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 colorful 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.

  • Page 1: Background
  • Page 1: Opening Pandora's Box for the second time
  • Page 1: Is this merely a fool's quest?
  • Page 1: Resulting renders
  • Experimenting with iterations and powers
  • Further exploration of the creature
  • Epilogue: Is this... the real 3D Mandelbrot then?
  • Programs to explore the Bulb

  • Click here to buy
    prints of the Mandelbulb

    Page 2/2 - Further exploration of the 3D Mandelbulb

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    Experimenting with iterations and powers

    Okay enough eye candy for now. Let's have a closer look at the structure of this beast. Firstly, the old 2D Mandelbrot can be represented as an evolution of iterations. At iteration 1, it's merely a circle, so we'll start from iteration 2. Click any picture to enlarge to 1920x1200 pixels.

    Iteration 2

    Iteration 3

    Iteration 4

    Iteration 6

    Iteration 10

    Iteration 20

    Iteration 50

    Iteration 5000

    The final stage (infinity iterations) is very similar at first glance to iteration 5000 (unless you zoom right in), as the shape converges to a shape comprised of tangent circles. Countless sizes of circles granted, but there's no real interesting detail left (unless you colour the background as is often done to produce nice psychedelic patterns).

    One interesting question is: Does this same phenomenon happen with our power 8, 3D Mandelbulb?

    Power 8 (zooming into this object produces all the eye candy on the previous page):

    Click any picture to enlarge.

    Iteration 2: One iteration is just a sphere, so we start with iteration 2. At the moment, we can only see a coarse bulbous pattern emerging.

    Iteration 3: Things are already beginning to take shape. The bulbs are beginning to grow buds!

    Iteration 4: Much better. Details appear very fast in the power 8 version on the formula. In fact, for the tasty pictures above, only up to 10 iterations were needed.

    Iteration 6: The object gets darker as the light gets lost in the cracks and crevices. This one approaches the limit of 'interesting detail' if you enlarge to the 1000x1000 version.

    Iteration 20: At this point, there's not much apparent change to the overall structure (unless you zoom in), but things look more 'foamy' and 'bitty'.

    Iteration 200: Unless you zoom right in, there's not much progress over iteration 20, even if you enlarge to the 1000x1000 pixel version.

    A higher quality image below, and a super-large 4000x4000 version is here for the patient.
    And once you zoom into that, you get the magic as shown before. Why doesn't the old square (power 2) version (which was featured at the beginning of this article) produce all the detail as the power 8 version just above? Let's take a look to see if we can find clues by delving into the various iterations:

    Squaring (power 2)

    Iteration 2: We start off with this odd looking shape.

    Iteration 4: Not much detail yet. Compare to iteration 3 of th previous power 8 version, which has much more detail.

    Iteration 6: We're beginning to the more familiar shape now. Detail is sparse over the object apart from a few nooks and crannies.

    Iteration 200: After you click the picture to enlarge, we're pretty much at the visible limit here without zooming in.

    Zooming in to the object above will produce mostly relatively dull patterns and maybe one or two surprises, but which still mostly have only 'whipped cream' style textures (see here for a 7500x7500 pixel render if you're patient). From the evolution of the iterations, it looks like any detail is being focused into specific points pseudo-randomly, rather than the nice spread and growth of detail we see in the power 8 version further above.

    Finally, let's take a look at power 3. Will it appear magical like power 8, or relatively boring like power 2? Or perhaps something in between?

    Power 3

    Iteration 2

    Iteration 4

    Iteration 6

    Iteration 20

    Click any to enlarge.

    Interesting. So it looks like we have a middle ground between the detail of power 8 and 2. But just to be sure, it might be worth our while to zoom into this thing. This below was the best I could find (2000x2000 resolution here), though you can see others here and here.

    Power 16

    It's easy to go to higher powers than eight. Results are beautiful, though somewhat predictable once you've seen the lower powers.

    And some zoom ins...

