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Author Topic: Complex 2-variable quadratic experiment  (Read 1839 times)
Description: Formula
0 Members and 1 Guest are viewing this topic.
Pauldelbrot
Fractal Senior
******
Posts: 2592



pderbyshire2
« on: December 31, 2012, 07:43:19 PM »

This should replicate my "Julia Foam" and some other recent gallery postings. It should work with the standard "Smooth (Mandelbrot)" coloring, TTBOMK.

The images I posted were Julia slices in the "z" plane, which the Julia version here will do. The Mandelbrot version plots in the "seed 1" plane, with the initial value set by the "seed 1" parameter (I don't know if this thing has anything analogous to the critical points of one-complex-variable systems). Adapting these to plot other "axis" planes shouldn't be difficult (though arbitrary slices of the 4D dynamic space and 6D parameter space, or even the 10D "Julibrot" space, would be much hairier to do -- one crude method would be to just add parameters to specify vectors for the unit vectors in the image space to map onto, leaving it up to the user to make them orthonormal or suffer the consequences in distorted images).

Because using the Mandelbrot to find good Julias might be difficult, and finding good Mandelbrot slices isn't especially easy either, interesting defaults are supplied. If UF has an "explore" or "mutate" search-feature for quickly exploring varied parameters, it'd probably be a good idea to use that (I've been doing something similar to find interesting parameters myself).

Code:
e_quad_1 {
init:
  z = #pixel
  complex w = @seed2
loop:
  complex ws = sqr(w)
  w = @seed3*w/z
  z = sqr(z) + ws + @seed
bailout:
  |z| < @bail
default:
  center = (0,0)
  maxiter = 10000
  title = "Experimental 2var Quadratic 1 Julia"
  param seed
    caption = "Seed 1"
    default = (-0.9064912128,0.1908320307)
  endparam
  param seed2
    caption = "Seed 2"
    default = (-0.043575,-0.092319)
  endparam
  param seed3
    caption = "Seed 3"
    default = (1,0)
  endparam
  param bail
    caption = "Bailout"
    default = 100.0
    min = 0
  endparam
}

e_quad_1m {
init:
  z = @seed
  complex w = @seed2
loop:
  complex ws = sqr(w)
  w = @seed3*w/z
  z = sqr(z) + ws + #pixel
bailout:
  |z| < @bail
default:
  center = (-0.75,0)
  maxiter = 10000
  title = "Experimental 2var Quadratic 1 Mandelbrot"
  param seed
    caption = "Seed 1"
    default = (0.366322,-0.127031)
  endparam
  param seed2
    caption = "Seed 2"
    default = (-0.043575,-0.092319)
  endparam
  param seed3
    caption = "Seed 3"
    default = (1,0)
  endparam
  param bail
    caption = "Bailout"
    default = 100.0
    min = 0
  endparam
}
Logged

fractalchemist
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Posts: 19



fractalchemist fractalchemist fractalchemist
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« Reply #1 on: January 01, 2013, 07:38:55 PM »

Saved and I am going to explore..:-) (Eveline)
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Pauldelbrot
Fractal Senior
******
Posts: 2592



pderbyshire2
« Reply #2 on: January 05, 2013, 03:20:10 AM »

Here are versions that use Brent periodicity checking, since UF's inbuilt will screw things up (in case z nearly stands still while w changes, say) and turning it off without supplying your own makes things slooooowww. Use with UF's inbuilt periodicity checking turned off.

As a side effect, all points color as "outside". To make inside points black requires using a special coloring plus the fact that these formulas will assign a very specific final value to z if the cycle detection trips. That coloring also works with normal Mandelbrots and gives you Fractint's "logmap=2" ability to logarithmically fit exactly one color gradient repetition between min iters and max iters found in image. It's after the first code block -- use transfer function linear and set "super transfer function" to log (or whatever you want) while ticking "fit gradient to range" and copying the minimum and maximum iterations shown in the General tab of the Statistics palette into the appropriate boxes.

I'll eventually also release a version of multiwave that adds the solid-flag check. Or you can modify your copy to wrap the entire innards of the (really looooonnnngg) "final:" section in IF (#z == (501,10)) #solid = true ELSE ... ENDIF. Unfortunately, UF hacking kind of requires some hoop-jumping, gimmicky tricks, and actual hacks, while also being the easiest way to make new fractals and colorings available to other people, at least for the time being. And I tend to have intermittent access to it to test stuff.

