## "Mandelbulb," a 3D Mandlebrot Construct, Discovered 255

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."*
## Actually, the Mandelbrot set is already 4D (Score:5, Informative)

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 (Score:4, Interesting)

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: (Score:3, Interesting)

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 [oldcomputers.net] 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 (Score:5, Interesting)

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

## Re: (Score:3, Informative)

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 (Score:5, Insightful)

## Re:Actually, the Mandelbrot set is already 4D (Score:5, Insightful)

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 (Score:5, Interesting)

artisticendeavour. 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: (Score:2)

They defined a way of multiplying points in 2-space equivalent to the "stretch and rotate" interpretation of complex multiplication. The formula for (x,y,z)^2 is given at the top of this [bugman123.com] page.

It doesn't have the same mathematical structure as the complex plane, but as the article suggested, it may be the case that the "stretch and rotate" property is all you need.

## Re: (Score:2)

Ugh, I meant 3-space, sorry.

## Re: (Score:3, Interesting)

{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 (Score:4, Insightful)

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

## Re: (Score:2)

> trying to extend the Mandelbrot set to 3D is ill-defined

Depends what you want to achieve. This could be said for all 4D objects that you want to project in a 3D space. Most fractal programs (that support quaternions) solve this by projecting three variables and varying the fourth through time.

## Re:Actually, the Mandelbrot set is already 4D (Score:4, Interesting)

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: (Score:3, Insightful)

## Re:Actually, the Mandelbrot set is already 4D (Score:5, Insightful)

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

## Re: (Score:3, Insightful)

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

## Re:Actually, the Mandelbrot set is already 4D (Score:4, Funny)

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

## Re: (Score:3, Insightful)

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 194

## Now do 4d and animate it! (Score:2, Interesting)

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

## Re:Now do 4d and animate it! (Score:5, Funny)

## Not a "true" 3D Mandelbrot (Score:5, Informative)

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 [skytopia.com]

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 (Score:5, Funny)

## Re: (Score:3, Informative)

## Re: (Score:3, Informative)

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.

## "Not a 'true' 3D Mandelbrot" misses the point... (Score:2)

While you may have

apoint, it is similar to complaining about Ampere's Law, before Maxwell's correction. Sure, it wasn'texactlyright, but it more or less had the same properties.This may not be the simplest function, but it retains the most fundamentally interesting properties of 2D fractals: infinite detail generated by a simple mathematical function. It is fascinating just the same, and is only a (very) minor modification of the original 2D function.

The Mandelbulb is awe-inspiring, and it is disappoi

## Re:Not a "true" 3D Mandelbrot (Score:5, Funny)

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.

## Elder feuds reignited? (Score:3, Funny)

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? (Score:4, Funny)

UID 3706 replies to UID 6544:I am not a number, you young punk! And get off my damned lawn!

## Re:Elder feuds reignited? (Score:4, Insightful)

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? (Score:4, Funny)

## Re: (Score:2, Funny)

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

## Re: (Score:2)

Once upon a time, Slashdot editors did read what they posted. I guess both 3706 and 6544 remember.

## Re:Elder feuds reignited? (Score:5, Funny)

*Burp*

And tasty they were, too.

## Re: (Score:2)

My pedantic nature is appeased. As you were, gentlemen.

## Ice Cream From Uranus? (Score:5, Funny)

That ruined it for me.

## Re:Ice Cream From Uranus? (Score:5, Funny)

Fry: "Oh. What's it called now?"

Professor: "Urrectum. Here, let me locate it for you."

## Re: (Score:2)

I came here to say that. Seriously great work until the 2G1C reference.

## Re: (Score:2)

Ice Cream From Uranus? In accordance with that very popular rule of the Internet... there is another picture for that... a more illustrative one too.

Tub Girl. Google It. You're Welcome... bwahahahahahhahahahahhhahah!

