TIMESTAMP 08/11/2009. The original Mandelbrot is an amazing object that has captured the public's imagination for 30 years with its cascading patterns and hypnotically colourful detail. It's known as a 'fractal' - a type of shape that yields (sometimes elaborate) detail forever, no matter how far you 'zoom' into it (think of the trunk of a tree sprouting branches, which in turn split off into smaller branches, which themselves yield twigs etc.).
It's found by following a relatively simple math formula. But in the end, it's still only 2D and flat - there's no depth, shadows, perspective, or light sourcing. What we have featured in this article is a potential 3D version of the same fractal. For the impatient, you can skip to the nice pics, but the below makes an interesting read (with a little math as well for the curious).
The article received a degree of popularity, but in the end, the problem remained open, and the Mandelbrot refused to relinquish its deepest secret. At times, it felt tempting to give up the search, but the potential for the thing is too great - we are talking about the Holy Grail of fractals after all.
Opening Pandora's Box For the Second Time
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 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.
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.
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.
Is this merely a fool's quest?
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:
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):
"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" Mini-Bulbs 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. It should be amazing to fly over and zoom into it. In the meantime, let's see what happens when if peek *inside* the Bulb.....
"Mandelbulb Spine" The inside is just as amazing as the out, as this zoom shows. A high definition, high res poster is available here (see this preview showing the detail of the 7000x7000 render).
"Ice Cream From Neptune" What, Neptune* 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 cornetto-esque dishes, sometimes in the shape you see above.
* Yes we renamed it. We mistakingly used the planet's old name that was in use until around 50,000 years ago. By another massive but convenient coincidence, they themselves renamed it for similar reasons to Professor Farnsworth's story from Futurama.
"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 definitely-really-true-and-we-mean-it-th
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). Added 13/11/2009 That version is low quality, but if you want the full size (7500x7500), high definition, high resolution version, you can buy the print from here.
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.
"The Honeycomb": The fantastic material combined with the different colour ranges give a real sense of depth in this picture created by David Makin.
"Siebenfach": The unusual material and ornate rope-like 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.