• May 08, 2021, 02:59:17 AM

### Author Topic:  3d Julia sets: True shape  (Read 1588 times)

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#### marcm200

• 3d
• Posts: 959
##### Re: 3d Julia sets: True shape
« Reply #15 on: October 17, 2019, 09:00:34 AM »
The first 3d Julia set showing multiple basins that has been found in a fully automated manner by a random walk through a degree 4 function family (TRIC4CPLXMULT), that is based on a standard complex exponentation (x+y*i)^4 (details below).

{x,y,z}new := { e*x^4+f*x^2*y^2+g*y^4 + h*x+j*z^4 , l*x^3*y+m*x*y^3  + k*y , n*z^3+o*y^3+p*x^3 } + { Cx, Cy, Cz }

Julia sets were automatically computed and judged by the direct RGB sum color filter I wrote about in one of my previous posts. In case there were at least two vastly different colors (judged by R, G, B independently being above 100, so 8 combinations in total), the set was actually saved.

The image below is: e=2.3 f=4.5 g=-0.8 h=-2.1 j=1.1 k=1 l=-0.2 m=0.7 n=-0.2 o=0 p=0, C={ 0.6,0,-0.6 }, at level 10 where interior cells emerged, computed in the 2-cube.

Technical note:
• Parameters e-p take real values and are mostly the coefficients of the complex exponentiation, randomly disturbed (+- <= 5 in 0.1 increments).
• A linear x-old-term was introduced into the new x part (italic, variable h - preliminary experiments showed that this was a good way to add structure) and a dependence on z-old (variable j) so every axis influences every other.
• Similarly I introduced a y-old-term (variable k) into the new y-coordinate.
• As for the z-term I use most of the time a sum of degree-1 or below powers of the old coordinates.
• The { Cx, Cy, Cz }-parameter was split into (4x4x4) values from -1.2 to +1.2 in each direction.
• I did not check whether the whole set is contained within the 2-cube, however interior cells if found are mathematically guaranteed correct no matter the escape radius. If a cell does not have a path to outside the 2-cube, the cell surely has now path outside a larger cube, so it will always be determined as interior if using the correct (larger) escape radius.
• White and gray cells however do not need to be accurate in this case as a cell might be prematurely judged as white in too small an escape radius. For those I need a valid escape radius  - e.g. using the jump method I posted some time ago.

#### marcm200

• 3d
• Posts: 959
##### Re: 3d Julia sets: True shape
« Reply #16 on: October 19, 2019, 11:21:29 AM »
"Periodicity"

I implemented the periodicity procedure from the 2D polynomial TSA (method described here: forum link) to see whether I could also find immediate basins in the tricomplex case.

Below are the two most "complicated" sets I encountered thus far: 3 fix points in the steak image, and a period 2 cycle in the basilica-like (Makin's formula) (left the full basins of attraction, right the immediate basins of the periodic points - colors are not necessarily comparable between left and right image).

Interesting is the similarity between the 3D basilica and the 2D polynomial form z²-1 as they both show a period 2 cycle reaching to the negative axis.

I wonder whether there are longer cycles than period 2 (does anyone know parameters for such a case?)

#### marcm200

• 3d
• Posts: 959
##### Re: 3d Julia sets: True shape
« Reply #17 on: October 21, 2019, 09:24:02 AM »
"Chlorophyllic DNA"

If DNA were built from chlorophyll I think it would look like this (left). Quite interesting from a topological view with all those holes.

Doug Bristor's formula shows a period-3-cycle (upper right). The interior still needs a long way to look like the point-sampled version (lower right, direct RGB sum colored).

The parameter was found by another brute-force walk, this time I computed small point-sampled 3D sliced Msets for a given formula and selected values that were almost on the boundary, but just inside (one layer of also-inside voxels around the chosen one, then at least one escaping).

I hope to get longer cycles by computing larger Msets and then trying to find seed values from deep within bulb-like structures (3D-centers of hyperbolic components?).

#### marcm200

• 3d
• Posts: 959
##### Re: 3d Julia sets: True shape
« Reply #18 on: October 24, 2019, 09:32:08 AM »
"A period-4 cycle"

Moving the seed out along the x-axis while keeping y,z=0,0 I hoped to hit smaller and smaller copies of the main Mset (if they exist, that is, which I don't know) thereby increasing the period of the original basilica (period-2). Since the -1,0,0-version is a very fast computable set with early emerging interior cells, I hoped that this might stay valid further out.

That led to this nice attracting period-4 cycle Julia set using David Makin's formula at c={ -42614129 , 0 , 0 ] * 2^-25 ~ {-1.27,0,0}.

Cycle being attracting
This was deduced by algorithmic properties of the detection method without the use  of the Jacobian.

Every geometrically isolated interior object (a "blob") was tested where its members point to in one iteration. It turned out that for a given blob all cells pointed to only one specific target blob - so no spreading out.

If I start with blob A and move to its target blob, to that one's target blob and so on and come back to A at some point, this sequence of blobs forms a cycle.

