An old formula revised

[M Benesi]

  About 40 minutes into the following documentary (PBS Nova- Fractals -  Hunting the Hidden Dimension), they do a little real world measurement of fractal like behavior of trees in a forest. 

  Probably what sparked my intuition....

<a href="http://www.youtube.com/v/LemPnZn54Kw&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/LemPnZn54Kw&rel=1&fs=1&hd=1</a>

[cKleinhuis]

i knew this video already, thank you for reminding me of that, more examples of usage of fractal dimension analysis for real lifers

[Alef]

As far as I understand, fractal antenna works not becouse it is fractal, but becouse in single wire it contains many different lenghts and effectively fills the space. Each wavelenght requires different lenght of antenna. So antennas can have not only fractal pattern.  

Probably trigonometry is so popular becouse people understand it. Throught for a practical aplication I would prefare Pythagoras. There should be fractal honoring Pythagoras.

This one rendered pretty fast. Alsou increasing bailout value to some 1.0E20 made it smoother like lowering maximal iterations.



Chaos Pro parameter:

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[M Benesi]

Nice job getting it to render smoothly. 

  Keep in mind, due to the nature of escape time fractals:

  less iterations= less time to calculate
  higher bailout= more time to calculate

[M Benesi]

i knew this video already, thank you for reminding me of that, more examples of usage of fractal dimension analysis for real lifers

  I can't remember where we post videos of fractal stuff (not in movies, obviously).  ??   derp derp  ??

  Update:  I've found that this formula allows adding in old x,y, and z values to the end of certain parts, and simply the x old x value after the Mandy part.  You can combine adding/subtracting the values in different ways (including using the absolute value of the old components).

  Here are a couple images using z^2 - z/2  for the Mag portion- it really works quite nice.  Second one I add in a portion of the yz pixel component- I like the way it works with this option.  A bit sharper.
click to ENLARGE

[M Benesi]

  As you zoom into one of the central "Chinese paper fans", the number of outer fan blades increases:



  Here's a random image, subtracting the beginning of the mag part from the end of it on the left.  On the right, it's adding the starting x component (before the mandy part) to the ending x component (after the mandy part).  The asymmetry is because I added in the YZ pixel portions during the Mandy part...  Make it have lots of little interesting details.
 

[jehovajah]

http://www.fractalforums.com/mandelbulb-renderings/mandelbulb-renderings/msg9603/#msg9603

Kram1032 produced this many moons ago. Any chance of reworking this with your magic? First you might need to see if Chaos pro can do the sphere inversions.

[M Benesi]

  I don't know how to do it.  An IFS, or maybe Indra's pearls 3d???  I'd think, from what Kram1032 said, it's some type of IFS.  If David Makin notices your post, I'd think he could tell you or at least point you in the right direction (I recall him talking about various IFS topics). 

  We'll see... someone will figure it out. 

  Jos Leys is the one who created the image.  He's got a pretty awesome website with lots of nice images:

http://www.josleys.com/index.php

Maybe... I don't think it is Indra's pearls.. but here is a link for you to peruse at leisure:

http://klein.math.okstate.edu/IndrasPearls/

[kram1032]

Uhm, jehovajah, I didn't produce that.
It's from somewhere on this awesome fractal site:

I put a collection of my Ultrafractal  efforts in 3D fractals (Mandelbulb and others).
See http://www.josleys.com

[David Makin]

  I don't know how to do it.  An IFS, or maybe Indra's pearls 3d???  I'd think, from what Kram1032 said, it's some type of IFS.  If David Makin notices your post, I'd think he could tell you or at least point you in the right direction (I recall him talking about various IFS topics). 
<snip>

I haven't done anything like these myself, at least not in 3D - but I think there may be something along these lines in reb.ulb for UF as I seem to recall Ron Barnett doing some sphere inversion formulas a bit like Jos Leys' ones - unfortunately I believe Jos is contract bound to keep his Indra's Pearls formulas private - something to do with a book he did ?

