How to use Ultrafractal for science

[gamma]

I wonder, how can I examine orbits in Ultrafractal?

For example, calculation of Julia(z^power+seed) set contains 3 relevant steps - sections in the formula file, (plus the variable definitions), followed by sections in coloring algorithms file.

...............................

init:
  z = #pixel
loop:
  z = z^@power + @seed
bailout:
  |z| <= @bailout

...............................

final:
  #index = 0.01 * #numiter

...............................

The most of Julia set' structure is based on the study of orbit (set theory, dynamics of complex maps...). Orbit is a plot of iterative function f(z) alone per starting point (init: z=#pixel). Distinguishing different kinds of orbit and hence, starting points z we are able to say where are fixed, periodic, or attractive points and which sets/areas are mapped onto other sets. It would be useful to have layers or use the F7 key functionality via switch parameter to see the orbit. Further, we could take records - animations of orbits along particular direction and so on.

[David Makin]

You'd need to write a formula that displays the orbits.

AFAIK the only one to do this at the moment is mmf4.ufm:Attractors which is specifically designed for viewing strange attractors but can be used to view any orbit from the fractal types included in the formula.

Note that because the formula renders orbits to an image array in the global section it suffers the same memory issues when rendering images larger than say 2048*2048 as does rendering flames in UF.

At some point I'll do a UF5 version that will let you do the same thing for any class formula.

[cKleinhuis]

could also be the julia mode used for that >?


but you are right, a global map has to be used for a rather simple result ... 

[David Makin]

could also be the julia mode used for that >?

I assume you actually mean UF's in-built switching ? That's how you use the mmf4.ufm:Attractors formula - you can choose whether the switch goes to a Julia view or an orbit view. Really it should have been written as 3 stages - Mandelbrot->Julia->Orbit but that's not so important now with UF's "Explore" feature available.

[gamma]

You'd need to write a formula that displays the orbits.

AFAIK the only one to do this at the moment is mmf4.ufm:Attractors which is specifically designed for viewing strange attractors but can be used to view any orbit from the fractal types included in the formula.

Note that because the formula renders orbits to an image array in the global section it suffers the same memory issues when rendering images larger than say 2048*2048 as does rendering flames in UF.

At some point I'll do a UF5 version that will let you do the same thing for any class formula.

This must be the second or third time you're responding to my questions with a ready made, high profile solution. Alas, this time its looks a bit Spanish and I don't understand it :-) DO NOT DO ANYTHING. I'll try to cheat the bailout condition with my function now and report back the results...

[David Makin]

Hi, try using the switch from this to view the orbits.

Code:

