News: Did you know ? you can use LaTex inside Postings on fractalforums.com!

## The All New FractalForums is now in Public Beta Testing! Visit FractalForums.org and check it out!

 Pages: [1] 2   Go Down
 Author Topic: Continuous Conformal Mandelbrots  (Read 17017 times) Description: 0 Members and 1 Guest are viewing this topic.
Fractal Molossus

Posts: 703

 « 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 1300 times.)  wholeCubeHighPersp.jpg (143.65 KB, 794x679 - viewed 1244 times.)  cubeScale3.jpg (155.56 KB, 798x660 - viewed 1281 times.) « Last Edit: January 31, 2010, 05:35:23 AM by Tglad, Reason: typo » Logged
Fractal Molossus

Posts: 703

 « Reply #1 on: January 31, 2010, 05:31:25 AM »

Close ups. Square, triangle and pentagonal surfaces.
 digitalSurfaceSmall.jpg (147.81 KB, 508x600 - viewed 1263 times.)  trianglesSmall.jpg (186.07 KB, 689x526 - viewed 1261 times.)  fiveFoldSymettrySmall.jpg (145.28 KB, 728x393 - viewed 1208 times.) Logged
kram1032
Fractal Senior

Posts: 1863

 « Reply #2 on: January 31, 2010, 01:14:49 PM »

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

Are those all on different sides of the cube or is it possible to find a border with some more chaotic behaviour?
 Logged
Fractal Molossus

Posts: 703

 « 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 1242 times.)  largeValleyS.jpg (188.81 KB, 647x612 - viewed 1298 times.) Logged
Fractal Molossus

Posts: 703

 « 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
 castleWalls.jpg (255.03 KB, 797x696 - viewed 1288 times.)  brickworkS.jpg (224.53 KB, 767x675 - viewed 1200 times.) Logged
kram1032
Fractal Senior

Posts: 1863

 « 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
 Logged
fractalrebel
Fractal Lover

Posts: 211

 « Reply #6 on: February 20, 2010, 09:09:18 PM »

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 1225 times.) Logged

Fractal Molossus

Posts: 703

 « 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 1210 times.)  cubeScale4.jpg (150.92 KB, 800x647 - viewed 1271 times.)  cubeScale6.jpg (137.67 KB, 800x650 - viewed 1295 times.) Logged
fractalrebel
Fractal Lover

Posts: 211

 « 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.
 Logged

fractalrebel
Fractal Lover

Posts: 211

 « 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 1271 times.) Logged

bib
Global Moderator
Fractal Senior

Posts: 2070

At the borders...

 « 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?
 Logged

Between order and disorder reigns a delicious moment. (Paul Valéry)
fractalrebel
Fractal Lover

Posts: 211

 « Reply #11 on: February 21, 2010, 09:00:54 PM »

Hi bib,

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

ConformalLattice_Julia {
::hmMZ2gn2tzR2OOuxx3Ng/HE07j2++wB8FvjthNWH4YvB5tMgjE1IODFpMFnr9rPV3VTN6gjK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}
 Logged

bib
Global Moderator
Fractal Senior

Posts: 2070

At the borders...

 « 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

Between order and disorder reigns a delicious moment. (Paul Valéry)
Fractal Molossus

Posts: 703

 « 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 1174 times.) Logged
fractalrebel
Fractal Lover

Posts: 211

 « 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 1182 times.) Logged

 Pages: [1] 2   Go Down