The Mystery of the REAL, 3D Mandelbrot Fractal
Timestamp 21/03/2008: Probably no eye has ever seen the wonders of the 3D Mandelbrot set. This page goes in search of the elusive beast  and what could be considered by some to be the Holy Grail of fractals.
When Mandelbrot first saw the shape's outline  that was in itself a curiosity  but nothing could have prepared him for what he saw when he zoomed in. The edge of the basic Mandelbrot shape gave way to millions of intricate details, which after further zooming, unveiled elaborate shapes of every type and description. Before I continue, here's a small collection for your viewing pleasure (created with Chaos Pro). Click any picture to enlarge. 
Type 1: "Spin me right round baby"Also known as the "vase lathe", this design looks neat, but is effectively a 2D Mandelbrot spun round its own axis. In fact, there are a few different ways of creating this shape (such as using Quaternion math). The object looks 3D, but the detail is still really 2D. 
By Jules Ruis. For full size, follow this link 
By Paul Bourke. 
By Marco Vernaglione. 
Type 2: "Mountain out of a 2 dimensional molehill"Okay maybe that trivializes some of the possibilities from this fractal type, but hey, this one 'cheats' too. The idea is to take the iteration level, and raise or lower the shape instead of coloring it (as in traditional Mandelbrot pics). The results can be stunning, but it's still 2D at heart. 
See here for the full Youtube animation. 
Mountain mandelbrot by Anton Feenstra. 
This beautiful image created by Aexion looks great, but it's not exhibiting fractal detail in all 3 dimensions. 
Type 3: "Bubblegum Flavored Whipped Cream"This type is based on fractals in 4 dimensions using Quaternion math. Despite the complex sounding theory though, we can see that the detail is effectively one or twodimensional with the 3rd dimension being turned into a whipped cream smoothie. I have my doubts that the 4th dimension fares any better. Looks tasty though. (Note that these images are in fact based on the Julia set, which is closely related to the Mandelbrot). 
Quaternion julia fractal by Paul Bourke.  Quaternion julia fractal shown at Wikipedia. 
By Godwin Vickers, this is in many ways more interesting than the previous two, though still has only 1 or 2 dimensiuonal detail. Taken from here 
Type 4: "Climbing Frame Madness"This more recent "Tetrabot" type was discovered around 10 years ago by Prof. Dominic Rochon. The variety of shapes are intriguing, approaching the Mandelbrot set in their subtlety. However, detail levels are somewhat poor (apart from at high iteration levels where unfortunately the 'framework' becomes so thin you can't see it). Still, this is a good attempt at creating something a little more 3 dimensional. 
Tetrabot fractals from 3dfractals.com/ and Youtube 
From here. 
From here. 
Created by Marco Vernaglione; part of a larger design. 


Okay if this looks familiar, that's because you'll probably recognize it as Romanesco Broccoli  the coolest looking vegetable in the world (and certainly the most fractallike). I added some colour in a somewhat successful attempt to spruce it up, but the original photo by Alfredo Matacotta is featured here. If you want to see more Romanesco Broccoli, then I suggest you Google it. 
Dr. Kazushi Ahara and Dr. Yoshiaki Araki first discovered this amazing Quasifuchsian fractal. You can see a video on Youtube, their gallery (from which this picture is taken), or their website. What's so amazing about this fractal is first how it contains detail in all 3 dimensions, but also how close it comes to the style and character of the 2D Mandelbrot set. 3D fractals are by and large very ordered, and more organic objects such as the Mandelbrot are incredibly rare in the 3D world. Unfortunately, as far as I know, the object will not have quite the variety of the Mandelbrot once you've zoomed into a section. Still the Quasifuchsian is definitely something I would like to explore further. 
This cool 3D fractal created by Gilles Pouliquen, and rendered by Thomas Ludwig, is an interesting attempt at creating the 3D Mandelbrot, and a lovely shape in and of itself. (but still no cigar...). The Mandelbrot set merges with the "Tricorn" fractal (or "Mandelbar" set) to form this interesting 3D hybrid created by Eric Baird. Love to zoom into this also, but once again, we can most likely expect a 'whipped cream' surface once we do. 
Is it really possible to compute a REAL 3D Mandelbrot? I've heard when rendered it's in the shape of an 'apple' with arms or tentacles growing off of it  I've been told this fractal was given it's own name, and that it's not called a "3D Mandelbrot".
Any info would be appreciated, with feedback via this Newsgroup & EMail. Thanks,  Lester 
> Has anyone tried to generalize the Mandelbrot set to include a third dimension?
Yes, look for references to Alan Norton. He explained them to me over greasy hamburgers in a cheap fast food restaraunt once. The Smithsonian Magazine had one of the little rootlike solids on their cover a long time ago... 
I saw this mentioned in the latest "Algorithm" journal...The method used was to vary the exponent (Z < Z^p + C) and use p as the third dimension. I didn't get a chance to carefully read the article, but.... 