Fractal1 {
::kS0kvgn2tZV2utuNQ03DQ+HEUfrBxWkaz+GwHSztBtANtBNfABMUUy8aKRVSqE7/+OcRWKpF
  wAa4scmFOzQ3qpMLV+trvKJxKsSOJ9xALUayHiG7BSVRWyBuo7glUsLLRSPz1GCyZBTzbEWD
  J9703FNJPRPKGuLfLOD+lt/uHBxct4YyLSxPOy190h7q3idKkhTv+KPUeXzojWhagk+LU2xO
  taaoJNRNSZC7ZCKLLpnbPoaI9TSrYkaMXfVPdcUM0FMnPY5aytZbqLqrryKQV1bz2kVhKyx4
  9odoikea3AZPqYDeXZNE9tKNgG1bfP9kwBAuMLZkrZH4sjEVbbSrQyHo9QVpvvtYzUbfqTfw
  d6zk0ne6xC6t3btuKmSbeWOZSTGf9doA5ymcgGy5mzkzcDQr0vJskBlDixXNHUfIGaVRh/YS
  KogQgcUr60cjh8nqBOc+NqQa5GL57CA6Ow7RE6pSpTIB9r3uPqoaySQcoCDnlnpjTDq3DwGU
  FKLlBz9nRE0GnqtNi+gatzqNTjIY/BJURceqKYeP1qFnC2wyIotOdYIS2WEQoV2gIgAuX/Zk
  rwYm6vk+QS2MxiKp0QjC5vV2beZq/mnDiA+DKuB6N4k/CuMg8jb+nJqmvUDduJGqiTaerkPD
  pw1UAqm5jLhm2ImMuMJYpYmTI91O7CkqGeMoagaOfEwHytNoInPEW2hLBgyAk+LJzRxorV9S
  0EA5NqkOw4RshGtDuBGkTtACMoZ36NMa6kh3LGCWDEe9DXAMlkk+T/GwY7vDtrU3ITaMdgQb
  EGWz8FNYSTKMzVNmhLJ/hi51HOTXqMBjpxpr0H8CSg7A3tFlO0BLEe4A81lhU7Z4mI9epM5e
  nET03UIihGxQETbpQ9LbjrdkGiJ3YAVJb8BHFCGXt+J6pNRz1rYl0CIMQENQk4C9PdJyixZs
  /IW/W7d2KvzW8Ob27s/j3/izZrcOsxIet9poIu4g8s6DuOAyA/jL9704kwbxvjO1I4FasXQ7
  rtDIClpMOIcnwkWpY01TNSuB+01w7IYf0DeXJPRSP9tHUDGb6cnjn/ZH/zJMFvtdj7eDsTz5
  rNl4Nj6zKocpVql1Q+DoNllbh1nuiVsDPE9gVa7c53Bij/F3bsI3S3dFVIv5lFF57mFB55mi
  68c3mYQ+ig8vgRx8uD3hy1HqmP0dZ/S38CGWHeFOd5kbn1oYOe7KvQtgU9FqdLTAsu9XUFty
  XoFS8iC5LkFrxAVuo+iDR1Lqvbhcxh4s1ggX8JGvQu4T8S+hXSQc1nApOcSMYgu5wr7a6gpF
  aFd7ru+K4Zi/fRJGlU0QKw7L2XVj3Xu6RwXgeoGquZzETu+hwX6hmoDpfpb/yrR4duldduFu
  gBeXa8W4f6Dmc8bmIoExQD/EMoCzrKNpqssMfPOytqIyGlnnjw5YHMBRoqdzyqqrrR7rjCwl
  1RBF511lRu5F5RulIAl4fzYdYBDyzhy8fCBXCva+vkbFijF=
}

[gamma]

Does not work yet....


(The only thing worse than this is waiting for International space station. I've been going out over and over to see a FLYBY following NASA's tracker which just keeps changing those transit times!)





;;;;; OrbitonMandel1 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;
;
;
;
;
;
OrbitonMandel1 {
;
;
;


init:

     complex  z = (0,0)
     complex  c = #pixel

loop:

     zold = z
     z = z^2 + c

bailout:

        ; which checks do you use?
        (|z - zold| > @condition) || (|z|>200)


default:
  title = "OrbitonMandel1"
  helpfile = "Uf*.chm"
  maxiter = 1000
  periodicity = 0
  center = (0,0)
  magn = 1.5

  param condition
    caption = "Orbiton Test Set Bailout"
    default = 0.00000001
$IFDEF VER40
    exponential = true
$ENDIF
    hint = "Bailout value; smaller values will cause more \
            iterations to be done for each point."
  endparam

switch:

  type = "OrbitonJulia1"

  c = #pixel

}
;
;
;
;
;
;;;;;;;;;;;; end of OrbitonMandel1











;;;;; OrbitonJulia1 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;
;
;
;
;
;
OrbitonJulia1 {
;
;
;
       $define DEBUG

global:

       ; the starting point is z_orbit[0]=c=#pixel that came from Switch F7
       ; z_orbit is 1D array with dimension #maxiter
       
       complex z_orbit[#maxiter]
       z_orbit[0]=@c
       int i=0
       while i<#maxiter
; error array index out of bound???
             z_orbit[i+1]=z_orbit^2+@c
             i=i+1
       endwhile
       
       ; orbit is now in array z_orbit[]
       ; at x=real(z_orbit[]) and y=imag(z_orbit[])
       ; every z_orbit needs to shine like a dot of color i

       i=0
       while i<#maxiter
             z_orbit=z_orbit
             i=i+1
       endwhile

       
       ;computer will remember z_orbit, but only withing this file/fractal... got classes?

init:

     ; checks per pixel, z is pixel
     complex z=#pixel
     int     count=0
     int     bailout=0

loop:

     ; Is current z on SOME z_orbit[]?
     ; z_orbit[] is all full of some values that are coordinates x,y...
     ; if they happen to be HERE on z, then color it with z[i COLOR]

     while count<#maxiter
           ; what if if several orbital points are piled up onto one location at z...?
          if |z - z_orbit[count]|<0.001
             bailout=1
          endif
          count=count+1
     endwhile

bailout:

        bailout > 0

default:
  title = "OrbitonJulia1"
  helpfile = "Uf*.chm"
  maxiter = 1000
  periodicity = 0
  center = (0,0)
  magn = 1.0

  param c
    caption = "Julia Seed"
    default = (1,0)
    hint = "This is the Julia seed, a constant parameter which \
            defines the shape of the fractal. Use the Switch feature in \
            combination with the Nova (Mandelbrot) formula to find interesting \
            values."
  endparam



switch:

  type = "OrbitonMandel1"

}
;
;
;
;
;
;;;;;;;;;;;; end of OrbitonJulia1

[gamma]

Hi, try using the switch from this to view the orbits.

Code:

Fractal1 {
::kS0kvgn2tZV2utuNQ03DQ+HEUfrBxWkaz+GwHSztBtANtBNfABMUUy8aKRVSqE7/+OcRWKpF
  wAa4scmFOzQ3qpMLV+trvKJxKsSOJ9xALUayHiG7BSVRWyBuo7glUsLLRSPz1GCyZBTzbEWD
  J9703FNJPRPKGuLfLOD+lt/uHBxct4YyLSxPOy190h7q3idKkhTv+KPUeXzojWhagk+LU2xO
  taaoJNRNSZC7ZCKLLpnbPoaI9TSrYkaMXfVPdcUM0FMnPY5aytZbqLqrryKQV1bz2kVhKyx4
  9odoikea3AZPqYDeXZNE9tKNgG1bfP9kwBAuMLZkrZH4sjEVbbSrQyHo9QVpvvtYzUbfqTfw
  d6zk0ne6xC6t3btuKmSbeWOZSTGf9doA5ymcgGy5mzkzcDQr0vJskBlDixXNHUfIGaVRh/YS
  KogQgcUr60cjh8nqBOc+NqQa5GL57CA6Ow7RE6pSpTIB9r3uPqoaySQcoCDnlnpjTDq3DwGU
  FKLlBz9nRE0GnqtNi+gatzqNTjIY/BJURceqKYeP1qFnC2wyIotOdYIS2WEQoV2gIgAuX/Zk
  rwYm6vk+QS2MxiKp0QjC5vV2beZq/mnDiA+DKuB6N4k/CuMg8jb+nJqmvUDduJGqiTaerkPD
  pw1UAqm5jLhm2ImMuMJYpYmTI91O7CkqGeMoagaOfEwHytNoInPEW2hLBgyAk+LJzRxorV9S
  0EA5NqkOw4RshGtDuBGkTtACMoZ36NMa6kh3LGCWDEe9DXAMlkk+T/GwY7vDtrU3ITaMdgQb
  EGWz8FNYSTKMzVNmhLJ/hi51HOTXqMBjpxpr0H8CSg7A3tFlO0BLEe4A81lhU7Z4mI9epM5e
  nET03UIihGxQETbpQ9LbjrdkGiJ3YAVJb8BHFCGXt+J6pNRz1rYl0CIMQENQk4C9PdJyixZs
  /IW/W7d2KvzW8Ob27s/j3/izZrcOsxIet9poIu4g8s6DuOAyA/jL9704kwbxvjO1I4FasXQ7
  rtDIClpMOIcnwkWpY01TNSuB+01w7IYf0DeXJPRSP9tHUDGb6cnjn/ZH/zJMFvtdj7eDsTz5
  rNl4Nj6zKocpVql1Q+DoNllbh1nuiVsDPE9gVa7c53Bij/F3bsI3S3dFVIv5lFF57mFB55mi
  68c3mYQ+ig8vgRx8uD3hy1HqmP0dZ/S38CGWHeFOd5kbn1oYOe7KvQtgU9FqdLTAsu9XUFty
  XoFS8iC5LkFrxAVuo+iDR1Lqvbhcxh4s1ggX8JGvQu4T8S+hXSQc1nApOcSMYgu5wr7a6gpF
  aFd7ru+K4Zi/fRJGlU0QKw7L2XVj3Xu6RwXgeoGquZzETu+hwX6hmoDpfpb/yrR4duldduFu
  gBeXa8W4f6Dmc8bmIoExQD/EMoCzrKNpqssMfPOytqIyGlnnjw5YHMBRoqdzyqqrrR7rjCwl
  1RBF511lRu5F5RulIAl4fzYdYBDyzhy8fCBXCva+vkbFijF=
}

It works! Black and white orbits are in the small window 'over to the right!

