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
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):
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.
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.
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 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):
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.
"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. This
reminds me of Romanesco Brocolli.
 |
"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.
Advert: Prints of this image (along with others on this page) are available here. |
[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" - A deep zoom into the picture on the left. |
"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.
(this shot shows the detail of the original 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-this-time-3D .... Mandelbrot.
 |
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 the high definition, high resolution print (7500x7500) is available 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.
|
Are gorgeous flyovers and parallax zooms now possible?
But
of course. This one from Youtube to the right is one such rendering
(Youtube decreases the quality and framerate a lot, so email me for the
original). Watch this space for future animations. Also, if you want to
make any yourself, and would like them to featured on this page,
let me know, or even better, post it to
FractalForums.com.
Also check out William Rood's
Mandelbulb section on his website,
bib993's video (especially
this one),
Iñigo Quílez's video, and
Softology's
video, David Makin's animations, such as the
degree 4 version of the Mandelbulb, his
Crater Lake Flyover, Krzysztof Marczak's
excellent rotational Mandelbulb variation, and this
computer generated romanesco broccoli created by Aleksandar Rodic (which is an IFS type fractal rather than the Mandelbulb we're exploring, but still really cool).
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 in this article are only a
preliminary look into the thing. The best stuff is surely yet to come!
The article and 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.
