"Mandelbulb," a 3D Mandlebrot Construct, Discovered

symbolset writes "Many know the beauty and complexity of the Mandelbrot set. For some years now a few enterprising mathematicians / rendering fiends have been seeking a true 3D Mandelbrot set. A month ago a solution was found, and it is awesome to behold."

Comments

Actually, the Mandelbrot set is already 4D

[maxwell demon]

While the Mandelbrot set as usually defined is 2D, each point has an associated Julia set, where instead of the additive constant, the starting point is varied (the original Mandelbrot set always uses zero as starting point). Together, they give a 4-dimensional set, where two dimensions are given by the starting point (zr, zi), and the other two by the additive constant (cr, ci). The original Mandelbrot set is a cut through this 4D set at the plane zr=zi=0, while the Julia sets are cuts orthogonal to theat, at planes with constant cr and ci.

Re:Actually, the Mandelbrot set is already 4D

[jhesse]

This.

You can find a picture of a "4-D" Mandlebrot set in a mid/late 80's issue of Scientific American.
I was generating pictures of this on a 286 pc. (with EGA graphics) 15 years ago, and the pictures
in TFA of z^2 look *nothing* like that did.

Re:Actually, the Mandelbrot set is already 4D

[Eudial]

While not a pure mandelbrot, but a buddhabrot rendering: For the curious, here's a nice 2D projection of such a (rotating) 4D fractal I whipped up a while back.

Re:Actually, the Mandelbrot set is already 4D

[caramelcarrot]

Also, trying to extend the Mandelbrot set to 3D is ill-defined as there is no good 3D algebra equivalent to the complex numbers (two, 1 and i) or quarternions (four, 1 and i, j, k) - hence you can't express the iteration formula in 3D.

Re:Actually, the Mandelbrot set is already 4D

[Garble Snarky]

I was following the fractalforums thread for a while, and IIRC that is what a lot of the discussion focused on - "how can we define the squaring operation in 3D such that the Mandelbrot iterative equation gives us something like our vague notion of what we want the Mandelbulb to look like?"

Site is down, but I got an email notification from fractalforums a few days ago, and they had some incredible results. The pursuit is at least as much aesthetic as it is mathematical, and in that respect they've succeeded marvelously.

Re:Actually, the Mandelbrot set is already 4D

[shadowofwind]

I had missed a lot of interesting aspects of the 4D Julia/Mandelbrot combo when it was discovered, since computers were so much slower. I wrote my first Mandelbrot program on a Kaypro in high school. Used to run it over night just to get a 100x100 or so image, with low iterations.

The Mandelbrot set has those hairlike strands coming off of it, particularly at high resolution near pi radians. Nearby Julia set fragments, so to speak, all connect through those strands. Since the strand is between 1 and 2 dimensional in the Mandelbrot plane (having infinite arc length within a finite area, the strand within the 4-D coordinates is less than 4-D. So you could almost see something interesting in 3-D there. (Projected to 2-D of course. People who say they see 3-D crack me up, since the back of the eye is a 2-D surface.)

By the way, I particularly like the logarithmic spirals.

Re:Actually, the Mandelbrot set is already 4D

[Demena]

It is not the retina that sees but the visual cortex. So I wouldn't laugh too hard.

Re:Actually, the Mandelbrot set is already 4D

[maxwell demon]

{0,0,1}^2 doesn't seem to be well-defined.
Not only isn't the formula well defined at that point (division by zero), it cannot even be continuously extended to that point, because
lim_{e->0} {e,0,1}^2 = {-1,0,0}
while
lim_{e->0} {0,e,1}^2 = {1,0,0}
and even
lim_{e->0} {e,e,1}^2 = {0,-1,0}

Re:Actually, the Mandelbrot set is already 4D

[maxwell demon]

People who say they see 3-D crack me up, since the back of the eye is a 2-D surface.

But most people have two eyes, and the parallax between them gives the third dimension.