[David Makin]

It works! Black and white orbits are in the small window 'over to the right!

I confess I'd forgotten the options in my own formulas - try this one instead:

Code:

Fractal1 {
::atuaTjn2lZ12uNuNQ03DQ+HE0+WDitI1N7NgPkmtBtANtB7+BEwQRJz1UkqkUJ2/9doIllyG
  ADohzlzchzM0tGKzRlf96rSScCnkTSfMwClm8uoxdgUVklcgL6O4IF7ySk0zcjlg8WwM8Ghz
  SS/G9NRTyT0jC1d5bxZwvs939IImbEHT+hU8zjcTPVdX9WsXhMc61XNB1krZ0BnQrIp/Olds
  zoHVNpJ6BKT4OTQZZJ9c3BdDpfU6EDUr96r6pDDCVXwcuyxNkbz2UXUXXlVgqq3mtJrCVkjx
  7R7QFJ90OFZPqYDeXZNE9taDgGdy+e6JhHAcZWyA3wOwZHJ622kWhkro9QVpvvtYzYbfqXfw
  dmzk0ne6xC6t37c+Km2YfWOaTTGe5NoA5zmcgGy5mzkzcLQrNvKcEl2DxwL2D63FqWdU4PHl
  CKIEIHM6OD3aJ/jWxhzvSFSH36IfTAQ3BePiQPVK9CJo/429RF1jOCiDVY4s8MdYUpfLAbQV
  osUGMf6Mig24VttR0HUrdWtZaEBPdQCVEvnqCm3TdGxpgNsMCarXHGiktFBEGtLICIg71fD5
  LM2x+LpPkkNjsoSaD0oQ+u2dzPG7v55gIgvSztQvBn8vwlBkfc7/NSN8lao3NxQVcywbl8ZI
  F+mCQ1sp4SYoNiRrPTCWKm5ESfj3uApuhHDqGomzHA8hcbDKy5dhjd4SAotA50lk9oYw3qeJ
  aCg8KVSVMeEboR7gfgB5VLgADa2dTGmNb8ol3LUB7BiJLCXBMtkk+l/EYs9hRzbU3ohnGTII
  4Ggx1spyGMrJF256Gzylk/Wzo+hM4Mdp2EMmGnvSvgaCcR4vyoUVHsV4hDwXfaSdnhrj07ly
  k79SsR3ThgG6GDBNtlCFxsN+eSaIsK8kaZzU8Rh4xXwfieaT0czKWJtACKioBiSf0/hbSWMU
  jNJxi4avzW5dWw75haXw7sP59fx5sVOHWbEv7+QUE3eQeW/O3EARxf/STPNOO8a87gXNCehG
  PJo9lWFiQZarHC/JMpVKG8NWDkbgPdN8OCeK6BvrlnIpn+6Dal1lO3+Mx/sn/5Emm32uxfvB
  2Z481mSmMjOlVQ5yo1L7imOg2UWuF2h6LWx28Q0DWZczlfPIe+Xcv1h8be3VUhmMvsoIf3sI
  IP3UUnn7XHDyXEk/LYUMvAxfoc9hq5DdXWy0Nvlh1hXhTXO52ZNKmj3uyLULIVfha3yQArb/
  FVRr8FahEvoQ+CZxaMQlLqv4QU9i67WIXcIObNI4FfixLkL+Evkf4lEEX9BQqDnEKL0NHei3
  QV2WoV0v066rg3K+sIpQxpmErWKaIF49F7rqx7L/0bhM5nfL8RtZ55w1PF6vsiDu/FMSd9Vd
  +1vgtT+22D9fHmeIEGhm2SBWIUN8TwELM4qNkqyyy894I3qiIbUeeOCnj9wEEhq2NLrquuGt
  vOKAXWHFUkXXXG5mXkH5WiAUi/pj1hFMRPHKz/lEcJ8G6/zqgguP
}

[David Makin]

Does not work yet....
<snip>

Hi, using your method which of course avoids the use of a global screen array here's a way to do it using the switch in just one formula - save it in a file called "Orbits.ufm"

Code:

MandelbrotOrbits {
;
; Formula for viewing orbits
; Note that orbit pixels are coloured as "Inside"
; with the orbit iteration count passed as real(z)
;
global:
  complex zv[@orbitcount]
  int i = 0
  if @orbit
    zv[0] = @start
    repeat
      i = i + 1
      zv[i] = zv[i-1]^@power + @seed
    until i>=@orbitcount || |zv[i]|>@bailout || |zv[i]-zv[i-1]|<@smallbail
  endif
init:
  int j = 0
  if @orbit
    while j<i && |#pixel-zv[j]|>@limit
      j = j + 1
    endwhile
    if j<i
      z = j/i
    else
      z = #pixel
    endif
  else
    z = @start
  endif
loop:
  if !@orbit
    z = z^@power + #pixel
  endif
bailout:
  (@orbit && j<i) || (!@orbit && |z|<=@bailout)
default:
  title = "Mandelbrot Orbits"
  center = (-0.5, 0)
  bool param orbit
    default = false
    visible = false
  endparam
  bool param orbit1
    default = true
    visible = false
  endparam
  heading
    caption = "Currently in Mandelbrot mode."
    text = "Use the switch to go to an orbit."
    visible = @orbit1
  endheading
  heading
    caption = "Currently in Orbit mode."
    text = "Switch to go to Mandelbrot mode."
    visible = @orbit
  endheading
  complex param start
    caption = "Starting point"
    default = (0,0)
    hint = "The starting point parameter can be used to distort the Mandelbrot \
            set. Use (0, 0) for the standard Mandelbrot set."
  endparam
  complex param seed
    caption = "Julia seed"
    default = (0.5,0)
    hint = "The constant c in z^p+c."
    visible = @orbit
  endparam
  complex param power
    caption = "Power"
    default = (2,0)
    hint = "This parameter sets the exponent for the Mandelbrot formula. \
            Increasing the real part to 3, 4, and so on, will add discs to \
            the Mandelbrot figure. Non-integer real values and non-zero \
            imaginary values will create distorted Mandelbrot sets. Use (2, 0) \
            for the standard Mandelbrot set."
  endparam
  float param bailout
    caption = "Bailout value"
    default = 65536.0
    min = 1.0
    hint = "This parameter defines how soon an orbit bails out while \
            iterating. Larger values give smoother outlines; values around 4 \
            give more interesting shapes around the set. Values less than 4 \
            will distort the fractal."
  endparam
  float param smallbail
    caption = "Small Bailout"
    default = 1e-5
    hint = "Convergent bailout test value."
  endparam
  int param orbitcount
    caption = "Orbit count"
    default = 1000
    min = 1000
    hint = "The max. number of iterations to use in orbit mode."
  endparam
  float param limit
    caption = "Orbit limit"
    default = 0.0001
    hint = "Sets the size of the orbit points. Larger values will produce \
            larger points."
  endparam
switch:
  type = "MandelbrotOrbits"
  orbit = orbit1
  orbit1 = orbit
  start = start
  seed = #pixel
  power = power
  bailout = bailout
  smallbail = smallbail
  orbitcount = orbitcount
  limit = limit
}

Example:

Code:

Fractal1 {
fractal:
  title="Fractal1" width=640 height=480 layers=1
  credits="David Makin;3/20/2009;Frederik Slijkerman;7/23/2002"
layer:
  caption="Background" opacity=100 method=multipass
mapping:
  center=-0.5/0 magn=1
formula:
  maxiter=100 percheck=off filename="Orbits.ufm"
  entry="MandelbrotOrbits" p_orbit=no p_orbit1=yes p_start=0/0
  p_seed=0.5/0 p_power=2/0 p_bailout=65536 p_smallbail=1E-5
  p_orbitcount=1000 p_limit=0.0001
inside:
  transfer=linear solid=4294967295 filename="mmf4.ucl"
  entry="MMF4a-ForAttractors" p_version=1 p_method=Real
outside:
  transfer=none filename="Standard.ucl" entry="Default"
gradient:
  smooth=yes rotation=1 index=0 color=6555392 index=64 color=13331232
  index=168 color=16777197 index=257 color=43775 index=343 color=512
opacity:
  smooth=no index=0 opacity=255
}

[David Makin]

Hi all,

Thought I'd post an example of a more interesting orbit.