    Further exploration of the creature

    There are so many questions we can ask about the structure and nature of the object (we'll stick mostly to the power 8 version for now):

  • What's the formula of this thing?
  • What does it look like on the inside?
  • Any interesting analogues to the 2D Mandelbrot set?
  • Is this thing really new?
  • What variations of the formula are possible?
  • What happens at infinity iterations?
  • Is the object completely connected, and maybe path, locally, or simply connected?
  • Why do lower powers exhibit less fractal-like detail than higher powers?
  • Does fractal detail really go on forever when you zoom in?
  • Any Julia equivalent for it?
  • What is its volume and Hausdorff dimension?
  • Are there near exact copies of the object deep inside the structure?
  • Any way to colour the object, to replace its mono-ish appearance?
  • Any decent 3D software to render the beast?
  • If the 2D Mandelbrot is the thumbprint of God,
    what does that make the 3D Mandelbulb?

  • What's the formula of this thing?

    There are a few subtle variations, which mostly end up producing the same kind of incredible detail. Listed below is one version. Similar to the original 2D Mandelbrot, the 3D formula is defined by:

    z -> z^n + c

    ...but where 'z' and 'c' are hypercomplex ('triplex') numbers, representing Cartesian x, y, and z coordinates. The exponentiation term can be defined by:

    {x,y,z}^n = r^n { sin(theta*n) * cos(phi*n) , sin(theta*n) * sin(phi*n) , cos(theta*n) }
    r = sqrt(x^2 + y^2 + z^2)
    theta = atan2( sqrt(x^2+y^2), z )
    phi = atan2(y,x)

    And the addition term in z -> z^n + c is similar to standard complex addition, and is simply defined by:

    {x,y,z}+{a,b,c} = {x+a, y+b, z+c}

    The rest of the algorithm is similar to the 2D Mandelbrot!

    Here is some pseudo code of the above:

    r = sqrt(x*x + y*y + z*z )
    theta = atan2(sqrt(x*x + y*y) , z)
    phi = atan2(y,x)

    newx = r^n * sin(theta*n) * cos(phi*n)
    newy = r^n * sin(theta*n) * sin(phi*n)
    newz = r^n * cos(theta*n)

    ...where n is the order of the 3D Mandelbulb. Use n=8 to find the exact object in this article.

    The formulae above is the higher power analogue to the first green object shown on the first page here, but also see the formulae on Paul Nylander's page, which corresponds to the second green object shown instead. Both produce amazing (and similar) objects when put to the higher powers. Also see this summary of formulas.

    There are also techniques to improve rendering speed as found by FractalForum.com members:
  • Distance estimation provided by David Makin.
  • Powers -2 to 4 and powers 5 to 8, without the trig, to increase speed, by Paul Nylander. Also see Karl's earlier version for power 2 to check for "divide by zero" errors for the y values. One may need to use that technique for the higher powers too.
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    What does it look like on the inside?

    We've already seen some deep zooms of the inside from some of the pictures further above, but what does the overall shape look like? The structure seems just as mysterious as the outside, but in a completely different way. For the below shots, I've cut the Mandelbulb in half to get a better look. By the way, the top left of the picture below is the inside of the brown tower picture to the upper right!

    Power 8 Mandelbulb - interior view:

    (click for larger size)

    Here's the inside again, but at a different angle. Notice how the detail changes according to the iteration level. Click each pic to enlarge:

    Iteration 7

    Iteration 15

    Iteration 40

    Also see just below for some inside zoom shots.

    Any interesting analogues to the 2D Mandelbrot set?

    Yes it would appear so. The left pictures are on the inside of the bulb by the way:

    A 3D Mandelbrot stalk (a deep zoom inside the Bulb)

    A 2D Mandelbrot stalk

    And yes, increasing the iterations will make the tip of the stalk grow in the 3D version, just like the 2D one.

    A 3D Mandelbulb spiral (found at a deeper zoom inside the "Ice Cream from Neptune" picture). Click to enlarge, and see this page of the thread for how it was found.

    A 2D Mandelbrot spiral

    Is this thing really new?

    As far as I know, yes.

    What variations of the formula are possible?