Re:Actually, the Mandelbrot set is already 4D

[fractoid]

This post needs more +insightful. What a lot of people are missing by getting wound up in the maths is that it is an artistic endeavour. Their definition of "a mandelbrot" (and yes, this broken terminology bugs the pedant in me beyond belief) is nothing to do with z^2+c, and everything to do with "a pretty looking blobby thing that maintains an aesthetically pleasing and visually interesting level of surface detail at all magnifications".

Re:Actually, the Mandelbrot set is already 4D

[SoVeryTired]

Good point. Hamilton was working on multiplying triples when he discovered the quaternions. Perhaps it can't be done in a sensible way.

Re:Actually, the Mandelbrot set is already 4D

[ObsessiveMathsFreak]

You don't even need a second eye, or at least, you don't need a parallax between them. Simply focusing on an object gives a good idea of its distance. To bring an object at a certain distance into focus, the eye muscles must contract "just so", allowing an estimation of that distance.

An then of course there is our brains, which interpret what we see. This is the reason why we can still have the illusion of 3D when looking at a truly two dimensional picture or TV screen. Of course, we can also be fooled, for example by cartoons or diagrams, into seeing 3D where none truely exists.

Our eyes really are designed to see in 3D. The grandparent appears to be suffering from chronic smartalecitis.

Re:Actually, the Mandelbrot set is already 4D

[Pedrito]

[i]People who say they see 3-D crack me up, since the back of the eye is a 2-D surface.[/i]

Really? A sphere is 2D? How are you enjoying things in flat world?

Re:Actually, the Mandelbrot set is already 4D

[maxwell demon]

Ever tried the "Magic Eye" pictures? There's exactly zero visual cues. Unless you manage to look at the image so that the left-eye and right-eye see it with a certain displacement (so different parts of the picture now match), you see not a single trace of the 3D figure hidden in it). The only depth information that is there is the displacement.

Re:Actually, the Mandelbrot set is already 4D

[Eudial]

Archive.org offers the full .avi file for download (the AVI version is about 4000 times more awesome than the flash version), and it's in public domain, so you are perfectly within your rights to go do it yourself.

Re:Actually, the Mandelbrot set is already 4D

[maxwell demon]

Thinking more about it, if we restrict ourselves to the unit circle, squaring a complex number is a continuous map from the circle on itself, which maps two opposite points of the circle to the same point. Now topologically, the pairs of opposing points of the n-sphere are equivalent to the n-dimensional projective space. The 1-dimensional projective space is topologically equivalent to the circle, so the continuous map is no problem. However, the two-dimensional projective space is not equivalent to a sphere. You can map it to a sphere if you map a whole straight line (i.e. for the original ball, a whole great circle, e.g. the equator) to a single point. To make that map, you can put a half-diameter sphere onto the equatorial plane of the original one, and then most diameters starting at a point on the original sphere cut the smaller sphere twice: once in the south pole (which is in the center of the original sphere) and then once more, which gives the image point. The diameters for the equatorial point pairs however only touch the south pole of the smaller sphere; all those pairs are then mapped to this single point. The result is continuous, but not any more 2 to 1 (since all the infinitely many points of the equator are mapped to the south pole). After that you can "blow up" the smaller sphere to the original size. Note that this map is continuous (the preimage of an open set is open; if the open set contains the south pole, the preimage contains the whole equator), however its "reverse" isn't (images of open sets don't need to be open; if the set contains part of the equator, the south pole is in the image, but is at the border of the image set).

I now think you can't make a map of a sphere onto itself which is both strictly 2 to 1 and continuous. I'm not completely sure, but I think whenever you have a 2-to-1 map of the sphere onto itself, it should be possible to apply a continuous bijection that moves those points to the opposite places of the sphere, so we end up in the situation described above, where it doesn't seem to work.

Of course a true mathematical proof (or disproof if I'm wrong) would be nice.