I just updated mmf4.ufm in the UF formula database, I've added a formula type to mmf4.ufm:Attractors that's based on the well known Mira formula that often produces strange attractors, example (requires the updated mmf4.ufm):

Code:

Mira3 {
; Copyright © 2009 by David Makin.
::OZ0rYgn29ZVzuRuNM47BIvDGu3aQcsk/b8WoDZT7iWgm2FbfACUkl9oGbJXJ5kZe7L1PesT2
  ieykUk8jkiky9aKzSH/01XlkYF2ROJ9RhmWkm8mozekUXmnckLGOa9kj0zcthgcqz08Oh1QS
  /Z6riukHpvIk/UxdY0d48820rvyrs3zM6sVokk0PTZvMoVLyu0E1MlJsnJo88kJu9oqjMtMa
  FzUj56rmozzC5Qwcu0y1kbLyqqwFoiDoq7QZFVtF1NHyrSmoDSSeWeLCXVXXVc9V9KN4Lq36
  J6JhzcUyMXzOyZvQU99J9iRukOBJ80UfZ2S/UqTbAK9ZoI84XKp3ev16qPKt5rjLm0k5neFS
  fXmUA0TUZ3ZiUBkK9zCL5M34cx8Tmjq3EyelTCw+3LjCaQxZtaQzNGyfokcg/ZqY0yNWynV2
  jRjnojjO5E0vcbbUH1CcBUVVUD8jnpzLS1rRvH0FS/qg9eeEBllDM9dipAy9rqtSjIYPzIUN
  cQVHMfia1iTBbY5E0dOdYIS+dQB8JtyGOCIg7zfE5KKmFHGBzhEsbhZXLLdQh/bK7N/1y0Nf
  NcEIXq4GongT+T4iASQu5fWoa+FnYcwEDVxJNvfkv6ShrZAUN3HXQranYx4ykgliVJh0X7sL
  Qq64xgqDq38Zw/QulhiSeTYZHvEAKDQ6vgMvImdto5XCnYZ/Z6IVy4RvDNZHdDD4gmBvwgGd
  bw48LeYxwnEyQoAEBr2sYkk+D/KI7ufD6ZpupGXFWAh4MMOm7LeaoDRYcjq0TZJQXYyjCZWa
  0FG+I53VMvtAP9dFMacSL9Bv0E4ex5fKVOAz+PcE+yD+haPDXQp3POmcv7Qf7PFieo7ME90e
  KUTzzc9o0Q85GLoqxOSMfoQw4uC8BKwq3z3DeQSEdQk4i7t4MmIxQdrthtHd2O0Zboz2hO79
  o/BwZ7AH2g42P80Hjh4eEibnortQC33+5gJQwJg+W4SF2AB3K83uMbQdTNBH8ccAaW9mbFUW
  TlzkqVJY/x9P1LREKTZ8kYS/oY2B3M5mgbG64DEsP9gAUNeike6TPokGb6mwzOhnTYKeffmT
  OYkmzJ41eLvWEvZUfaD1TtabiW5mVq8RYRcQMMW4CZclbapxF6g5a/8Zc5gzhbpsnF5s5QZN
  KkvllFHWPCy5symiiGwfw5bHU8BfUS25/q9M1rMDXWQNsuhiNg35nhCytraUGnBYDVXo28Uz
  FqD76BHavoKaHWoNS8mCFbkl79BqaT9NARNbqfYjcDQc+eng3wEj3I3wEvlf4tEEX/On0E4E
  SD06He1XTlmeo50vw7jvKyGTf/TifRp3eVMuyZ9RR/gRYo9bwg61XBva994MKkcqOxoGFdkS
  cbZbdDut6/C6v7B53j+/H0Du3BAb9Ybmge7j+F3CZH/EsHF2To0k6DHOgxlRpIUTUeVR7h8D
  VR54a/TrhjgO2WUdBOeWRb76B1NNNYEoa8Pb2DNMhti86/9grAV/XEnp5fJ=
}

[gamma]

Does not work yet....
<snip>

Hi, using your method which of course avoids the use of a global screen array here's a way to do it using the switch in just one formula - save it in a file called "Orbits.ufm"

Code:

MandelbrotOrbits {
;
; Formula for viewing orbits
; Note that orbit pixels are coloured as "Inside"
; with the orbit iteration count passed as real(z)
;

Excellent work.

 :