However I was thinking about trying a method I used in 2D but using quaternions or hyoercomplex/bicomplex or triplex etc. - specifically a full or restricted escape-time IFS where one or more of the transforms is a julia set and others are just affine - if the julia is the equivalent of z^2+zero then obviously the result will be made from spheres (or equivalent if using bicomplex).

[jehovajah]

Respect JosLeys!

Maybe JosLeys could work with you on this Matt ?

Twinbee has seen the spheres at the extreme limit of the Mandelbulb power8, and that in itself is satisfying confirmation of the correlation of the process for both 2d and 3d. The extra degrees of freedom in the 3d case are what have to be sifted through to find the holy grail, but some way of clearing the mist of spheres has to be in place, and yours is the only one I am currently aware of.

It is not just a question of surface determination, or pixel  connectedness, a subject we discussed briefly many months ago now. I think that we have good grounds to expect small spherical regions to be obscuring the larger connected mass of spheres even close to the main body. This is why I find your results interesting.

[M Benesi]

unfortunately I believe Jos is contract bound to keep his Indra's Pearls formulas private - something to do with a book he did ?

  I don't know- the link to the "Indra's Pearls" site above is from Jos Leys' website, and it has links to a book on Indra's Pearls, in addition to a LOT of information about them.  There is even a course that I am checking the notes of at the moment:
http://klein.math.okstate.edu/IndrasPearls/Wonders/

 The course is from the main Indra's Pearls site:

http://klein.math.okstate.edu/IndrasPearls/

  The link for which I found on Jos Leys site here:

http://www.josleys.com/links.php

However I was thinking about trying a method I used in 2D but using quaternions or hyoercomplex/bicomplex or triplex etc. - specifically a full or restricted escape-time IFS where one or more of the transforms is a julia set and others are just affine - if the julia is the equivalent of z^2+zero then obviously the result will be made from spheres (or equivalent if using bicomplex).

  Definitely sounds like it would work.

  I think I remember someone doing a 3d Julia of spheres a while back as well.  Can't recall though... perhaps a "3d julia spheres" search is in order. 

[M Benesi]

Respect JosLeys!

Maybe JosLeys could work with you on this Matt ?

 He is probably busy doing important stuff.  

It is not just a question of surface determination, or pixel  connectedness, a subject we discussed briefly many months ago now. I think that we have good grounds to expect small spherical regions to be obscuring the larger connected mass of spheres even close to the main body. This is why I find your results interesting.

 I do like the spheres.  

  The following is an interesting idea.. but it will cut out many sections of the secondary fractal.  It might do it in an interesting way.
 I think our best bet (to the grail) is taking a rather complicated route.  We'd need to calculate each yz plane independently, using the radius of a 2d Mandelbrot at that point.

  By yz plane, I mean each point on the x- axis is an individual yz-plane.  

  at each x,  we would calculate the first  R that was within the 2d Mandelbrot.      z^n = (x + i R)^n

  Start at   R=2, and subtract from R incrementally until the point (x,R) is within the set.  


  Divide our yz plane components by R and calculate the whole plane (y + iz)^n.  It will be one flat 2d Mandelbrot.  Keep all points within the set- we will remove interior points later.

  Starting from that R, subtract from R incrementally until the point (x,R) is outside of the set.

  Divide our yz plane component by the new R and calculate the whole plane.  All of the points within the 2d Mandelbrot we calculate with the new R are OUTSIDE the 3d set.  

  Starting from that R, subtract from R incrementally until the point (x,R) is inside the set.

  Divide out yz plane component by the new R and calculate the whole plane.  Put all these points IN the set.

  Starting from that R, subtract from R incrementally until the point (x,R) is outside the set.

  Make all points within the new yz/R fractal outside the set.  Next R will be inside the set (you see the pattern?).

  Keep on repeating until R=0.  At this point, move on to the next yz-plane (the next x point, determined by whatever your x resolution is...).  Now repeat from the top.  


   This will work because dividing the yz components by a small number (say at x=-1.8, you will have a small R) will make them a lot larger.  This means you get a much smaller fractal for that particular plane (yz plane at x=-1..  You will get the same exact fractal (y+ iz)^n, it will just be scaled down.  