Re:Actually, the Mandelbrot set is already 4D

[DrVomact]

You can find a picture of a "4-D" Mandlebrot set in a mid/late 80's issue of Scientific American. I was generating pictures of this on a 286 pc. (with EGA graphics) 15 years ago, and the pictures in TFA of z^2 look *nothing* like that did.

Hah, I can beat that! I used a Compaq portable with an 8088 processor, 256 K of RAM and 2 floppies! I wrote a C program based on that original Scientific American article, and then had a Basic program read the results and display it. I think the C program took a week to run.

The joke, of course, is that the Compaq didn't have a color screen—it had a small grayscale monitor built in. But I still thought it was really cool.

Re:Actually, the Mandelbrot set is already 4D

[DrVomact]

No matter how many eyes you have, or where they are placed, you still see only surfaces.

That's interesting. As I think about it, I wander over to my aquarium and stare pensively. The water looks clean, the guppies seem as happy as guppies get. The seaweed is wafting gently back and forth. But wait, do I really see my aquarium? Or am I only staring at its surface?

Suddenly seized by philosophical doubts, I hold my hand in front of my face. Can I see my hand? —or only my palm?

Your remark is similar to one made by the British philospher G.E. Moore, in a paper published some time in the 1940s (I think). Can't remember the title at the moment...might have been "A Defence of Common Sense".

Now do 4d and animate it!

[Anonymous Coward]

Or would that open up a Lovecraftian dimension better left to slumber?

Re:Now do 4d and animate it!

[JoshuaZ]

No, Bob Howard at the Laundry already confirmed this one was ok. However, this is perilously close to the Turing-Lovecraft theorem which the public isn't supposed to know ab n34pnt!@!$ *NO CARRIER*

Not a "true" 3D Mandelbrot

[HEbGb]

It's definitely nifty, the pictures are beautiful, and the creator deserves praise, but the author himself says it's probably not a "true" 3D Mandelbrot:

http://www.skytopia.com/project/fractal/2mandelbulb.html#epilogue

As exquisite as the detail is in our discovery, there's good reason to believe that it isn't the real McCoy. ... ...
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.

So, Slashdot, I know this is asking a lot, but can you PLEASE at least read the article before posting? Thanks.

Re:Not a "true" 3D Mandelbrot

[Anonymous Coward]

You must be new here.

Re:Not a "true" 3D Mandelbrot

[symbolset]

There is a subtle difference between "a solution" and "the solution".

But yeah, I was selling it a bit because the pictures are so lovely.

Re:Not a "true" 3D Mandelbrot

[Tumbleweed]

So, Slashdot, I know this is asking a lot, but can you PLEASE at least read the article before posting?

No! I hate everything you stand for.

Re:Not a "true" 3D Mandelbrot

[Sebilrazen]

Looking at their UID I'd say they were here when here was new.

Ice Cream From Uranus?

[Anonymous Coward]

That ruined it for me.

Re:Ice Cream From Uranus?

[SeNtM]

Professor: "I'm sorry, Fry, but astronomers renamed Uranus in 2620 to end that stupid joke once and for all."
Fry: "Oh. What's it called now?"
Professor: "Urrectum. Here, let me locate it for you."

That thing looks like all of my nightmares.

[Anonymous Coward]

You could put it in a horror movie and make it pulsate.

Poorly-defined problem

[Brian Gordon]

What are they trying to do, make up some 3D fractal that just looks like the mandelbrot? This mandelbulb seems pretty arbitrary, and the whole point of the story seems to be that they've found a good one, not that they've found any kind of "true" solution.

Re:Poorly-defined problem

[Anonymous Coward]

They're trying to make a particular kind of 3d fractal, ie: has no simple edges. I'm sure these images look neat to the old and computer illiterate, but if publishable math has become "wow check out my graph!" then it's a sad day indeed.

Looks like a big sea slug.