Let me know by PM or in this thread if either of these gives any problems.

Code:
e_quad_1j_brent {
init:
  z = #pixel
  complex w = @seed2
  complex oz = z
  complex ow = w
  int i = 0
  int nexti = 1
  float micro = 1/@bail
  bool periodic = false
loop:
  complex ws = sqr(w)
  w = @seed3*w/z
  z = sqr(z) + ws + @seed
  periodic = ((|oz - z| + |ow - w|) < micro)
  i = i + 1
  IF (periodic)
    z = (501,10)
  ELSEIF (i == nexti)
    nexti = nexti * 2
    oz = z
    ow = w
  ENDIF
bailout:
  |z| < @bail && !periodic
default:
  center = (-0.75,0)
  maxiter = 10000
  title = "Experimental 2var Quadratic 1 Julia (Brent)"
  param seed
    caption = "Seed 1"
    default = (0.366322,-0.127031)
  endparam
  param seed2
    caption = "Seed 2"
    default = (-0.043575,-0.092319)
  endparam
  param seed3
    caption = "Seed 3"
    default = (1,0)
  endparam
  param bail
    caption = "Bailout"
    default = 1000000000000000000000.0
    min = 0
  endparam
}

e_quad_1m_brent {
init:
  z = @seed
  complex w = @seed2
  complex oz = z
  complex ow = w
  int i = 0
  int nexti = 1
  float micro = 1/@bail
  bool periodic = false
loop:
  complex ws = sqr(w)
  w = @seed3*w/z
  z = sqr(z) + ws + #pixel
  periodic = ((|oz - z| + |ow - w|) < micro)
  i = i + 1
  IF (periodic)
    z = (501,10)
  ELSEIF (i == nexti)
    nexti = nexti * 2
    oz = z
    ow = w
  ENDIF
bailout:
  |z| < @bail && !periodic
default:
  center = (-0.75,0)
  maxiter = 10000
  title = "Experimental 2var Quadratic 1 Mandelbrot (Brent)"
  param seed
    caption = "Seed 1"
    default = (0.366322,-0.127031)
  endparam
  param seed2
    caption = "Seed 2"
    default = (-0.043575,-0.092319)
  endparam
  param seed3
    caption = "Seed 3"
    default = (1,0)
  endparam
  param bail
    caption = "Bailout"
    default = 1000000000000000000000.0
    min = 0
  endparam
}

