Choosing the squaring formula by location

[TedWalther]

Mike, what would a four-fold symmetry look like instead of the current 16-symmetry?

It looks to me like you've found it.  You've really found it.

Ted

[msltoe]

Ted,
 
  The "golden mean" seeded Julia, for sure, is an important breakthrough. The mandelbrot-like objects you see though are just good illusions generated by lots of symmetry. The four-fold M-sets began with one the first posts on this thread. The original four-fold J-sets are about ten posts up.
 I can't believe I've literally spent half my little daughter's life (whose now 8 months old) on this problem.   At least, we've found some new shapes/ideas along the way...
 
-mike

[fractalrebel]

Hi Msltoe,

I think you have a cool formula which makes a giant step towards the goal of a 3D Mandelbrot and/or Julia. I have put some experimental code into my UF Raytracer and can duplicate your Julia that has a seed of (-0.8,0,0). I read earlier in the thread that there were problems with the y z plane, or at least something involving the y and z variables. If I take you Julia and use a seed of (-0.8,0.1,0) I start to see fissures and cracks in the object. Increasing the y value even more will create a real mess. I also found something very interesting. If after decrementing psi and psi2, psi is subtracted from psi2, the resulting image is a quaternion! This will happen for virtually any symmetry value. A quaternion can also be obtained by increasing the symmetry to a large value, say 512. Here are some illustrative images:

[msltoe]

Fractalrebel: Thanks for implementing and evaluating the formula. I've seen the fissures, too. I guess that means that the formula is not continuous for b or c != 0. Perhaps, the true formula is? The psi/psi2 thing you see sounds like the infinite symmetry version, which is *too* roundy.

 What puzzles me is why the Julia b & c =0 sets look so good yet the M-set version doesn't. It could be that the formula is only asymptotically correct for b & c = 0. The fissures might be removed by making the symmetric elements continuous at their intersection points. Think back to the simplest "z*z<y*y" version: The two parts visually merge at the two planes z*z=y*y, but the formulas do not. The quaternion is one such solution but it's too smooth. Some of the quaternion variants might also satisfy this condition but probably would look identical w/ and w/o the symmetry operator.

[fractalrebel]

I agree that making the elements continuous at their intersection points will probably solve the problem. Then we will have the true 3D Mandelbrot/Julia. Do you have any thoughts on how to do that?

[reesej2]

I'm not sure if continuity alone will do it, actually... this feels like a "cusp" problem. I think it needs to be not only continuous, but with a continuous derivative, too. That way we won't have any corners in the iterating function, which will probably smooth out the result, too.

[fractalrebel]

Here is another Julia, this one with a seed of (0.375,0,0).

[fractalrebel]

Reesej2,

You are probably right that continuity for both the function and the derivative are necessary.

[msltoe]

fractalrebel: Nice zeolite! As long as b=c=0, things look good.

reesej2: I, too, agree that continuous derivatives are important. In fact, I think the joining at y*y=z*z is continuous. I tried using an interpolation function like 1/(1+exp(-beta*(alpha-0.5))) where alpha = z*z/(y*y+z*z) and it always seems to have some artifacts. I've also looked at symmetrizing the y and z formulas, but so far nothing works. This is a tough problem, but well worth it.

-mike

[twinbee]

msltoe, that big zoomed in pic looks ace. Until I get the formula up and running myself, is there any chance of that (or even more zoomed in), but with less iterations? I'm curious as to what the surface would look like. Same or bigger resolution would be great too.

[msltoe]

Similar closeup of the Julia x8 with 15 iteration limit

My renderer currently gives a "muddy" look.

[twinbee]

Interesting, and thanks.

Here's an idea which might be worth exploring; trying to create a similar Julia, but in 2D, and then try to see the differences between that and the 2D Mandelbrot, and then try to use that information to find the difference between this 3D Julia and a potential 3D brot.

[msltoe]

The 2-D Julia looks a lot like the 3-D one. The best 3-D Mandelbrot version I can find is by alternating signs of z for even/odd iterations using the same formula.

The thing that's so special about this 3-D Julia is that the surface is always positively indented. This occurs because I've symmetrized out the poles and because only one coordinate, x, is being translated at each iteration. The M-set at low iterations gets "teared" which tells me that it is isn't right. Something about the way the y and z coordinates interact allows this to happen. It's not obvious how or whether this problem can be resolved. But, considering how far we've come, anything's possible.

[kram1032]

I think, somehow this still wasn't really tried but it was proposed a few times...
how would a default Mbulb look like, which has *8 for the second angle, rather than *2?

r² phi*2 theta*8...
or alternatively
r^8 phi*2 theta*8

[cKleinhuis]

ehrm, stumbeld on this thread because of a recent gallery posting, doesnt this qualify as the holy grail ?
why havent we accepted it ?!
http://www.fractalforums.com/theory/choosing-the-squaring-formula-by-location/msg14371/#msg14371