• May 08, 2021, 04:07:10 AM

### Author Topic:  3d Julia sets: True shape  (Read 1591 times)

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#### marcm200

• 3d
• Posts: 959
##### Re: 3d Julia sets: True shape
« Reply #30 on: January 18, 2020, 12:44:23 PM »
"Pretender"

A very nice shape of this level 13 Julia set - no interior found, and it's not clear if any will ever appear. But it looks a bit like small Enterprise star ships next to a giant space station.

On the right of the image below are three smaller greenish point-sampled pictures of the same set: non-escaping points, escape radius 100; 100 max it (upper), 125 (middle), 150 (lower).

The object "loses" interior points, but to what extent? Empty interior? Cantor dust? Some very small volume-bearing mini-objects missed by point-sampling?

{x,y,z}_new := { 6x²yz-2z²y² , 4x²y²+4x²ys+s²-z² , x4-y4-z² } + { 0.25 , s , -0.25 } with s=-1

With interior present the gray area at some point reflects roughly the shape of the underlying non-escpaing objects. So it is interesting to see how the gray disappears when presumably no interior is there.

#### marcm200

• 3d
• Posts: 959
##### Re: 3d Julia sets: True shape
« Reply #31 on: January 20, 2020, 11:42:23 AM »
A period-1 cycle in a very strange shape. I wonder whether there is branching-out at the tips (or at the thick part) of the threads here.

I hope I can transfer in the future the periodic-point detection method from 2D to this case to see whether here actually a fix-point exists or if it is more like a strange attractor's set of unstable but contained orbits (by construction, every turquois voxel iterates always in the turquois connected object).

{x,y,z}_new := { x4-y4-z² , 2xz-yz , y+3x²y-y3 } + { 0.5, -0.5 , -0.25 }

The formula is quite simple and very fast to compute (double precision was still sufficient in level 13).

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