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Author Topic: Continuous Conformal Mandelbrots  (Read 9682 times)
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Tglad
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« on: January 31, 2010, 05:07:42 AM »

Inspired by msltoe's fractals at "http://www.fractalforums.com/3d-fractal-generation/sierpinski-like-fractals-using-an-iterative-function/" I've started this topic for displaying fractals that are continuous (no tearing), conformal/anti-conformal (small shapes don't distort, but can reflect), and lastly apply the +C so they are a map of a suite of Julia sets.
This means an iteration can only consist of reflections, translations, rotations, scale and inverse (multiply radius by 1/radius^2).

The first of this kind uses the following iteration code:
Quote
 int count = 0
            repeat
              if (point.x > 1)
                point.x = 2 - point.x
              elseif (point.x < -1)
                point.x = -2 - point.x
              endif
              if (point.y > 1)
                point.y = 2 - point.y
              elseif (point.y < -1)
                point.y = -2 - point.y
              endif
              if (point.z > 1)
                point.z = 2 - point.z
              elseif (point.z < -1)
                point.z = -2 - point.z
              endif
              Vector diagonal(1,1,1)
              diagonal.Normalise()
              float dot = point.Dot(diagonal)
              point = point - 2*dot*diagonal
            until (count = count+1)==2
              
            float k = 3
            point = point * k + C
Which creates this fractal-


* fullCube.jpg (201.17 KB, 783x772 - viewed 826 times.)

* wholeCubeHighPersp.jpg (143.65 KB, 794x679 - viewed 775 times.)

* cubeScale3.jpg (155.56 KB, 798x660 - viewed 805 times.)
« Last Edit: January 31, 2010, 05:35:23 AM by Tglad, Reason: typo » Logged
Tglad
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« Reply #1 on: January 31, 2010, 05:31:25 AM »

Close ups. Square, triangle and pentagonal surfaces.


* digitalSurfaceSmall.jpg (147.81 KB, 508x600 - viewed 786 times.)

* trianglesSmall.jpg (186.07 KB, 689x526 - viewed 777 times.)

* fiveFoldSymettrySmall.jpg (145.28 KB, 728x393 - viewed 741 times.)
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kram1032
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« Reply #2 on: January 31, 2010, 01:14:49 PM »

So, it's actually pretty interesting and worth an exploration smiley

Are those all on different sides of the cube or is it possible to find a border with some more chaotic behaviour?
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Tglad
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« Reply #3 on: February 01, 2010, 01:25:43 AM »

The changes in surface are on the same side of the cube, here's some pictures of such changes.


* changeInSurfaceS.jpg (228.93 KB, 737x617 - viewed 743 times.)

* largeValleyS.jpg (188.81 KB, 647x612 - viewed 790 times.)
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Tglad
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« Reply #4 on: February 01, 2010, 01:35:54 AM »

Also, a couple more shots.
The next fractal in this family is going to get its own post, I think you'll all like it  cheesy


* castleWalls.jpg (255.03 KB, 797x696 - viewed 785 times.)

* brickworkS.jpg (224.53 KB, 767x675 - viewed 722 times.)
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kram1032
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« Reply #5 on: February 01, 2010, 04:31:20 PM »

I think, a lot of details get lost of aliasing, so, due to noise which actually *is* fine detail, but too fine to actually recognize it as such. Poorly it harms the kinds of detail which are just above the noise-level...

It would be really nice to see them Antialiased smiley
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fractalrebel
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« Reply #6 on: February 20, 2010, 09:09:18 PM »

Tglad,

I hope you have more great formulas coming. After successfully creating an Amazing Fractal plugin for my 3DFractalRaytrace(UF5) formula, I have done the same for Conformal Mandelbrot, and a variant of Timeroot's approach worked quite well for calculating the derivative. Hopefully the UF Database will be back up soon, so I can upload the upgrades to my UF library file. Here is an image:



* StoneCarvings.jpg (165.11 KB, 640x512 - viewed 746 times.)
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Tglad
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« Reply #7 on: February 21, 2010, 01:29:24 AM »

Excellent. This is actually a pretty interesting fractal, especially how it shows these different shape tiles.. I wouldn't be surprised if there was a hexagonal part somewhere.
You can see it get more undulating as the scale increases, so scale 4 or 6 might be interesting to take a closer look at too.