Code:
smooth4 {
;
; This coloring method provides smooth iteration
; colors for convergent and divergent fractals and logmap and powermap options.
;
; Variant allows maxit and z_values array to have different size.
;
; Formulas may also be designed to assign some pixels solid color when this coloring is used.
; The formula simply leaves the complex value (501,10) in z when it bails out to do this.
; (There's a vague resemblance between "50110" and "SOlID". :))
init:
  complex il = 1/log(@power) ; Inverse log (power).
  float lp = log(log(@bailout)) ; log(log bailout).
  complex z_values[@zmax]
  int i = 0
  float sum2 = 0.0
  complex zold = (0,0)
loop:
  IF(@convergent)
    z_values[i] = #z
    i = i + 1
    IF (@esm)
      sum2 = sum2 + exp(-1/cabs(zold-#z))
      zold = #z
    ENDIF
  ENDIF
final:
  float ix = 0
  IF (#z == (501,10))
    #solid = true
  ELSEIF (@esm && @convergent)
    #index = sum2
  ELSE
    IF (@convergent)
      IF (@alt)
        int j = @zmax - 1
        IF (i < @zmax)
          j = i
        ENDIF
        i = 0
        bool found = false
        WHILE ((i < j) && !found)
          IF (|z_values[j] - z_values[i]| < (1/@bailout))
            found = true
          ELSE
            i = i + 1
          ENDIF
        ENDWHILE
        int k = i + 1
        found = false
        WHILE ((k < j) && !found)
          IF (|z_values[j] - z_values[k]| < (1/@bailout))
            found = true
          ELSE
            k = k + 1
          ENDIF
        ENDWHILE
        int p = k - i
        ix = k/p + @perfix*(log(@bailout) + (log(|z_values[i] - z_values[j]|)))/log(|z_values[j] - z_values[i]|/|z_values[j] - z_values[k]|)
      ELSE
        int j = @zmax
        IF (i < @zmax)
          j = i - 1
        ENDIF
        i = 0
        bool found = false
        WHILE ((i < j) && !found)
          IF (|z_values[j]-z_values[i]| < (1/@bailout))
            found = true
          ELSE
            i = i + 1
          ENDIF
        ENDWHILE
        int k = i + 1
        found = false
        WHILE ((k < j) && !found)
          IF (|z_values[j]-z_values[k]| < (1/@bailout))
            found = true
          ELSE
            k = k + 1
          ENDIF
        ENDWHILE
        int p = k - i       
        float tween = abs((sqrt(1/@bailout) - cabs(z_values[j]-z_values[i])))/(abs(cabs(z_values[j-p]-z_values[i-p]) - cabs(z_values[j]-z_values[i])))
        tween = sin(tween*#pi/2)
        tween = tween^(0.73)
        IF (tween < 0)
          tween = 0
        ENDIF
        IF (tween > 1)
          tween = 1
        ENDIF
        ix = i - tween*p
      ENDIF
    ELSE
      ix = real(#numiter + il*lp - il*log(log(cabs(#z))))
    ENDIF
  ENDIF
  IF (!@esm)
    IF (ix < 1)
      ix = 1
    ENDIF
    float mn = @fitminit
    float mx = @fitmaxit
    IF (@transfer == 1)
      ix = ix^(1/@transpower)
      mn = mn^(1/@transpower)
      mx = mx^(1/@transpower)
    ELSEIF (@transfer == 2)
      ix = log(ix)
      mn = log(mn)
      mx = log(mx)
    ENDIF
    IF (@fit)
      IF (ix < mn)
        ix = 0
      ELSE
        ix = (ix - mn)/(mx - mn)
      ENDIF
      ix = ix * @fittimes
    ELSE
      ix = 0.05*ix
    ENDIF
    #index = ix
  ENDIF 
default:
  title = "Smooth (Generalized) 2"
  param convergent
    caption = "Convergent"
    default = false
    hint = "Check this for convergent attractors, uncheck to color divergent points."
  endparam
  param esm
    caption = "Use exp smoothing"
    default = false
    hint = "Applies convergent exponential smoothing as the \
            standard formula of that name, except that fractals \
            can make some areas solid-color."
    visible = @convergent
  endparam
  param alt
    caption = "Use alternate method"
    default = false
    visible = (@convergent && !@esm)
  endparam
  param perfix
    caption = "Period fix factor"
    default = 1.0
    visible = (@convergent && @alt && !@esm)
  endparam
  param zmax
    caption = "Maximum iterations to record"
    default = 100
    hint = "Determines how many iterations to look ahead for a convergent attractor. If \
            a convergent attractor is not found, the solid color is used for the pixel. \
            Normally you want to use the maximum iterations you selected on the Formula \
            panel, but if you're getting 'out of memory' errors you'll need to set this \
            lower. There's generally little to gain by making it greater than 1,000,000."
    visible = @convergent && !@esm
  endparam
  param power
    caption = "Exponent"
    default = (2,0)
    hint = "This should be set to match the exponent of the \
            formula you are using. For Mandelbrot, this is 2. \
            Only needed when coloring divergent points."
    visible = !@convergent
  endparam
  param fit
    caption = "Fit Gradient to Range"
    default = true
    hint = "Check this to spread the gradient out over the range of iteration values."
    visible = !@esm || !@convergent
  endparam
  param fittimes
    caption = "Number of repetitions"
    default = 1.0
    min = 1.0
    hint = "Repeats gradient the specified number of times over the range of iteration values."
    visible = @fit && (!@esm || !@convergent)
  endparam
  param fitminit
    caption = "Start iteration"
    default = 1
    min = 1
    hint = "Gradient begins at this iteration number. It is best if it's approximately the lowest \
            actual number of iterations in the image. You can find the exact number by looking at \
            Statistics after generating the image once."
    visible = @fit && (!@esm || !@convergent)
  endparam
  param fitmaxit
    caption = "End iteration"
    default = 1000
    min = 1
    hint = "Gradient fitting is based on this range of iterations. Can be profitably made lower than \
            maxiter -- try reducing it by factors of 10 until the gradient doesn't fit well, then raise \
            it by a factor of 10 once."
    visible = @fit && (!@esm || !@convergent)
  endparam
  param transfer
    caption = "Super transfer function"
    enum = "Linear" "Power" "Log"
    default = 2
    hint = "Linear distributes gradient evenly over iterations. \
            Power weights gradient towards lower iterations for powers > 1. \
            Log weights gradient towards lower iterations."
    visible = !@esm || !@convergent
  endparam
  param transpower
    caption = "Transfer power"
    default = 3.0
    hint = "Larger values weight gradient more towards low iterations. \
            3.0 with a regular transfer function of Linear and a super transfer \           
            function of Linear with a regular transfer function of CubeRoot \
            produce the same results."
    visible = (@transfer == 1) && (!@esm || !@convergent)
  endparam
  param bailout
    caption = "Bail-out value"
    default = 100000.0
    hint = "Larger gives smoother coloring for divergent points, and more accurate and smoother for convergent ones."
    min = 1
    visible = !@esm || !@convergent
  endparam
}
Logged