Since there are only finitely many detected blobs, starting with a yellow blob B and following its target blob and so on must eventually lead to a blob D that has already been visited before. The blob sequence between the first and the 2nd occurance of D then forms a cycle.

The sole cycle detected is the turquois period-4 one, so every yellow blob finally lands in that cycle.

#### marcm200

• 3d
• Posts: 959
##### Re: 3d Julia sets: True shape
« Reply #19 on: October 28, 2019, 10:14:54 AM »
"Chinese tower"

Many of the interior showing sets I could compute so far look more or less like fields of asteroids of some shape floating around in space (and I can't go to higher refinement levels yet to "close the gaps"). So I was pleasantly surprised that this one has quite a different shape.

A period-2 cycle at level 11 in a degree-5 family (based on complex exponentiation (x+i*y)^5 with the usual additions):

{x,y,z}new := { f*x+h*z2 + x5-10x3*y2+5x3*y2 , e*y+5x4*y-10x2*y3+y5 , z2-y2-x2 } + { Cx, Cy, Cz }

here with e=f=h=0, C={ -0,71875,-0.71875,-1.71875 }.

#### marcm200

• 3d
• Posts: 959
##### Re: 3d Julia sets: True shape
« Reply #20 on: November 05, 2019, 10:30:45 AM »
"Period 8"

A very interesting Julia set using David Makin's formula, not from the shape point of view (it's just another collection of asteroids), but the first with period 8 (turquois/yellow as always, only 7  immediate basins visible, see details below) and with interior cells emerging at level 14.

David Makin's formula has the interesting property, that, when used with {x,y,0} and {cx,cy,0} it reduces to the classic quadratic z²+c case.

Makin:
{x,y,z}new := { x²-y²-z² , 2xy , -2z*(x-y) } + { Cx, Cy, Cz }

when z, Cz=constant 0:
{x,y,0}new := { x²-y² , 2xy , 0 } + { Cx, Cy, 0 }

So I was interested to see how the 2D corresponding Julia set behaves in the TSA.

The 2nd image shows at level 14 also a period-8 cycle (yellow) - and black interior emerges at level 14 here as well. So the 2D properties translate into the higher dimension world and form a 3D object. Maybe I can do that for other interesting shapes as well.

Technical details:
• I implemented a new data structure that works especially well with sparse objects: allocating memory per row only encompassing the gray region, while still preserving direct array access and keeping memory fragmentation at minimum.
• The 8th immediate basin in the 3D version (far left) is not visible in the image here which is a trustworthily 8-fold downscaled version, but it was computationally determined by the TSA. Currently my cube-viewer cannot yet handle cubes of side length 16k (working on it).
• The C value was found exploiting the reduction property looking for small Makin Mset copies and their geometric centers (as described in post #18).

#### marcm200

• 3d
• Posts: 959
##### Re: 3d Julia sets: True shape
« Reply #21 on: November 10, 2019, 03:16:20 PM »
"Cord loop"

This filled-in Julia set shows some winding ropes, a phenomenon which I do not encounter very often. It's very filigree, as compared to the more common asteroid rocks.

I'm using David Makin's formula as a base currently - changing exponents, adding some terms etc, hoping to find new shapes while retaining its fast computation property and relative abundance of attracting periodic points.

{x,y,z}new := { x4-y4-B*z² , A*x+2xy , z4-x²*y² } + { -0.5 , -1 , 0 } (named MakinExp4).
with A=-0.25, B=-1

The image below is at level 13 (tw. downscaled 4-fold), level 14 would need just a bit more memory than I have available currently.

The initial point-sampled small image showed a nice color separation (direct RGB sum), which corresponds to the immediate basin found. Although by far not all sets show such a separation, it is usable as a priority queue for computation order.

#### marcm200

• 3d
• Posts: 959
##### Re: 3d Julia sets: True shape
« Reply #22 on: November 12, 2019, 04:09:34 PM »
A degree-5 formula that shows a nice evolution of interior and its fix point: New regions appearing at every step, and, as also often seen in the 2D case, formerly non-immediate (brown/yellow) regions getting flooded over by the immediate red/cyan basin (e.g. the objects to the left and right of the front eigth symbol). It looks a bit like a lion's cage is being closed with a red lid.

{x,y,z}new := { x5-y5-B*z5 , x4-A*y²*z² , 2x*(z-y) } + { 0.375 , -0.96875 , 0.96875 }
with A=0 and B=0.5

Technical details:
• Levels 9 to 11: Immediate basin = turquois, attraction basin = yellow.
• At first I couldn't compute level 12 as it reached an implementation limit: more than 2^16 gemometrically separated blobs.
• Devising a new cycle detection method (storing only the current orbit and the immediate basins of all cycles), this set was now computable.
• The current method's limit is 2^16 separated Fatou components in an orbit to a cycle. I wonder when I reach that barrier.

#### marcm200

• 3d
• Posts: 959
##### Re: 3d Julia sets: True shape
« Reply #23 on: November 28, 2019, 01:41:42 PM »
In the 2D images the gray region is always visible and gives a good impression of how the overall shape of the Julia set looks like.