[Ethanol-fueled]

I wonder if we'll ever reach the point where we will be able to define, with equations and rules, a sea slug using the principles of cellular automata?

Re:Looks like a big sea slug.

[wizardforce]

I wouldn't doubt it a bit. A sea slug is already defined by known rules and equations, it's just a matter of doing the math. Their genomes aren't terribly extensive compared to other organisms so it should be quite possible to simulate one quite accurately with a few simple equations and basic rules of chemistry and physics.

Re:Looks like a big sea slug.

[scheme]

It's all chemistry, physics and math.

Has anyone actually done this? With even a ''simple'' organism ( yes, those are air-quotes ), like a paramecium? It sounds easy in theory, but I bet when we actually get down to it, there'll be a few speedbumps and unexpected obstacles in the way.

Things are not even close. Look at vcell to see what's close to the state of the art in cell simulation. Right now, it's a matter of trying to get a few reactions and cell compartments working correctly. I don't think anyone has even come close to modeling any type of complete cell.

Re:Looks like a big sea slug.

[Artifakt]

Remember the film, Jurassic Park? They applied some simple math to make flocking behavior in their dino models look realistic. It works - just about everybody says the dinosaur flocking looks just like real flocking. Of course real biologists who have been trying to find the math behind real flocking have tested those equations the film makers used, and found some trivial little problem like you need to have faster than light telepathic communication between animal brains if you don't want the animals to get into a ridiculous gridlock once you add in some real environment modeling, but it sure looks like it's real flocking.
      And I'm sure we'll get paramecium models or mitochondrion models, or whatever, which 'look just like' the real thing, but turn out to be built on math that has fundamental problems with the rest of reality and uses some cheap hack like omitting surface roughness or gravity to gloss over that part, many times before anyone gets an actual model. We'll see 'accurate' models of atomic nuclei that build all 13 stable elements (or all 1047). 'Accurate' models of natural selection that show only plants should evolve eyes will follow. Eventually, your sea slug will act just like a real one does when the liquid it swims in is molten Sodium, (but not, unfortunately, in water).
      People will probably work some or most of these out. Accurate computer modeling of some events has happened, and many more will probably happen with advances in technology. Claiming that all of them will definitely work makes about as much sense as claiming all computer based aircraft models can safely skip the wind tunnel test stage of development.

Flashback

[Tx]

Weird, I definitely saw that thing after taking acid once, in fact I floated though it for quite a while. It may look all pretty on your screen, but that shit put me off drugs for life, man.

Re:Flashback

[Korbeau]

Weird, I definitely saw that thing after taking acid once, in fact I floated though it for quite a while. It may look all pretty on your screen, but that shit put me off drugs for life, man.

Modded informative?!?

What, is seeing the "mandelbulb" the mathematical incarnation of "this man" http://thisman.org/?

All I see is a big white rectangle

[Anonymous Coward]

With a message saying Page cannot be displayed. Not that impressive.

Re:All I see is a big white rectangle

[MobileTatsu-NJG]

With a message saying Page cannot be displayed. Not that impressive.

Did you try zooming in?

Re:All I see is a big white rectangle

[symbolset]

Try the mirror. (It needs sound and it takes a while to cache.)

Re:All I see is a big white rectangle

[The Dark]

Did you try zooming in?

It's 404s all the way down.

Video games need these now

[sayfawa]

Picture Half-life's Xen, Doom's Hell, or some Final Fantasy dimension rendered with these. Awesome.

Katamari Mandelrot

[Riddler Sensei]

I imagine if they included Mandelbrot fractals as something you can roll up in Katamari, then there would no longer be ANY need to experiment with psychedelic drugs ever again.

Zooming

[Spy Hunter]

Here's a 7500x7500 (56 megapixel) image of the fractal: http://seadragon.com/view/fnr.

Re:Zooming

[Sir_Lewk]

I love how ontopic your signature is.