Alef
Fractal Supremo
*****
Posts: 1174



WWW
« Reply #3 on: January 15, 2013, 09:26:09 AM »

IMHO, this is the best new thing in 2D in years. Maybe ultracomplex shapes somewhat don't go with ultracomplex but not very smooth colour methods, but it generates enermous variance of baroquesque /marine shapes.

Could I upload it to UF database?
Is there any name suggestions?

Here I a bitt modified for end users (inside switch, few mutions to play around and turned seed3 into real factor (anyway it's not a seed and less is more pro user.), and seeds put that it gives atractive symmetric ovewall shape. IMHO (1,0) (1,0) 0.5 have better patterns, but then are barely noticable. Maybe I alsou should add few addition modifications, such as tricorn and burning ship. )


Code:
Fractal1 {
::70eiSgn2lF1yuxNMMw7Gw/DC6eWL5N2dTL0h2D5QAKQ/DWwYJ7ltySGS0Zj/7L9j0WgeyQcG
  OzQP9JojA/nLLECCJvzIfefkWKujW6m5ctSczhD3IT9jKhHWcpsRvuRXyZRKbkfNbhxvor0N
  V1K9ZZZxGtNZ7gJCjBj8bQ3vGSx5gVKiTQHSLGtSVWMCTTYYYntLQuk5B1pmKlYEGCrW1HTj
  zeYjxI8OuShXV0jeXAG5UTuMdaufUKYBSLG5PgZv19aKSSx01szZNqqtUv/q2oZDmuevnP3Y
  ywGyveFQfcmWFXpOp24S0ykz8dIYd+V9O04OSd3+A9lZPC80fu+l16cl6Uro/a/co76UMTG0
  yBbfVM8mLRcgCxyCMkZo9CIBhcPfahYwVWw54/h8YwBpyihEYRWxN08YMyN1iLLQOlvbUiuo
  nPqLfS3Wf5yx01/Y7z1t6a9lmmDg61WQcg9YbzT6Dgz/djmWOR7t2/6ZI+HL/oTrbaKL+NAV
  D5LD
}