## Re: (Score:2)

[Harry has just had an alien removed rectally]Dr. Allison Reed:It's over, it's over. You did great! Do you need anything? Can we get you anything?Harry Block:Ice cream... I'd like an ice cream please.Dr. Allison Reed:Okay, what flavor?Harry Block:It doesn't matter. It's for my ass.## That thing looks like all of my nightmares. (Score:5, Funny)

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

## Re: (Score:2)

Yes, in so many ways yes.

## Poorly-defined problem (Score:4, Insightful)

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: (Score:2, Insightful)

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.

## A sad day indeed... (Score:5, Insightful)

## Re: (Score:2)

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

## In nature - I give you, Brassica oleracea! (Score:4, Informative)

Some of it, at least, has already happened: see this fine example of Brassica oleracea [ubcbotanicalgarden.org], 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,

## Looks like a big sea slug. (Score:5, Interesting)

## Re: (Score:3, Interesting)

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: (Score:2)

## Re: (Score:2)

Not quite. We do know the rules of chemistry well enough to model proteins, the problem is that the amount of sheer number crunching is enormous. As for water, that's also not quite true. We have the equations for interactions between molecules worked out it's just a matter of doing the math which is a lot... It took weeks to simulate proton jumping using similar equations in superacids for a time period of less than a microsecond. There's a lot of math involved but it's math that we know how to do.

## Re: (Score:2)

If only

## Re: (Score:2)

A sea slug is already defined by known rules and equations, it's just a matter of doing the math.

Really? What are they?

## Re: (Score:2)

*assignment of codons to specific amino acids (conversion of the genetic code into polypeptides)

*energy minimization of protein structure (protein folding and interactions)

*capacitor electronics (nervous system)

It's all chemistry, physics and math.

## Re: (Score:2)

And this is, of course, worse than saying that a computer is just a pile of transistors. The environment in which the object grows, its diet, the concentrations of salt and oxygen and the temperature of its environment all affect the most basic of its functions. Worse, it ignores the mitochondria and the environment of the egg in which it was hatched.

It may all be chemistry, physics, and math, but a lot of the most critical functions are completely unsoluble and intractable if you treat them this way. And t

## Re: (Score:2)

Not really. Our computors are advancing rapidly and from what I've seen in the field, there's significant room for efficiency improvements in the way we do the calculations. Protein folding for one may benefit from some newer algorithms being developed in the field. The mitochrondria are very simple by comparison to the rest of the organism. That doesn't pose much in the way of being an obstacle. Then there's the part where you aren't forced to model every single molecule in the cell at once. Just the

## Simulating a sea slug (Score:2)

It's still just physics. You don't have to do any energy minimizations or understand how protein folds. Just solve it the way Nature does: brute force. Stick some atoms together and plot their movement over time. If you want to include the slug's environment and food, then expand the box to include those things too.

The only problem is that your computer isn't fast enough. You can't simulate a slug. You can't simulate a slug's heart. You can't simulate a single cell. You can't simulate a strand of DNA. The b

## Re: (Score:2)

It's all chemistry, physics and math.That's just handwaving.

No one fully understands the complete physics and chemistry of the simplest forms of life, let alone a sea slug.

## Re: (Score:2)

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.

Once we get the sea slug calculation going, anyway, how do we test it? You know, to see that these formulas actually create a phenomena that mimics a sea slug, beyond just looking like one, cellularly? The environment that a sea slug lives in is probably orders o

## Re:Looks like a big sea slug. (Score:4, Informative)

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 [uchc.edu] 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. (Score:5, Insightful)

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

allof them will definitely work makes about as much sense as claimingallcomputer based aircraft models can safely skip the wind tunnel test stage of development.## Re: (Score:2)

The basic quantum physics formulas that cover the interactions of protons, neutrons and electrons.

Given sufficient RAM and processing capability we could simulate practically anything via a brute force approach. I doubt that the worldwide total of either one is enough to fully simulate a sea slug down to the subatomic particle level, but we already know algorithms that could do it.

## Re: (Score:2)

The basic quantum physics formulas that cover the interactions of protons, neutrons and electrons.

