Apollonian fractals

[ericr]

Thank you again for everything but when all find time to sleep
Please may I have the param Vocan figure make in this  with topical Calyey
and called clayey
ERICR

[DarkBeam]

It was a test fmla I deleted ... maybe I'll recover

I will fix some formulas currently buggy whenever possible, later

[Jesse]

Couldnt resist to do some asm optimizations for the apollo2d, but because of the trig functions it is only ~15% faster...
Have attached it, if you like, take it or not. 

[DarkBeam]

  the trig functions essentially make this (see the crappy drawing)

1. find the quadrant of x,y
2. If the quadrant isn't the first, rotate x&y back to the 1st quadrant
3. if the quadrant is 2, 4, 6 ... then y = -y

Surely it's possible to optimize it at least for the case polyfold=3 (the classic gasket) but formula is less general

Also for a "true3D" version, we should try to check if x,y,z belongs to a "pyramid" over the origin, in this case rotate z-axis back to x-axis to make appear another branch in the z-axis. Else, do the normal formula + a lil rotation of xz (more or less) (too complicated huh ... I did not dare to try this!)

important; cayley transform flips x&y but it does NOT work - so before cayley I flipped them again (and maybe changed some signs)

[DarkBeam]

param for ericr& friends


Mandelbulb3Dv17{
V.....S....O/...w....2....UbGNdVFhR9.XuZvv.21X2EtGT33QpgJzXiM6P.ha77.h2pCW0C56/k
................................sEUWPU/ZMz1........A./..................y.2.....
................/M.0/....6kr....f0....E2.....YpvK9RqqSsD...........m/dkpXm1....U
z.....kD12../..........wz..................................u1....y1...sD...../..
.w1...sDwUx2eDdm1yX1w/Lcq3QSzoeMMf/DJ4pDu8pgCLcblx9gdeQefh4RzYYBTU0k/CsDiznTzTmG
nxXx4PsVhWFUzq4Fcl13QerD......YP/.............kD.2....sD..kz....................
.............oAnAr1...sD....z.lRo4.grFP.EV5h/.DSo4.YtFP.kc5h/.hSo4......w....k1.
..................kz.wzzz1.U..6.P....M4...EB....W....k1....F....8/...I1.....SF52
...U.qFG9yzb2zzzRYoWzz7lz16.mc..zXCc..k18XGQeGyD.sIRhrVAkz1..........2.28.kFrA0.
.Ub96aAIVz9.1se7Umvxz0........../6U0.wzzz1................................E.0c..
zzzz.................................2U.8.kzzzD.................................
/6U0.wzzz1...................................2CcN/UvNP6.eeWCNq0.yRii.EJJUk1f..XR
S1.mx3CcN/UvNP6.QsLsUa3.ibhV..bTV1OK.sSq40.ly3CcN/UvNP6.MwLsUa3.ibhV.kqTV1OK.sSq
40.kz3CcN/UvNP6...EsUa3.eeWCNq0.IJ36wk8.wyLsUa3.................................
E....6....E.....I....c....EEkx4PgxqA2pGG4B3..............................MU/46..
................................UJCoWgRyXz1.....................................
...0.BmqtvbOwGvDXcoS1y0HWzHCoWgRyydoz...........................................
.....................2.....3....8....6qPsZYFH/EPZZYFH/..........................
....Cs....E.0.................zD........kz1........wz.........zDsuFVf53iSy1.....
..............................................zD................................
................................}

 

[ericr]

thank you for the formula and thank you again for everything
you go too fast I have trouble keeping up in the last days

[Jesse]

the trig functions essentially make this (see the crappy drawing)

1. find the quadrant of x,y
2. If the quadrant isn't the first, rotate x&y back to the 1st quadrant
3. if the quadrant is 2, 4, 6 ... then y = -y

Surely it's possible to optimize it at least for the case polyfold=3 (the classic gasket) but formula is less general

The only thing that comes in my mind (yet) is to limit the polyfold up to.. lets say 32, and have the cos and sin values stored in a lookup table.
Edit: The table would be to rotate the x,y values back to the first quadrant, but you need n-1 sin+cos values for each polyfold order, so a limit up to a order of 10 must be made...


The integer power rotation would stretch the metric, maybe there is some other math trick for polyfolding?

Also for a "true3D" version, we should try to check if x,y,z belongs to a "pyramid" over the origin, in this case rotate z-axis back to x-axis to make appear another branch in the z-axis. Else, do the normal formula + a lil rotation of xz (more or less) (too complicated huh ... I did not dare to try this!)

I know, i am an expert in spending many hours with similar tickling investigations without having any results  
(I let you do it, and if the result is ok i might do some asm optimization)  

[DarkBeam]

No trick ... The fact is that polyfold is just a folding but based on regular quadrants instead of rectangular (infinite) slices. so it is strictly conformal in 2d, 3d, 4d...

[ericr]

hello, I have a book about measures of fractals and hyperbolic geometry (the circle of Apollonius)
did you think of the extent of Beiscovitch, a Lebesgue measure (made ​​for fractals)
distance poincare used geometreie hyperbolic
J hope to help
ERICR

[DarkBeam]

hello, I have a book about measures of fractals and hyperbolic geometry (the circle of Apollonius)
did you think of the extent of Beiscovitch, a Lebesgue measure (made ​​for fractals)
distance poincare used geometreie hyperbolic
J hope to help
ERICR

That is the theory, but for a real formula you must take into account not only one complex transform (like the theory says) but a minimum of 3 (possibly forming a sym pattern, for keeping things simple - even if in general you know, an Apollonian gasket can be made of N circles of different radius. Images; http://en.wikipedia.org/wiki/Apollonian_gasket#Integral_Apollonian_circle_packings  .But it was already hard to make the basic version!...   ). Remind that you must stick to non-probabilistic methods! So blabla... In the end the only way to get there in MB3D is to use a polyfold (kaleidoscope) transform and after it a Mobius one.