test.ufm:Pauldebrot {
;Complex 2-variable quadratic system
; by Paul Derbyshire aka Pauldebrot.
; http://www.fractalforums.com/index.php?topic=14541
::XLXT4in2Fa12OutNQ03Fg+HG48idjXZ7tJ9htRou5iBcR3ggmUkHLolorYXKRFRKLbj8x3ZI
  puYbtIGYBEnhzcm5MX4KKEmHCDCDSU5lS+RIJeZ/hGNdKMY7GY6aN3YOVyjjn8IrIlL3VpMT
  m5VKKOwrM8U8cS8qFvoUckLpvXvvuI5fKVaz0EU5H+zP/BS8zfhP++tbCDOHj4xTJwJTuKA+
  japg9cYv+/It9Q1du1zQL4nHC+5ewDDAojCgYwGK3HGIVqyHIlNaUq+bVTbmZPinaWc+nmuu
  ZPLxoqsSP7vz5ZwLJLeJkEGsjJkqaj1Lf/83h3Ar9iCDS57Z1SnuEehhXheY6dLje98lWP+r
  5sjCn4VLxfkMjwI5ogJfCtNlbLKUGkxZpii/FJAWpRoKorsRVlXLZwblqknwbxLS7uGAlsKW
  O4YdA/NwwHJ3CfGVNxpzHqU8tc+qZOhHEaxObw88NLAggaR6SIv/GMJ0A1e4r3C5KPfcrr88
  /NObjV84uD53xdWXh5Kn9Wn8bcktmEt0JOXQ3d51em+bsSjtje0CTYQLLZZ0OriRGyAkoJdt
  O00KvoO/CW37cnb9BSnTbEmksrdNFQONgR17dKFd+32TMGC9tA7ZSN/SIbHpcQ7nK7yGHD47
  xaznpLj+55YF6XmN03juKYISjxwPWbY036b4XaHA0tIYQI9JaNBkzKLxbavBJfQ4JSxx0pI4
  ZiCT8kN+bAoFSB2+mWXRWSzsWsBBuuVwkCt9YE8ReDupRVhhCzw1UDqd3ERgenNdM9zigvkJ
  0QSd1BUcBOdmMH2VjNEZcoUyK4zBEvmMBWE3Xht/MJgGkWxaKmDaFqVeCsJAkfHynkWMVh94
  cit2Qdc6I4v4JiSoRV9kGYIiiqEsIY35q5ROyki1+mq29xDo/Jb9ywCs5iOqN/udNtjA/CG8
  aWOfEgg/uQKei7pn2aihIBHea4d28qF9LzKk4GkUT+xtLbL2Tt+G+RMuenvz9+7OwqEMqL8b
  1sUqomA6TaDH3WcCoVww75V7OpzEVYW8ED6XLHBZGT5DLW000E5LJIZXnrjwJjFCcc6YUZW5
  vZUlC8hsX96XtC2agM2BOsTZy6yAM5whPMvanAtyigBb5w1SJYxVDZqGIvGbC08EFaUfczxP
  dNlaoWj2fzghbHA+ekdW7yXYcxS/bp+llxdLNbrDx9tDtTtXt15ijxdi7mIjHMb6XwH3tony
  8u/bC7LJx+3T+fAOn+MW
}

Code:
Pauldebrot {
;Complex 2-variable quadratic system
; by Paul Derbyshire aka Pauldebrot.
; http://www.fractalforums.com/ultrafractal/complex-2-variable-quadratic-experiment/
init:

complex c=0
complex ws=0

IF (@settype=="Mandelbrot")
IF (@inverted)
c=1/#pixel
c=@func_post(c)
ELSE
c=#pixel
c=@func_post(c)
ENDIF
z=@seed

ELSEIF (@settype=="Julia")
IF (@inverted)
c=1/@julia
ELSE
c=@julia
ENDIF
z= #pixel
z=@func_post(z)
ENDIF

  complex w = @seed2
loop:
  ws = sqr(w)
  w = w/z*(@wfactor)
  z = sqr(z) + ws

;now adding c and introducing tricorn and burning ship versions
IF (@zfunction=="None")
z=z+c
ELSEIF (@zfunction=="Tricorn")
z=conj(z)+c
ELSEIF (@zfunction=="Burning Ship")
z=abs(z)+c
ENDIF

bailout:
  |z| < @bailout
default:
  center = (-0.5,0)
  ;maxiter = 10000
  title = "Pauldebrot"

heading
caption = "Formula Block"
endheading
  param seed
    caption = "Mbrot Seed"
    default = (0,1)
    visible = (@settype=="Mandelbrot")
  endparam
  param seed2
    caption = "Seed of W"
    default = (1,0)
  endparam
  param wfactor
    caption = "Factor of W"
    default = 0.5
  endparam
  param bailout
    caption = "Bailout"
    default = 10000.0
    min = 0
  endparam
 
  heading
caption = "Julia Block"
endheading

param settype
caption="Set type"
default=0
enum="Mandelbrot" "Julia"
endparam

param switchsettype
caption = "switch to"
default = 1
enum = "Mandelbrot" "Julia"
visible = false
endparam

complex param julia
caption="Julia Seed"
default=(0.3, 0.6)
visible = (@settype=="Julia")
endparam

