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Author Topic: Choosing the squaring formula by location  (Read 13050 times)
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msltoe
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Posts: 187


« on: November 29, 2009, 04:46:57 AM »

Hi All,

In the last two weeks, I've been spending way too much of my free and (not so free) time trying to find the holy grail of 3-d mandelbrot sets. Up to today, I really hadn't found anything original to share. An idea I came up with is to choose the squaring formula based on location of the (x,y,z) coordinate. This includes changing that choice for each iteration as the triplet is being squared (cubed, e.g.) and translated.
Here's two results I've gotten so far.

One I call Mandelettuce - this uses the cubic White-Nylander form, but only phi (asin(z/r)) is being tripled, not theta. The distance criteria is distance to the axis. The closest axis defines which one to define phi.

The second one uses the standard quaternion form which normally just produces axial symmetric 2-D mandelbrot. Which axis X, Y, Z will be the dominant (i.e., X-axis: x-x-y*y-z*z) is determined by which one is the shortest distance to the axes again.

It seems like this new method is "cheating" as it may be similar to a symmetry operator that might be used in an IFS. But, I guess you could say this a hybrid method.

One of the issues to resolve is how best to choose the proper axis and how to position the axis into geometric shapes (e.g. tetrahedral symmetry)

-mike


* mand.lettuce.v2.small.png (233.84 KB, 500x500 - viewed 911 times.)

* mand.3-attractors.small.gif (109.91 KB, 600x600 - viewed 926 times.)
« Last Edit: November 29, 2009, 04:56:11 AM by msltoe » Logged
TedWalther
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« Reply #1 on: November 29, 2009, 08:25:57 AM »

Wow, Mike, we're thinking along similar lines.  Beautiful images.

Here is one more thought I had today, while considering those possibilities.

Try viewing it like this.  In 2d, Z is being rotated away from [1,0,0].  I think of it as kicking a ball.

To generalize that concept to 3d, what about this?

Take your x and z components.  "kick the ball" separately in relation to the x and z axis, then add the two resulting vectors together.  The magnitude of the two vectors would be in the same proportion as that between x and z.  So when z=0 in C, you get the standard mandelbrot in the xy plane.  Although to be honest, I suspect you'd also get the standard mandelbrot in the zy plane.  Would look like a twin hydra or something.

Sorry I'm so busy at work, I didn't have time to turn that into a formula.

Ted
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msltoe
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« Reply #2 on: December 05, 2009, 02:50:00 AM »

   I found another species of Mandelettuce.

  I'm using "triplex algebra" with two possible attractors:
  The standard "z^2" if y*y>z*z ,
  otherwise, perform "z^2" operation with flipped y and z for theta and phi formulas.


* mand.lettuce2.gif (218.95 KB, 697x689 - viewed 1042 times.)
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kram1032
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Posts: 1863


« Reply #3 on: December 05, 2009, 03:17:52 AM »

wow cheesy
Great idea!
How does that whole thing look zoomed? smiley
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TedWalther
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« Reply #4 on: December 05, 2009, 09:06:41 PM »

Very pretty pic.  That just begs for some of twinbees renders with an iteration of 100.
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msltoe
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« Reply #5 on: December 05, 2009, 11:38:25 PM »

When I zoom in, there's some nice features but it seems like it's far from an ideal fractal as there is some chaos in 1-D but not 2-D.
I dialed up the iterations to 50.


* mand.2-attract.closeup.gif (230.78 KB, 704x578 - viewed 853 times.)
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msltoe
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« Reply #6 on: December 06, 2009, 03:34:51 AM »

I just wanted to show what happens if you simply apply a flip of y and z coordinates if the absolute value of one is bigger than the other using the standard z^2 Mandelbulb. It's nothing special, but maybe it'll help us visualize better the solution.



