"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

Now do 4d and animate it!

[Anonymous Coward]

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

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.

[Anonymous Coward]

No. read this (quite interesting):

http://en.wikipedia.org/wiki/Digital_physics#Pancomputationalism_or_the_Computational_universe_theory

and go from there. The fact that we know the equations doesn't mean we can simulate anything.

For example : say a slug was a spherical shape. Just to represent pi, and therefore its volume would require infinite precision. There is a massive difference between knowing the ideal equations and simulating an organism.

Video games need these now

[sayfawa]

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

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.

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.

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: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/?

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

[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

[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.