Re:Zooming

[Anonymous Coward]

Is that one word now? Is its associated quality or state "ontopy"?

Slashdotted

[Kaladis Nefarian]

Seems to be slashdotted, cached version: http://www.skytopia.com.nyud.net:8090/project/fractal/mandelbulb.html

Or something like...

[denzacar]

Langoliers remake.
Those things already look like they are made of teeth. Endless rows of teeth that devour the world.

A sad day indeed...

[symbolset]

If that's the case, it's been a sad day since at least 1984. These things teach us interesting things about numbers and are interesting in and of themselves. As a way of making math more visually beautiful they also serve to draw the interest of youth to a field ordinarily seen as dry and boring.

w00t

[lycium]

cool, nice to see my images linked on slashdot :) hopefully we'll have some gpu-accelerated results to show you all soon (and for those with opencl supporting cards, executables).

btw interested parties might like to check out my 3840x2400 resolution render of the 7th degree version here: http://lyc.deviantart.com/art/siebenfach-139038934 (it's buried deep in the thread, and fractalforums is creeking a bit)

Yes...

[Anonymous Coward]

Looks like a Yes album art generator...

a great leap forward

[circletimessquare]

for scientific screensaverology

Fraqtive

[nephridium]

A very nice open source app, available through the Ubuntu/Debian repositories. The author's page even got a windows version.

It supports multi-core CPUs, i.e. if you really want to tax each of your CPU's core to the limit, just use the app to browse through the mandelbrot set. It also supports a 3D extrapolation of the 2D set (OpenGL and software).

Strangely enough it doesn't seem all that popular, as the forum doesn't seem all that populated..

broccoli

[oliphaunt]

and here I thought I was coming to read a post about Romanesco Broccoli (link goes to gis for "romanesco"). Seriously, it's like eating math.

Animated quaternion

[_bernie]

The common Mandelbrot set is really a 2-dimensional slice of a 4-dimensional object identified by both the combination of the complex numbers Z0 and C in the canonical Zn+1 = Zn^2 + C. The mandelbrot set lives in the plane where Z0 = 0 + 0i, while the Julia sets live on infinitely-many-squared orthogonal planes in the remaining two dimensions, each one intersecting Mandelbrot's plane in a single point of complex coordinates C.

Visualizing this hyperspace monster was made easy by POV-Ray. It took my computer two week of computation to render 80 seconds of animated 3D slices of a the quaternion. Check out the scene source.

/me looks forward for a real-time Julia4D explorer.

Elder feuds reignited?

[Mathinker]

UID 3706 replies to UID 6544:

> No! I hate everything you stand for.

From my almost 7-digit standpoint, your feuding looks a lot like cyber-mythology! Is there a deeper story here? Were you both swallowed and subsequently regurgitated by a 3-digit UID?

Re:Elder feuds reignited?

[Tumbleweed]

UID 3706 replies to UID 6544:

I am not a number, you young punk! And get off my damned lawn!

Re:Elder feuds reignited?

[Per Wigren]

I'm caught in between here :-(

Re:Elder feuds reignited?

[Tumbleweed]

but do you even had computer in the 4 digit era? or was slashdot some sort of paper mail based discussion forum?

Gawd, don't they teach you brats anything in school these days? It was all vacuum tubes back then. Of course, it's all ball bearings, now. We would've _killed_ for ball bearings back in the day!

Re:Elder feuds reignited?

[Rubinstien]

Me too. Hmm. Newton's Method with UID's?

Re:Elder feuds reignited?

[Bill Currie]

*Burp*

And tasty they were, too.

In nature - I give you, Brassica oleracea!

[zooblethorpe]

These particular fractals remind me of things I hope never to see in nature.

Some of it, at least, has already happened: see this fine example of Brassica oleracea, for instance.

Then again, you might have been referring to some of the fractal images that call to mind the work of H. R. Giger... < shiver >.

Cheers,