* mandelbulb_2_yz.png (154.99 KB, 452x452 - viewed 932 times.)
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msltoe
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« Reply #7 on: January 08, 2010, 02:43:30 AM »

Here's another variant.
Six possible attractors chosen on distance to axis and whether the coordinate of that axis is positive or negative. Attractor is quaternion Mandelbrot. It seems to have infinite detail, although here I cap at 50 iterations. The closeups are detailed, almost noisy, if anyone's interested.

There's actually a term for what the basic shapes are in my multi-attractors: spherical polyhedra.

-mike


* mand.ball_of_life.v3c.png (250.81 KB, 500x484 - viewed 1132 times.)
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msltoe
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« Reply #8 on: January 18, 2010, 04:24:43 AM »

Similar variant as above but with ambient occlusion and max iterations = 10.
The problem with this species is that the features seem to get very thin.


* mand.6-way.quat2.jpg (249.79 KB, 547x547 - viewed 842 times.)
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Calcyman
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« Reply #9 on: January 18, 2010, 08:22:44 PM »

Quote
There's actually a term for what the basic shapes are in my multi-attractors: spherical polyhedra.

Wow -- can you arbitrarily select the coordinates of the attractors? If so, please try out the following twelve coordinates. (  phi = 1.6180339887, or (1+sqrt(5))/2  )

(phi,0,1)
(phi,0,-1)
(-phi,0,1)
(-phi,0,-1)

(1,phi,0)
(-1,phi,0)
(1,-phi,0)
(-1,-phi,0)

(0,1,phi)
(0,-1,phi)
(0,1,-phi)
(0,-1,-phi)



This generates a shape with icosahedral/dodecahedral symmetry. It has an automorphism group of order 120 (60 rotations * 2 reflections). By comparison, those cubic fractals have an automorphism group of order 48 (24 rotations * 2 rotations). A dodecahedral fractal would be one of the most elegant and organic geometrical objects in three dimensions, resembling pollen grains, radiolaria or Herpes viruses. The golden ratio is also designed to look aesthetically pleasing, so you might win an art competition by rendering it! (The human body, Parthenon and sunflower phyllotaxis all incorporate the golden ratio, to name a few.)
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msltoe
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« Reply #10 on: January 18, 2010, 08:52:14 PM »

Calcyman - the reason I've only done cubic so far, is because it's simplest to do. To do tetrahedral or the dodecahedral as you suggested I will need to rotate (x^2-y^2-z^2,2*x*y,2*y*z) to the various points, and I will need to calculate distances to each one on the points to determine which formula to use.
The thing that puzzles me is why I don't get bulbs at each attractor. It could have to do with the position and size of each attractor.
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msltoe
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« Reply #11 on: January 18, 2010, 11:56:20 PM »

By request, the icosahedral attractor. I found a simple shortcut instead of computing the rotated quaternion attractors. The idea is that z^2 can also be thought of as a reflection of the pole [e.g.,(1,0,0)] to (x,y,z) with lots of sphere normalizations.


* mand.ico.jpg (361.5 KB, 800x800 - viewed 844 times.)
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kram1032
Fractal Senior
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Posts: 1863


« Reply #12 on: January 19, 2010, 12:21:39 AM »

nice cheesy
Like a lot of flowers!
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Calcyman
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Posts: 38


« Reply #13 on: January 19, 2010, 08:33:12 PM »

Impressive! Much less 'artificial' than the cubic counterpart -- it's a shame that polyhedral symmetry and Mandelbulb-esque detail are almost mutually exclusive.

A quaternionic fractal could have the even more impressive icosian symmetry, which has 14400 automorphisms. The icosians are a group of 120 vectors, which are closed under multiplication. In the context of fractals, this means that squaring any icosian will produce another icosian. In a geometric context, the icosians are the vertices of the 600-cell, or the centres of the faces of the 120-cell (dodecaplex).
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msltoe
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« Reply #14 on: January 24, 2010, 03:16:02 AM »

Using multiple "attractors," I've found several variations.
This one is interesting:


* mand.ico_crop.small.png (483.5 KB, 518x510 - viewed 702 times.)
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