  If you want the claimed "Grail" look, you'd need to calculate the YZ 2d Mandelbrot as a z^5.  I always thought there would be an additional asymmetry due to the additional axis (the +/- x asymmetry, and an additional  +/- extra axis asymmetry), which would compound the asymmetry of the whole beast.

  I suppose someone has to implement this somewhere... I wouldn't mind seeing it, but it's  rather hard in ChaosPro.

  I think I'll have to repeat yz plane R calculations for EVERY pixel.  It would be MUCH easier if I could do the whole plane at a time- which I can't.  

[Alef]

   I think our best bet (to the grail) is taking a rather complicated route.  We'd need to calculate each yz plane independently, using the radius of a 2d Mandelbrot at that point.

  By yz plane, I mean each point on the x- axis is an individual yz-plane.  

  at each x,  we would calculate the first  R that was within the 2d Mandelbrot.      z^n = (x + i R)^n

  Start at   R=2, and subtract from R incrementally until the point (x,R) is within the set.  


  Divide our yz plane components by R and calculate the whole plane (y + iz)^n.  It will be one flat 2d Mandelbrot.  Keep all points within the set- we will remove interior points later.

  Starting from that R, subtract from R incrementally until the point (x,R) is outside of the set.

  Divide our yz plane component by the new R and calculate the whole plane.  All of the points within the 2d Mandelbrot we calculate with the new R are OUTSIDE the 3d set.  

  Starting from that R, subtract from R incrementally until the point (x,R) is inside the set.

  Divide out yz plane component by the new R and calculate the whole plane.  Put all these points IN the set.

  Starting from that R, subtract from R incrementally until the point (x,R) is outside the set.

  Make all points within the new yz/R fractal outside the set.  Next R will be inside the set (you see the pattern?).

  Keep on repeating until R=0.  At this point, move on to the next yz-plane (the next x point, determined by whatever your x resolution is...).  Now repeat from the top.  


   This will work because dividing the yz components by a small number (say at x=-1.8, you will have a small R) will make them a lot larger.  This means you get a much smaller fractal for that particular plane (yz plane at x=-1..  You will get the same exact fractal (y+ iz)^n, it will just be scaled down.  

  If you want the claimed "Grail" look, you'd need to calculate the YZ 2d Mandelbrot as a z^5.  I always thought there would be an additional asymmetry due to the additional axis (the +/- x asymmetry, and an additional  +/- extra axis asymmetry), which would compound the asymmetry of the whole beast.

  I suppose someone has to implement this somewhere... I wouldn't mind seeing it, but it's  rather hard in ChaosPro.

  I think I'll have to repeat yz plane R calculations for EVERY pixel.  It would be MUCH easier if I could do the whole plane at a time- which I can't.  [/size]

IMHO if something is very hard for Chaos Pro compiler it would be almoust impossible for more restricted Mandelbulb3D or much harder language of Fragmentarium.

But this method sounds as the most realistic way to get to 3D Mandelbrot. Maybe this could be simpified using Distance Estimation or some analytical method like that. Maybe this would allow to get rid of some of the most complex steps. Never coded any distance estimated fractal, and had forgot all of derivatives, becouse they are pretty useless in real life aplication, but having some sort of non-iterating formula for needed variable could make it more easy. Arrays of stored variables or doble loop would be overkill.

Throught some ideal naturalistic pattern xxxbox fractal could be as good target as 3D mandelbrot;)

[M Benesi]

  I'm not even sure the method will work.  I've taken fractals of different sizes and overlapped them- end up with lots of discontinuities.  I'll eventually implement a 2d version for quick tests of the algorithm, then extend that up to 3d (easily, as the algorithm is a 2d slice of a 3d object). 

  As to programming difficulties-

  In ChaosPro the difficulty is in being able to address a single yz plane at a time.  It takes a lot of time to recalculate the various Rs for every pixel in a yz plane.  In other words, it would be quicker to calculate the Rs one time for the whole plane.

  <I assume> Fragmentarium can be programmed to do so.  I just need to learn to program it- perhaps doing a dry run with the formula from this thread.