OVERVIEW: Basically, the Mandelbrot can be plotted on a graph as shown below. Let's test a point at x=0.2, y=0.4 (shown as blue dot below). The computer has to decide whether the point at x=0.2, y=0.4 is part of the Mandelbrot set or not. We know it is, because the blue dot is inside the Mandelbrot shape, but the computer only has the numbers x=0.2 & y=0.4 to go by. How can it decide? 
STEP 1: Rotate the point round by the same number of degrees again (0 degrees is 3oclock Eastwards, and the pivot (to rotate around) is the big white dot at x=0, y=0): This creates a new point at about x=0.27, y=0.36: 
STEP 2: Find the distance from center to the original point ( squareroot(0.2*0.2 + 0.4*0.4) = 0.447 (Pythagoras theorem)), and make the newly rotated point (x=0.25, y=0.36) the same angle, but at the new distance (0.447^{2}). This makes a third point (x=0.2, y=0.16), colored bright white below: 
STEP 3: Now, move the new point by as much as the initial vector was worth (the blue arrow or x=x+0.2, y=y+0.4). The yellow star represents the final position! ' 
STEP 4: Finally, perform the above 3 steps again, but this time with the yellow star point replacing the old blue point (however, make sure you always use the ORIGINAL blue arrow for the final movement in step 3 (i.e. always move it by x=x+0.2 y=y+0.4). Once you've gotten the new "yellow star" point, do the same again, and AGAIN. In fact, keep doing it, until the "yellow star" point either flys away to infinity, or closes in on the center at x=0,y=0. If it DOES close in to the center, then the original blue point (at x=0.2, y=0.4) is colored black (i.e. is in the Mandelbrot set). Otherwise, if it flies off to infinity, 0.2,0.4 is colored white (and is not part of the Mandelbrot set). STEP 5: Now do all of the above 4 steps, but not just for the point at x=0.2, y=0.4, but for every point inside the rectangle ranging from x=2 to x=1 ...and... y=1 to y=1. CONGRATS  You've just drawn a mandelbrot, and understood complex arithmetic without even knowing it! Steps 1 & 2 are actually complex multiplication, and step 3 is complex addition! 
Best of the Web  3D Fractal Gallery  
"(0,0.5)" Out of all the fractals on display here, these first three (created by Dr. Kazushi Ahara and Dr. Yoshiaki Araki), come closest to characterizing the original 2D Mandelbrot set, but in 3D! Click picture, or see their fractal gallery for high resolution versions. 
"(0.949,0.275)" Another classic "Quasifuscian" fractal by Dr Ahara and Araki. The originals are all similarly coloured, so I changed them slightly for variety here. Gotta love the encrusted 'microbial' look in the many layers of detail. 
"(0.046,0.801)" The final pick of the gorgeous spongelike spheres. If you want to see the animation, visit the YouTube site here. 
"Cipater" Created by Thomas Ludwig (aka Lycium), this unique fractal architecture combines order and chaos in its sweeping, fiery subsurface balconies. Also see his excellent frosty "Snow" picture, similar in style to this. 
"The Return of the Sunset Castle" This jawdropping fractal created by Aexion looks like something straight out of science fiction. The subtlety of shade, the positioning of elements, the way each tier decorates the rim on the underside, all make this fractal a sight to behold. About the only thing I can fault is the lack of colour. Also see inside the sunset castle or his earlier nearmonochrome Sunset Castle fractal, both of which also look amazing. 
"Chaotic World" Also created by Aexion, this fractal uses some kind of circle packing technique around a sphere, but thanks to the slight randomization, crater textures, use of colour, it is so much more than that. Visit his site to see full size! 
"X01021" This inspiring design by Joseph Presley (aka fractart.net) comes under the class of the "iterated function system" fractal. But what separates this from the rest is the judicious use of contrast between larger and smaller objects, along with the nicely shaped contours of the fractal 'shell'. Other pictures of note in his gallery include these ones. 
"Kid in Menger Sponge" The menger sponge is back, and it has some tricks up its sleeve, including full raytracing, reflection, and a snazzy perspective (other artists take note!). Created by Paul Bourke. 
"Blobopolis Closeup" There are literally hundreds of these type of fractals on Deviantart.com, and it's no wonder why  it's fairly easy to create good looking results. However, this surpasses 99.9% of them through clever use of light, shade and pattern. We can thank piritipany for creating this fantastical paradise dome. Here is a variation on the fractal. 
"Field of Depth" The creator of this masterpiece won a contest on deviantART.com and it's not hard to see why. The way the 'fabric of spacetime' is dented by the spheres looks good enough, but Zueuk creates fantastic drama in this molecular apocalypse through the use of clever lighting and motion effects. See his gallery for full size and more cool pics like this. 
"Dreams Of Loneliness" One of the best Apophysis renders ever, this gem created by Xavier Hemon is quite simple, but effective due to the mesmorizing patterns of neon light and design of lower levels. 