Well, I think you're over-simplyfying it. We do know the physics and equations for those, but just plugging them into a computer won't give us sea slugs. ( I think the domain of possibility of DNA life is infinite). We need to actually specify the 'sea slug' program -- as another poster pointed out, the DNA that specifically is a sea slug.

## Flashback (Score:4, Funny)

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: (Score:2)

A few of those things look like magnified pictures of pollen...

## Re: (Score:2, Interesting)

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/ [thisman.org]?

## Re: (Score:2)

Hi I'm the author of the article. Curious, which image was it that created the flashback?

## Movie possibility? (Score:2)

## Or something like... (Score:3, Funny)

Langoliers remake.

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

## All I see is a big white rectangle (Score:2, Funny)

## Re:All I see is a big white rectangle (Score:5, Funny)

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

Did you try zooming in?

## Re: (Score:2)

Yeah. Nothing changed. :-(

## Re:All I see is a big white rectangle (Score:5, Funny)

Did you try zooming in?

It's 404s all the way down.

## Re: (Score:2, Funny)

## Video games need these now (Score:3, Interesting)

## Re: (Score:2)

You mean like this? http://en.wikipedia.org/wiki/Rescue_on_Fractalus [wikipedia.org]!

## Re: (Score:2)

## Katamari Mandelrot (Score:3, Insightful)

## Zooming (Score:4, Informative)

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

## Re:Zooming (Score:4, Insightful)

I love how ontopic your signature is.

## Re: (Score:2, Funny)

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

## Re: (Score:2)

Nifty but it seems to be a Microsoft thing so can't actually be good. :) Seems to work with javaScript and not Flash or Silverlight!

## Re: (Score:2)

Though they do say "for better performance, install Silverlight."

## Slashdotted (Score:5, Informative)

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

## w00t (Score:5, Informative)

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 [deviantart.com] (it's buried deep in the thread, and fractalforums is creeking a bit)

## Re: (Score:2)

## Re: (Score:2)

Thanks for this. It takes me back to 1991 or so when I discovered fractals and suddenly didn't detest math any more.

## Re: (Score:2)

## ...and in other news (Score:2)

...and in other news: Shares in printer ink manufacturing companies rose significantly tonight, and a spokesperson for local schools' IT said they hoped this development would now give them something to finally replace that picture of the cartoon duck smashing the computer with a large mallet, provided the aged blue tack hadn't fused the original printout from 1998 permanently to the computer room walls.

## What if the "true" set is more mundane? (Score:2)

I found that hot chocolate (not too watered-down) in a white ceramic mug leaves a very rudimentary but easily discernible "Mandelbrot" set. At least the classic image (I have no way to zoom in to great detail on the side of my mug.). The set is left over from "chocolate bubbles".

Is it possible that the lines of the Mandelbrot set are simply outlines of colliding bubbles? The 3D version of this, while cool - would be significantly less impressive than the images from the article.....

-CF

## Broccoflower formula? (Score:2)

## Yes... (Score:2, Funny)

Looks like a Yes album art generator...

## a great leap forward (Score:5, Funny)

for scientific screensaverology

## Fraqtive (Score:5, Informative)

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 [mimec.org] doesn't seem all that populated..

## Ow my sanity (Score:2)

Amazingly cool maths though!

## Re: (Score:2)

They are nightmarish images alright...

The whole creature looks very malicious.

## Nature imitates art (Score:2)

Compare the images to Louis Sullivan's late 19th and early 20th century ornamentation:

http://en.wikipedia.org/wiki/File:Van_Allen_Column_Capital.jpg [wikipedia.org]

http://en.wikipedia.org/wiki/File:Van_Allen_3.jpg [wikipedia.org]

http://www.harboearch.com/getProject.php?projname=sullivancenterc [harboearch.com]

## broccoli (Score:3, Interesting)

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

## Animated quaternion (Score:4, Interesting)

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 whereZ0 = 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 [povray.org]. It took my computer two week of computation to render 80 seconds of animated 3D slices of a the quaternion [sugarlabs.org]. Check out the scene source [sugarlabs.org].

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

## Sea urchins and diatoms (Score:2)