    Just by adjusting the formula slightly, new and interesting variations are possible. Let's take a look:

    To create these two 'hilly' and 'smooth' versions above, use a lower or higher power instead of the usual square root, when calculating the radius in the conversion from Cartesian to polar 3D.

    Instead of assigning: newy = radius*sin(theta)*cos(phi), this one uses: newy = radius*sin(theta)*tan(phi).

    Instead of assigning: newx = radius*sin(theta)*cos(phi), this one uses: newx = radius*cos(theta)*cos(phi).

    Rudy Rucker's original formula (see here for details) has a small 'mistake', but it still produces great results when put to a higher power, as these two pictures show. This one is shown in the XY plane. Click on the pic for the 7000x7000 version, or see here for a zoom in.

    Same object as the left, but viewed in the YZ plane.

    Shown here is 1100 cross sections of the Gateau mountain. Tunnels sprout from one place and end up somewhere completely different! .

    Is the object completely connected, and maybe path, locally, or simply connected?

    It's incredibly difficult to say if the object is connected because the object's surface gets very 'foamy' near the surface. As an experiment I rendered a gigantic high resolution with around 100 iterations further below.

    Despite the appearance of island like structures and foam from the higher iterations, I've seen no real evidence that it isn't connected, though there are 'bridges' and other interesting holes such as the ones shown in David Makin's picture here.

    What happens at infinity iterations?

    See above. At the limit, the object starts producing lots of foam, or more specifically, countless spherical shapes of all sizes, in an orbiting satellite pattern around each other. Sometimes. Other times, the spheres will form a path in the shape of a sine wave or similar. It's all pretty bizarre. Take a look at the below image for instance. It's the same camera view as the picture in the gallery earlier. Except it uses around 100 iterations, instead of 9! See here for 7500x7500 resolution. Due to speed and complexity concerns, the lighting model used isn't perfect however, so check back later for a more accurate image.

    Why do lower powers exhibit less fractal-like detail than higher powers?

    I don't know - it's a surprising find to begin with. It could be that higher powers partially correct an error in the basic quadratic formula, because of the repetition of the calculation for higher powers.

    Does fractal detail really go on forever when you zoom in?

    Probably. That video further above zoomed in to around 10,000,000x, and detail lasted all the way (apart from the end, but that was only because I used 9 iterations).

    Any Julia equivalent for it?

    Actually yes. We've already seen a couple further above. But here are a couple more. Visit the respective links for full resolution and other pictures.

    Created by Krzysztof Marczak.

    Created by Paul Nylander.

    What is its volume and Hausdorff dimension?

    I'll leave it to others to work out a closed solution (which has got to be a least a little bit daunting), but for now we can always count voxels! For the quadratic version, the bounds are around: x: -1.1459 to 0.9693; y: -0.8275 to 0.8233 and z: -1.4142 to 0.8261. Due to the nature of fractals though, and the fairly small amount of time I spent, I wouldn't trust those figures 100%. The volume appeared to converge to around 0.869 (only accurate to 2 or 3dp by the looks of it).

    The power 8 version on the other hand has bounds just over radius 1 strangely enough (x, y & z from +-1.02 to +-1.04 roughly). The volume appeared to converge to around 2.945, significantly higher than the quadratic version.

    No idea about its Hausdorff or fractal dimension.

    Are there near exact copies of the object deep inside the structure?

    Not as far as we've seen so far. However, x-sided star shapes appear frequently though (according to the exponent in the formula). There may be something lurking deep within though!

    Any way to colour the object, to replace its mono-ish appearance?

    Other than a simple gradient across say, the Z-axis, or a radial mapping, there are a number of ways to colour the object, which I'll explore later. One approach may be translucency, or basing the surface colour on the average of the iteration escape value of the points around it. Along with perspective, it could easily yield exquisite results, surpassing anything even on these pages.

    For now, here are some brots with radial mapping (click for larger size):

    Any decent 3D software to render the beast?

    Not from what I've seen. I originally looked for a 3D program, but nothing out there seems to render arbitrary functions. Perhaps POVray comes closest with its isosurface approach, but even then one can't use local variables or constructs such as While loops.