"jOrb" Worlds within worlds of exquisite detail, Christopher Payne (aka Khan71) did a sterling job in using Apophysis' powers creatively. 
"Rise" What happens when you join numerous Sierpinski tetrahedrons to form an icosahedron? Something similar to the above, as is demonstrated by Thomas Ludwig (aka lyc). Beautifully raytraced, this object looks like some kind of foam I want to squish. As well as the Quasifuscian objects, I want real life versions of these. 
"Cubism" And here we witness the Cantor Dust Fleet joining the borg collective to unite as one. This curious fractal by Stefan Vitanov is a neat twist on an old trick. 
"Fractal Reflections" Okay, this one's a bit simplistic, but the polished material, spherical building material and general camera position helps make the otherwise ordinary Sierpinski pyramid a little more special. Created by JRJay. Also see his excellent flash animation. 
Fractal Structure And Some Red (Feb 03 2007) If any of you have ever played the brilliant game  Marble Madness, you'll feel right at home with this fractal by Rune Skovbo Johansen. Nice concept, and execution. A little more variety of shape would be nice though. As usual, click picture to see full size. 
"Ivory Tower" Stefan Vitanov (aka Fractalexperience.com) has created many variations on the fractal tower (see his gallery for more), but the three presented here are arguably his best. Despite the lack of colour, the fogging, 'rotational' shape recursion, and atmosphere make this one not to miss. One day, all buildings will look something like this. 
"Temple of Shadow" Great shot again by Stefan Vitanov. I want more colours and higher resolution though please! 
"City (2)" Our last pick from Stefan Vitanov's fractal temple series. 
"Inside the Sierpinski Temple" David Makin (aka MakinMagic) has also created variations of the fractal temple  this one's on the inside. Here's another view. Also see his brilliant animation inside the temple. 
"Spherism" This just made the list thanks to the cool texture, and somewhat unusual fractal extrusion style. Taken from Yanekthedarkangel's gallery. 
"Sphere packings in 3D  Tetra 1" A treasure of tasty spheres packed into one. Good for a snack or if you want to fill in between meals. There's plenty more where this came from. Also see his excellent sculpture gallery. 
"Rising Apophysis III" Combining sculpture with fractal goodness, this image created by Michael (aka Gibson125) uses vast plumes of smoke to carve the shape you see in the horizon. Also see the Twisted Cube created by greenhybrid which has a similar atmosphere to this. 

"Fractal tree Plate b 3" The best fractal tree I could find during my search. Let us know if you find a better one! Created by A.Kitaoka (?). 
"Faire" This iterated function fractal by Bob Q (aka gateman45) just gets better the more I look at it. Very elaborate detail, but see full size to fully appreciate it. 
This expertly crafted abstract structure by Francesco Mai is barely a fractal, but had to be included, if only for the delicious translucent and golden material. 
"Furnace" Another winner by Aexion (aka rfractals.net), and looking as though it came straight from the Matrix. Spectacular lighting and camera ensure this image gets its place in our top collection. 
"Opengl15" For this collection, we generally frown upon mountaintype fractals. But this image created by Jason Rampe (aka Softology) has enough going for it for inclusion  from the whirpool like imagery, to the vivid coloring. Also see Gibson125's amazing mountain fractal  "3D Fractal Landscape VI", and Duncan Champney's "Douaday's Easter Rabbit". 
"Orbital Fern" Similar in style to Gateman's "Faire" work, this organic IFS fractal created by Phidelity begins to rival the Romanesco brocoli in its awesome detail and style. Food would taste much better if it looked like this :) See his other IFS fractals like this one, and if nothing else, see his AMAZING animation. 
"Sensible Extensibilities" This excellent image was also created by Phidelity. Also see his variation on the Xenodream shell. 
"Xeno_Dream_Test_Run" This image is actually based on one of the example routines in the XenoDream fractal program, but f__l__A__r__k_s has done a good job in bringing the most out of it with his use of colour. 
3D Fractal Elephant Wow, this is gorgeous  the Aztecs would have been proud I'm sure. The combination of curves and corners along with the colour scheme gives a real sense of style and mystery to this piece. Created by Duncan Champney, see his 3D fractal gallery for more. 
Fractal Monoliths Our second pick from Duncan's gallery. Looking like a giant set of strange dominos, I love the perspective view in this one  the sharp angle gives a real sense of depth. See his excellent Fire and Ice themed 3D fractals, also created by Duncan's "FractalWorks" software. 
Sphereflake level 9 Created by the user 'fpsunflower' down at ompf.org, this object has the characteristic Mandelbrotstyle sphere on sphere idea. See the forum for a larger version, and an excellent 'night' version as well. 
3D Fractal Broccoli Created by Aleksandar Rodic, this stunning visual is not real Romanesco broccoli, but rather a 3D adapatation. I prefer it to the original vegetable in many ways, not least because of the variety in colour that's possible. See his website for a great video and 3D anaglyph version. 