I tried a conformal mandelbrot using just 4 or 8 sphere inversions plus rotations etc but just got dusty sparse sets that weren't too nice.
Making them is straightforward, you just do some reflections (e.g. if (y>r)y = r + r-y), scaling (point = point * k), rotations (e.g. temp = x; x = z; z = -temp), sphere inversions (as in the mandelbox), and lastly translations.
But to show nice patterns the formula should probably be quite symmetrical and simple. I don't think you can fold solids other than cuboids without breaking up the symmetry.


* cubeScale2.jpg (130.71 KB, 800x648 - viewed 765 times.)

* cubeScale4.jpg (150.92 KB, 800x647 - viewed 790 times.)

* cubeScale6.jpg (137.67 KB, 800x650 - viewed 791 times.)
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fractalrebel
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« Reply #8 on: February 21, 2010, 05:07:29 AM »

I tried a little variant. I generalized the normalized diagonal vector (1,1,1) to be any 3D normalized vector, and I get some really interesting results. Also get some interesting stuff with the Julia variant. Bedtime now. I will post some images tomorrow.
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fractalrebel
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« Reply #9 on: February 21, 2010, 07:53:50 PM »

The Ultrafractal database server is up and running again. I have updated my formulas, so Conformal Mandelbrot and Conformal Julia are now available for use with 3D Fractal Raytrace (UF5). As I mentioned in my last message, the diagonal vector has been updated to take any 3D vector. The formula automatically normalizes the vector so you don't have to worry about that. Here is a Conformual Julia image. The Julia seed is (1,2,-3) and the 3D vector is (1.5,2.5,-3). Hexagonal-like structures are seen.


* ConformalLattice_Julia.jpg (198.4 KB, 700x700 - viewed 811 times.)
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bib
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« Reply #10 on: February 21, 2010, 08:09:05 PM »

Here is a Conformual Julia image. The Julia seed is (1,2,-3) and the 3D vector is (1.5,2.5,-3). Hexagonal-like structures are seen.
Nice! Could you please post the upr? I would be very intersted in exploring this one. Do you think some octogonal symmetries could be done?
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fractalrebel
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« Reply #11 on: February 21, 2010, 09:00:54 PM »

Hi bib,

Here is the upr. Make sure you update reb.ulb.

ConformalLattice_Julia {
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}
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bib
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« Reply #12 on: February 21, 2010, 09:10:22 PM »

Thanks fractalrebel!
I've always used mmf and just trying reb ;-)
First impression :
- less parameters to control position and precision so maybe less powerful but easier to understand
- calculation time seems better overall (at least on Tglad's amazing fractal), but the big drawback is that the first preview appears after 5 to 10 seconds of calculation whereas in mmf it's immediate
- the color presets are nice
I will have to test this more deeply, but again, the time to display the first preview might be a big barrier for anyone used to play a lot with parameters and especially for animations previews.
« Last Edit: February 21, 2010, 09:37:10 PM by bib » Logged

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Tglad
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« Reply #13 on: February 21, 2010, 10:49:22 PM »

"Do you think some octogonal symmetries could be done?"
Here's an older pic using the 1,1,1 rotation axis, it shows 4 octagonal tiles down the middle.


* floorTiles(1).jpg (248.84 KB, 795x764 - viewed 721 times.)
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fractalrebel
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« Reply #14 on: February 22, 2010, 03:08:54 AM »

A Stone Face emerges in this Conformal Julia. There are also numerous almost circular structures. The Julia seed is (1,2,-3) and the 3D Vector is (3,2,-3)


* StoneFace.jpg (185.32 KB, 640x640 - viewed 739 times.)
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