I tried the same here in the 3D case, experimenting with transparency of the gray region.

Here's a period-3 cubic triplex Julia set (magenta is the immediate basin, yellow the attraction basin at that level):

{x,y,z}new := { x3-3xy²+0.5z² , 3yx²-y3 , 2xz-2yz } + { -0.265625 , -1.265625 , 0 }

I'm trying to find a setting that - if one has e.g. a magenta-colored interior region deep inside a gray region (a "rock"), that the magenta shines through to the surface. But I'm not quite there yet.

Usually my virtual screen that goes through the cube picks colors up - either a non-transparent full color RGB value (than it stops travelling for that screen pixel). Here I'm adding small gray amounts per encountered gray cell in the journey path (i.e. gray is not fully transparent) until the interior is reached and then overlay the two color values. But I either get a gray smear (added gray is too thin) or a white area (added gray is too thick).

Is there a better way of getting a "shine-through" result?

#### marcm200

• 3d
• Posts: 959
##### Re: 3d Julia sets: True shape
« Reply #24 on: December 23, 2019, 01:55:55 PM »
A period-4 attracting cycle in a rather complicated iteration formula, based on Tyler Smith's idea of two consecutive rotations. Here the first step is a power-3 complex exponentation of x,y_old, followed by a classical power-2 rotation with y_temporary,z_old.

The interior when emerging does not look like rocks but rather multiple-armed stars. The yellow non-immediate basins can grow together and merge with the periodic basins, but the object looks here at level 11 already quite similar to what the point-sampled version shows.

formula CPF3:
{x,y,z}_new := { x3-3xy² + B*z² , 9x4y²-6x²y4 + y6 - z² , 6x²*yz-2y3z } + { Cx, Cy, Cz }

#### marcm200

• 3d
• Posts: 959
##### Re: 3d Julia sets: True shape
« Reply #25 on: January 04, 2020, 06:18:25 PM »
An interesting fix-point (purple): The two "arcs" have a very thin connection, but are geometrically one object. This is actually only the 2nd example I have encountered that has very anfractuous Fatou components.

{x,y,z}_new := { x²-y²-z² , x3 - 3xy² , 4yx3 - 4xy3 } + 2^-25 * { 20132659 , -33554432 , -6710887 }

(It's the first result of mixing different component functions between already used iteration formulae.)

#### marcm200

• 3d
• Posts: 959
##### Re: 3d Julia sets: True shape
« Reply #26 on: January 09, 2020, 09:06:17 AM »
A nice shape, although no interior found till level 12. It seems to have a ever-smaller getting elephant trunk rolling in.

This is one of so far only a few examples where the object actually resembles a 3D twisted version of a quadratic Julia set (although parameters are not related, z²+(0.3-0.42i) ).

Level 13 is currently too time-expensive.

{x,y,z}_new := { 2xy , -2xz , x²-y²-z² } + { 0 , 0.5 , 0 }

#### marcm200

• 3d
• Posts: 959
##### Re: 3d Julia sets: True shape
« Reply #27 on: January 10, 2020, 09:52:34 AM »
"Romulan warbird"

Lately I find some strange shapes, this one looks like a space ship from planet Romulus. It consists almost entirely of the large immediate basin of a fix-point (purple), I wonder whether the small yellow not-yet/non-immediate basins will eventually all merge into it.

{x,y,z}_new := { 4xyz+2zs , 4x²y²+4xys+s²-z² , x²-y²-2xy+z } + { 0.25 , s , 0 } with s=-0.75

The x/y_new terms are based on Tyler Smith's 2-step rotational formula.

#### marcm200

• 3d
• Posts: 959
##### Re: 3d Julia sets: True shape
« Reply #28 on: January 12, 2020, 10:14:03 AM »
This is the most completed set I've calculated thus far, until it exceeds my memory: Level 13, period-2 cycle (bright green) (With my current brute-force combinations I find a lot of fix points and period-2, but so far nothing longer).

The set shows an interesting evolution. with the arc-parts emerging quite early. - otherwise I might have missed them thinking "boring rocks again".

{x,y,z}_new := { x²-y² , 4x²y²+4xys+s²-z² , 2xz-yz } + { -1 , s , 0 } with s=-1

#### marcm200

• 3d
• Posts: 959
##### Re: 3d Julia sets: True shape
« Reply #29 on: January 15, 2020, 11:23:24 AM »
One of the most (rotationally) symmetrical sets I've found so far. I guess the symmetry stems from the y_new term, which, again, is based on Tyler Smith's 2-step formula. Unfortunately it owns again just a fix-point (red).

{x,y,z}_new := { 6x²yz-2z²y² , 4x²y²+4xys+s²-z² , x4-x²y² } + { -0.5, s ,-1 } with s=0.5

It's the first set computed with a new version of juliatsa3d (to be released) which uses a low resolution reverse cell graph as in the classical 2D software. That led to quite a substantial reduction in memory transfer to the cache (and some fewer bounding box calculations). So for images not exhausting the memory completely, that works quite well as a speed-up.

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