    In the end, I had to create my own renderer (as did the other peeps at FractalForums.com)


    If the 2D Mandelbrot is the thumbprint of God, what does that make the 3D Mandelbrot?

    Er, the heart? Brain? Or the liver maybe? Ask a stupid question...

    What music are you currently listening to?

    What the heck has that got to do with anything? Since you obviously must know, I'm listening to the music from Phenomena Enigma, one of the best tunes of all time, let alone for a demo. Plus it's got a Mandelbrot in it (well a Julia at least!).

    Is this... the real 3D Mandelbrot then?

    As exquisite as the detail is in our discovery, there's good reason to believe that it isn't the real McCoy. Sure, there are incredible patterns, and I for one could be fooled at first glance. However, it would seem that the real thing will have even more exquisite detail, surpassing even the pictures we've seen! (That's if it exists, but hey, there seems less doubt about that now!)

    Evidence it's not the holy grail? Well, the most obvious is that the standard quadratic version isn't anything special. Only higher powers (around after 3-5) seem to capture the detail that one might expect. The original 2D Mandelbrot has organic detail even in the standard power/order 2 version. Even power 8 in the 3D Mandelbulb has smeared 'whipped cream' sections, which are nice in a way as they provide contrast to the more detailed parts, but again, they wouldn't compare to the variety one might expect from a 3D version of Seahorse valley.

    That means the biggest secret is still under wraps, open to anyone who has the inclination, and appreciation for how cool this thing would look. For sure I'll still keep looking. For those people who take up the search, I wish you the best of luck. Until then, we'll still have great fun exploring this object to the right I think!

    Programs to explore the Bulb

  • Fractal Lab You can immediately start exploring the Mandelbulb and other objects in your browser with this super fast renderer based on OpenGL GLSL. Included is a text editor so you can create and compile your own shapes!
  • Fragmentarium Also based on Open GLSL which again means it uses the GPU (if you have one) for amazing speed. Like Fractal Lab, you can also create your own custom functions/objects.
  • Mandelbulber - Amazing Mandelbulb/box renderer created by Krzysztof Marczak, and available for Linux and Windows. Also see the forum page for help.
  • Mandelbulb 3d - Excellent Mandelbulb/Mandelbox renderer by Jesse. Contains neat navigator mode.
  • Tacitus Beta by Xyrus02 - Nice interface and decent rendering for the Mandelbulb (and other objects potentially).
  • 3D Mandelbulb Ray Tracer - Nice realtime renderer by Subblue. It works either through the Adobe Pixel Bender Toolkit (free of charge), or through Adobe Photoshop (Photoshop is expensive, though this approach is easier to use).
  • Chaos Pro - The latest version (4 at time of writing) can now support Mandelbulbs and Juliabulbs.
  • Visions of Chaos - Created by Softology, this one creates some of the nicest renders I've seen yet. See these links for examples.
  • MathFunction Renderer by Thomas Grip.
  • Mandelbulb Explorer by Dmitry Brant.

    Other Mandelbulb related links

  • Mandelbulb (Images des mathématiques) - Further exploration of the Mandelbulb by Jos Leys, including orders less than eight.
  • Mandelwerk - A giant 3.3 Gig pixel zoomable Mandelbulb render - the biggest so far, created by Johan Andersson.
  • News coverage at New Scientist, Slashdot, Wired, to name a few.

    Other fractal links:

    Mandelbox home page - Tom Lowe's recent discovery - the incredible 'Mandelbox'; a 3D fractal with a more 'mechanized' look to it than the Mandelbulb.
    If you have a meaty GPU, try this small program created by Jan Kadlec to explore the Mandelbox in realtime.
    rfractals.net - Still some of the best fractals I've ever seen, created by Ramiro Perez (aka Aexion).
    Rena Jones - Open Me Slowly Video - Created by Kris Northern, probably the best 3D fractal animation I have seen. What are you waiting for, see it now!
    Evolving Fractal Animations - Also amazing. Very high details animations which warp and evolve fractals, created by Jock Cooper.
    3D Fractals - Tetrabrot - A cool program to explore a different kind of 3D Mandelbrot set.

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    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.