Very simple formula for fractal patterns

[Kali]

@ Kali

do you think that this can be done also with M3D ?

Sorry but there's no unconditional circlefold transform in M3D...

maybe if Luca or Jesse are reading...  


@element90 - thanks for replying for me when I don't have the time to do it myself
nice images btw

[Alef]

Fractint formula lacks colouring revealing patterns. Without additional colour method only z=abs(z)/c+c julias will give some result.

For so popular in FF formula(s) it is suprising, that there is no good Ultra Fractal implementation of this formula. There are just some formula, who needs to load another formula 

[Ryan D]

I'm a latecomer to this, and I'm old fashioned as well - I still have more fun with Fractint than anything else.  So, I've followed along with this and tried to implement a Fractint version of the Kaliset fractal.  The original formula (z = real(x) + imag(y)) can be set up in Fractint's formula editor to come up with images similar to what was posted by Kali, but many of the others don't come up with anything like what was posted.  At least, not the way I've written them up.  Oh well.

In any case, in the process of debugging the Kaliset formula, I noticed how important the bailout amount was.  I created a short animation with the bailout varying logarithmically from 0 to about 170 (modulus squared).  Here it is - the pattern of the lakes is interesting, especially the way they suddenly appear.  This is not an artifact of having the number of iterations insufficient to meet a higher bailout - I calculated the image with 100,000 iterations and the lakes remain exactly the same as they are at the end of this video.

Ryan

<a href="http://vimeo.com/moogaloop.swf?clip_id=49573142&amp;amp;server=vimeo.com&amp;amp;fullscreen=1" target="_blank">http://vimeo.com/moogaloop.swf?clip_id=49573142&amp;amp;server=vimeo.com&amp;amp;fullscreen=1</a>

[Ryan D]

To stay with Kali's original intent on his Kaliset formula (no bailout, using inside colouring), I took a stab at a Fractint version without a bailout.  Fractint has limited choices available to it for inside colouring, and not too many of them were interesting.  For the majority of the Kaliset Julias, using Fractint comes up with an image that is just too busy.  I started off animating a path using relatively high iteration count, and the images were nice, but there were far too many tiny strands and the aliasing was awful.  So, here's a stab with the iterations reduced to 64.  The path starts near the origin and heads off to the outer edge of the Kaliset along the negative X axis at a constant rate, with the Y values located barely on the negative side.  It gets very busy towards the end - the rapid bifurcations come ever quicker, so perhaps an inverse exponential rate of travel would slow things down and show interesting views.

Ryan

<a href="http://vimeo.com/moogaloop.swf?clip_id=49677881&amp;amp;server=vimeo.com&amp;amp;fullscreen=1" target="_blank">http://vimeo.com/moogaloop.swf?clip_id=49677881&amp;amp;server=vimeo.com&amp;amp;fullscreen=1</a>

[Pauldelbrot]

To stay with Kali's original intent on his Kaliset formula (no bailout, using inside colouring), I took a stab at a Fractint version without a bailout.  Fractint has limited choices available to it for inside colouring, and not too many of them were interesting.  For the majority of the Kaliset Julias, using Fractint comes up with an image that is just too busy.

You can use any inside coloring you can imagine, with this trick:

First, in the loop section of your formula, if you're about at the iteration limit (say, one less) calculate the coloring you want and assign it to z as a real number. Something like

Code:

i = 0
z = whatever:
i = i + 1
IF(i == maxiter-1)
  z = calculate-coloring-from-z
ELSE
  z = whatever-fractal-function-of-z
ENDIF,
1 == 1

Then use inside=real coloring.

You can, for example, implement exponential smoothing that way.

[Kali]

Nice video!

I was about to suggest what Paul did, I used that trick for doing exp.smoothing with Fractal Explorer

[Fractal Ken]

Here's a small generalization for one of Kali's formulas from earlier in the discussion: z = abs(z/c₁) + c₂

Example: c₁ = (0.5, 0.5), c₂ = (-0.8, 0.2)


Coloring is by an exponential smoothing variation where the magnitudes of the z's real components are subtracted, rather than the magnitudes of the z's themselves.

Edit: Zoomed versions of this fractal are here.

[Kali]

Wow, excelent image Ken... such a clean render, good coloring. So you used only reals for the exp.smoothing? Nice result, I'll try that.

[Fractal Ken]

Wow, excelent image Ken... such a clean render, good coloring.

Thanks, Kali. The image is sharp because I was able to do 100 iterations.  On most of these non-escaping fractals, that many iterations leads to a noisy looking picture, but not here. The orbits are somehow well-behaved; it's probably related to the tree structures.

So you used only reals for the exp.smoothing? Nice result, I'll try that.

I often use just the real components or just the imaginary components.

[spongman]

these are fantastic!

i made some webgl animations i find them quite mesmerising to watch  

• z = abs(z)/m + c
• z = abs(z /(c1 + c2))/m + c
• z = (c + mouse) / abs(z)^2 + c

(click the 'hide code' button and set the resolution drop-down to '1' if your video card can handle it)

[Kali]

Very nice, spongman! I should get into webgl to do some stuff... this will help me for learning, thanks for sharing!

[spongman]

Very nice, spongman! I should get into webgl to do some stuff... this will help me for learning, thanks for sharing!

thanks! although, be aware that you're only going to get medium precision floats with webgl, so no deep zooming. but it's good for demos.

one question, though. i don't quite understand why sometimes these animations get quite noisy (grainy), for example on http://glsl.heroku.com/e#5654.6 just after the background turns a black/white checkerboard. is that just the nature of the fractal given those parameters, or maybe something wrong with the way i'm doing the coloring?

[kram1032]

Those are some really neat animations

[Hiato]

Just thought I'd write in here and say that I (accidentally) found what turned out to be a rotation-ish of the Kali set. Well, it's actually a variation on the formula (arrived at accidentally) which, if the point is rotated by 90 before you add c then you get the Kali set, so I suppose this is a 'different' fractal:

ax = abs(p.x); ay = abs(p.y); k = x*x+y*y;

p.x = ay/k;
p.y = ax/k;
//if you rotate here by 90 degrees then you get the Kali set
p += c;

Enjoy  

EDIT: For some nice trees, try the Julia version of this formula at -1.08410,-0.21192 and its neighbourhood.

[spongman]

Just thought I'd write in here and say that I (accidentally) found what turned out to be a rotation-ish of the Kali set.

very nice.

i extended this to an arbitrary rotation:

z = abs(z . exp (i.theta)) / |z|^2 + c

here's an animation of the M-set with a continuously varying theta:
http://glsl.heroku.com/e#5742.1


and here's a navigable julia of the same rotating/capsizing ship thing:
http://glsl.heroku.com/e#5758.0