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Author Topic: I've got a new way to view fractals... maybe?  (Read 1948 times)
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« on: October 09, 2007, 11:49:24 PM »

I believe the mandelbrot fractal is merely a single perspective of a 3D structure. That is, all current fractal programs are drawing a 3D structure in merely 2D, a single 2D perspective. But you could produce many many more 2D images by retooling the fractal program to interpret the fractal function as 3D.

Now I know there have been some '3D' fractal programs out there that visualize the mandelbrot set in a three dimensional form. As if the image it produced was a terrain height map, black is high, white is low.
This is not what I mean, this is merely using the same fractal single perspective 2D interpretation like all others as a heightmap. It is showing the outcoming 2D image in 3D, it is not interpreting the fractal as 3D.

Now what exactly do I mean when I say interpret it as 3D?

It is quite simple. When I see recursion getting increasingly smaller, I do not see a 2D image getting recursively smaller. I see it as perspective.

See what I mean?

Imagine if, the recursively smaller portions of the fractal actually did not get recursively smaller. But rather, they were just moving away from you in 3D space. So one swirl, in 3D space, is the same size as it's proceeding swirl, it's just, farther away from you.

This is more than just constructing the fractal into 3D space as shown here http://www.fractalforums.com/index.php?topic=717.0
Which btw is an awesome image, I'm not saying this stuff cause I don't think current fractals are great, I'm just curious to see what this might open up. But as you notice, that 3D fractal still interprets recursively smaller as physically smaller. I want to see what it would look like to interpret recursively smaller as further away in 3D space.

So think of current fractal programs as interpreting the visual image onto X and Y plane, 2D. With iterations as a controller of the detail of that image. I want to see what it would like if the mandelbrot set used the variable of iteration to interpret the placement of voxels along the Z axis.

So sample the mandelbrot set with iteration of 1. Place the sampled points in 3D space varying the position in the X and Y plane based on the output of the evaluation, but all of these voxels retaining a z value of 1. Voxel = point in 3D space
Move to iteration of 2, same as before, position the output of sampled voxels varying the X and Y, but all the voxels with a Z value of 2.
Then to iteration of 3... etc. Basically iterations = Z position.

Has anyone done this?

I come here to find anyone who is familiar enough with the math and possibly knows the basics of an open source fractal program or something to possibly implement this. Because I am not familiar enough with the math or the existing programs to do this in any efficient fashion. But perhaps, some programmer is willing to try a radical new method of visualizing the mandelbrot? Is this a new method? Or has someone else already thought of it and done it?

Now I need some help also possibly figuring out the math. Because I am not sure if just iteration = Z is exactly what I'm thinking.

For example, this is the relation I am looking for between the mandelbrot fractal and the 3D interpretation. The bottom of this image representing the mandelbrot. Like the mandelbrot is at the bottom looking up into a tree that have uniform width branches. So maybe some math needs to be in place to counteract the reduction in size of things to keep it all uniform in size?

Where the big circle in the center of the mandelbrot, is really not a 'big circle'. But rather just the bottom of a tree that has uniform branches from the trunk.

Z=iteration, maybe?

anyone care to help?

I do 3D renderings for my job, so if anyone does manage to help me get this into 3D model format. I could render it in all sorts of ways.
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Posts: 238

« Reply #1 on: October 10, 2007, 12:15:58 AM »

Interesting idea.  If I understand correctly, then what you want to do is to plot the orbits of many pixels in 3D space, where the third dimension (vertical in your illustrations) is the time or iteration dimension.  If I have that correct, then the math is fairly straightforward and a list of iterates can easily be generated.  However, what you see may not be what you expect, for a couple of reasons.  Firstly, the Mandelbrot set is a plot of parameter space, not of variable space.  That is, the Mandelbrot set essentially catalogs the fate of each pixel, not the dynamics of how it got there.  Secondly, there will be large jumps from one iteration (vertical level) to the next.  For example, the point (-2,0) will be at (-2,0) on the first iteration and (2,0) for second (and each successive) iteration, a jump of 4 units in one step.  Not that these are problems, but characteristics of which you should be aware.

Please keep us posted with what you find!

Fractal Supremo
Posts: 1158

« Reply #2 on: October 10, 2007, 02:48:03 AM »

that method is the very first i used to render 3d fractals smiley here's the result for a julia set (reducing to cantor dust):

unfortunately, it is not very interesting: if you consider the formula z^2 - c and the bailout condition being |z| > some constant, it's easy to see that the first iteration is just going to be a circle. the second iteration produces a quartic polynomial, defined by substitution:

z1 = (z0)^2 + c
z2 = (z1)^2 + c


z2 = ((z0)^2 + c)^2 + c = z0^4 + 2c*z0^2 + c^2 + c

which is a very smooth boundary. as you go on iterating the boundary becomes increasingly fractal, however there is a further problem: you can only go up in integer steps. sure, there are formulae for recovering a continuous iteration count, but those are special cased for the above equations, which don't really lead to interesting behaviour. using c = z0 (mandelbrot instead of julia set) only slightly "complexifies" the result.

a somewhat related way to define a volume of fractals is to interpolate (along height) between julia sets with different c-constants; this doesn't suffer from the resolution problems you get with integer iteration counts, and can produce some reasonably interesting resuls. unfortunately i don't have any renders from this method left over.

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« Reply #3 on: October 23, 2007, 11:55:19 AM »

I believe the mandelbrot fractal is merely a single perspective of a 3D structure.   ......
Now what exactly do I mean when I say interpret it as 3D?   ......
When I see recursion getting increasingly smaller, I do not see a 2D image getting
recursively smaller.  I see it as perspective.
Has anyone done this?
anyone care to help?
I do 3D renderings for my job, so if anyone does manage to help me get this
into 3D model format.  I could render it in all sorts of ways.

Greetings, and Welcome to this particular Forum!!    smiley

Have you had any luck yet in progressing with your idea and rendering??  I would be interested in how you are doing and what images you come up with.


Posts: 81

« Reply #4 on: November 12, 2008, 06:39:23 PM »

Hello from far far away. Even after a year a new reply might help :-)
I completely understand what you're saying probably because of inner mechanism in the brain. I have this artistic impression that is just like yours, but I studied it before and came to an answer!

Lets call it the human geometric sense. We are seeing all around common fractals repetition of one or a few shapes. The shapes diminish in size while leaving an impression of spatial dimension, or distance associated with it. If we check the fractal set globally we can see that it would not fit an artistic painting based on modern perspective because of multiple or irrational interpretations in space. In principle algorithm should be able to check for perspective in the space. Without tools we evaluate by hand and a little reason. Some fractal art can be deliberately chosen to resemble a building or landscape stretching afar and meeting infinity or even bending beyond infinity.

Repetition of shape in common fractals is a closer in analogy to mirrors. Mirrors also give impression of distance because the math behind it is same for looking at real distances so eyes have taught brain already (which ever way it goes) about distance estimation. Mandelbrot set has a little more information in it than it can be explained with just mirrors. A very simple fractal, Newton's method fractal for z^3-1 (all programs mention this one) has very little geometric information within a shape that is repeated via shortest path and compressed in the finite area of plane so that it can literally be made just from the mathematical rules for mirrors. Of course that theme is mentioned on Internet.

Here's one lousy little page http://classes.yale.edu/fractals/Labs/OpticalGasketLab/OpticalGasketLab.html
I'm trying to search for Christmas balls fractals...
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