn't a true analogue to a complex field in three dimensions (which is true), and so there could be no 3D Mandelbulb. But does the essence of the 2D Mandelbrot purely rely on this complex field, or is there something else more fundamental to its form? Eventually, I also started to think that this was turning out to be a Loch Ness hunt. But there was still something at the back of my mind saying if this detail can be found by (essentially) going round and across a circle for the standard 2D Mandelbrot, why can't the same thing be done for a sphere to make a 3D version?Our story continues with mathematician - Paul Nylander . His idea was to adjust the squaring part of the formula to a higher power, as is sometimes done with the 2D Mandelbrot to produce snowflake type results . Surely this can't work? After all, we'd expect to find sumptuous detail in the standard power 2 (square or quadratic) form, and if it's not really there, then why should higher powers work?But maths can behave in odd ways, and intuition plays tricks on you sometimes. This is what he found (also see forum thread , and the full size pic at the 'Hypercomplex Fractals' page of his site ):Okay... now this is starting to look interesting. We're already starting to see buds growing on buds. Could.... this... object still hold any fractal detail if we zoomed in far enough?! More of Paul's work can be found here Then something amazing happened.Another fractal explorer, computer programmer David Makin was the first to render some sneak preview zooms of the above object, and this is what he found:WOW! Okay, now we're talking. These are deep zoom levels (the first being over 1000x), but fractal details remain abundant in all three dimensions! The buds are growing smaller buds, and at least to the picture on right, there seems to be a great amount of variety too. We're seeing 'branches' with large buds growing around the branch in at least four directions. These in turn contain smaller buds, which themselves contain yet further tiny buds.Remember, these pictures are not created from an iterated function system (IFS), but from a purely simple Mandelbrot-esque function!Even the picture on the left is interesting, and is reminiscent of the Romanesco broccoli vegetable . But glance at the top right of the left picture - there also seems to be a leaf section in the shape of a seven sided star. Does this hint at a deeper variety in the object than we can possibly imagine? What the heck have we stumbled upon here?Because of the lack of shadows, it's difficult for the renderings to give justice to the detail, but what we have here is a first look into a great unknown.Eager to get a better look at this thing, I set about trying to find software to render it, preferably with full shadowing and even global illumination, and at least something that was fairly nippy. But it turns out that there are probably no 3D programs out there on the market that can render arbitrary functions, at least not with while loops and local variables (a prerequisite for anything Mandelbrot-esque!). So I set out to create my own specialized voxel-ish raytracer. Results could be slow (perhaps a week for 4000x4000 pixel renders!), but it'll be worth it right?At first, I implemented 'fake' lighting based on the surface angle, and this produced a further glimpse into this incredible world (this one again from the power 8 version of the Mandelbulb):And here is the beast itself (power 8 version). All of the above images come from this object below (giant 4500x4500 pixel version available here ).