Opening Pandora's Box For the Second Time
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 sciencefiction 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):
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 Mandelbrotesque. 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. 
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 nothing truly fantastic. The best shot I could find was this view from the YZ plane (found just before this article was published actually):
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 halfturn 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
Quasifuchsian fractal (see right), but it turned out that even that doesn't have the variety of the Mandelbrot after zooming in.
Is this merely a fool's quest?
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):
Mandelbulb (order 8)
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 Mandelbrotesque 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 Mandelbrotesque!). So I set out to create my own specialized voxelish 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):
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.
"Mandelbrot Crust" You don't have to zoom in far before you get to see this. 
"A Slice of Mandelbrot Gateau" This picture is a deeper zoom of the previous picture (see its far right, just below vertically center  this one's near there.) 
"Cave of Lost Secrets" This ancient half mile high cave still exists (now underwater) from a planet several billions of light years away from Earth. It was built by a (now extinct) intelligent race of beings who also discovered the 3D Mandelbulb we are witnessing on this page. Inside the cave however, lies  amongst other technological and mathematical secrets  the last remaining scroll which contains the much deeper secret of the even more incredible real 3D Mandelbrot formula (giant structures of that were also built at a later stage, but were apparently destroyed for reasons unknown).

"Magic Broccoli" MiniBulbs from the set don't just sprout uniform smaller bulbs, but rather twist and shear to create surprisingly strange patterns. 
"Mandelbulb Garden" Zooming in yet further to the left, we visit the Bulb's horticultural centre, and still, we see no sign of detail degradation. 
[below] Zooming out again, we see some of the main support structures holding up the giant 40 mile wide Colosseum at the top right of the picture. Seriously, this universe has got to be quite messed up to be harbouring math secrets capable of this kind of Baroquian beauty. 
"Christmas Coral Egg" Red, green and blue lighting combine to create all the other colours in this chrimbo themed fractal. 
"Christmas Coral Reef" 
"Honeycomb Heaven" It's easy to think of a bee hive when seeing this. Little did I know at the time about the honeycomb pattern below the hive. 
"Mandel NightShade" Rumour has it that one sniff of this plant and you're turned to dust. A little more deadly than usual then. 
"The Mandelbulb" Here's the whole thing, with some perspective this time. Let's see what happens when we dive *inside* this thing..... 
"Mandelbulb Spine" The inside is just as amazing as the out, as this zoom shows. 
"Ice Cream From Uranus" What, Uranus from our solar system? Neah, we're talking about a planet unique in all but the name, near the distant edge of the universe. They have advanced food making equipment, and an eye for detail, so they regularly consume attractive cornettoesque dishes, sometimes in the shape you see above. 
"Caramelized Hazelnut Swirl" Objects take on a completely different character inside the 3D Mandelbulb, as these surrounding pictures testify. 
"Shell Life" More chaotic scenes appear too. Whenever fractals are involved though, it's never going to be completely random. (...well unless we're talking about Quaternion Julia fractals [runs and ducks for cover] ;) ) 
"Hell Just Froze Over" 5 minutes ago, possibly because someone found the definitelyreallytrueandwemeanitthistime3D .... Mandelbrot.

And here is the beast itself (power 8 version). All of the above images come from this object below (giant 7500x7500 pixel version available
here, but please don't hot link to it ;) ):
It's not just myself of course who's rendered stunning shots of the Bulb. The guys have over at Fractal Forums have also rendered mouth watering pictures and animations, some of which I'll show below. It's a real honour to present them on this page  please visit their websites to view more of their creations.
This jaw dropping image created by Krzysztof Marczak uses many iterations to achieve a more fragmented surface texture. View full size, and you'll notice countless 'satellites' around other bigger satellites. 
"Alien Coral": Subtle forest colours combine with a dash of blue to good effect in this excellent render by David Makin. 
"Siebenfach": The unusual material and ornate ropelike detail evokes a more mysterious atmosphere in this stunning render from Thomas Ludwig. Full resolution available from here. 
Gotta love the luminous sorbet style texture of the quadratic version of the Mandelbulb, created by Paul Nylander. See his Hypercomplex Fractals page for a bigger view. 
"Asteroid National Park"  This degree 4 behemoth would have certainly made an interesting replacement for the asteroids used in "Armageddon" and "Deep Impact". Excellent render by David Makin; view full res to see it in all its glory.

Created by Garth Thornton, a special variant of the Julia formula is used in combination here to create this amazing fossilized design. See the thread here for further interesting variations.

Keep an eye out for the latest news and renders on this page, and over in the
3D Mandelbrot thread over at Fractal Forums. As of 05/11/2009, the pictures here on this page are only a preliminary look into the thing. The best stuff is surely yet to come!
Most pictures on this page are copyright Daniel White 2009 onwards, apart from
where attribution is made  where they are copyright of their respective owners.