heading
caption = "Mutations"
endheading

func func_post
caption="Pixel mapping function"
default=ident()
hint="Function applied during iteration initialisation. New coordinates of pixel = function(coordinates of pixel). This curves not c, but the plane, on which fractal is drawn, so only ident m-set is map for julia sets. Recip works as circle inverse."
endfunc

param zfunction
caption="Z function"
default=0
enum="None" "Tricorn" "Burning Ship"
endparam

param inverted
caption ="Inverted Set"
default = FALSE
hint="The same as circle inverse. Unlike pixel mapping this inverts C, so inverted m-set is map of julia sets."
endparam

heading
caption = "Info"
text="Complex 2-variable quadratic system by Paul Derbyshire aka Pauldebrot. http://www.fractalforums.com/ultrafractal/complex-2-variable-quadratic-experiment/ It have both julia seed and mandelbrot seed. Factor of W decides how much second quadratic equation is used."
endheading

switch:
type="Pauldebrot"
julia = #pixel
bailout=bailout
inverted=inverted
settype=switchsettype
switchsettype=settype
func_post=func_post
zfunction=zfunction
wfactor=wfactor
seed=#pixel
seed2=seed2
}
« Last Edit: January 15, 2013, 09:44:53 AM by Alef » Logged

fractal catalisator
Pauldelbrot
Fractal Senior
******
Posts: 2592



pderbyshire2
« Reply #4 on: January 15, 2013, 02:29:21 PM »

Thanks!
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fractalchemist
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« Reply #5 on: January 16, 2013, 04:13:20 PM »

 Cantor Dance Thanks!!! For the switch, I already experimented with the M/J separately, and it is very cool!

See my FB album ( it's public)

http://www.facebook.com/media/set/?set=a.10200274158001280.2196453.1373252680&type=1

Evie
« Last Edit: January 16, 2013, 04:15:40 PM by fractalchemist, Reason: wrong URL » Logged

Alef
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« Reply #6 on: January 19, 2013, 03:34:22 PM »

In UF mailinglist some folks asked to include your smooth4  formula, so I uploaded it as Pauldebrots Smooth. Just added proper credits and a sigmoid function n=3*n/(3+abs(n)) as transfer function.
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fractal catalisator
Pauldelbrot
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pderbyshire2
« Reply #7 on: January 19, 2013, 08:42:29 PM »

No problem there. Consider anything I post here to be CC-BY. smiley
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Pauldelbrot
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pderbyshire2
« Reply #8 on: January 24, 2013, 09:50:21 PM »

The formula contains "minibrots" after a fashion, but they can be hard to find (lacking a critical point as normally defined). Here is how to find some.

First, a normal Mandelbrot image showing some buds and minibrots:



Note how one minibrot is straight out from the bud along the line from its base through the biggest sub-bud that's on roughly the opposite side. The much larger and more prominent one is off to one side.

This suggests how to try to find minibrots in the formula.

Here is a shallow image of this formula, sliced in parameter space.



There are areas where arcs of buds are discernible. We would prefer one where there's a lot of dense foamy tangles and not just a few stray dendrites about, as a few stray dendrites might all miss the minibrot that lurks in a neighboring dimension, so we zoom into the arc inside the hollow, at the hollow's lower-left, where there's a lot of foam:



The color scheme has been chosen so black contrasts fairly strongly with it.

Note at position 1, a ghostly darkened shape that may be a minibrot. It has approximately the correct shape. Its relationship to the buds nearby is different from what's typical in the plain M-set, though. It may be worth investigating but there are more promising targets.

Position 2 is lined up along the axis through a bud, similar to one of the minibrots in the plain M-set image discussed earlier. There are a few specks that seem darker than the other bluish bits in the same vicinity at this spot. (A high res display may really help spot this sort of thing!)

Position 3 is a slight darkening that's just past another bud, related to it similarly. It's less promising in one respect: it's so faint and small it may be nothing. It's more promising in another: the bud, and the position's relationship to the bud, are much more exactly matching normal M-set buds in form, relative positions, and proportions.

We choose position 2. Zooming at position 2 reveals this:



A field of small black nuggets scattered about near the image center, surrounded by blue and yellow foam. The nuggets are shaped like small filled-in quadratic Julia sets -- this is what happens when the foam occupies the same space as the dimensionally-displaced minibrot, thus making the minibrot visible, after a fashion, and providing interesting shapes to explore.

Turning all escaping points white reveals a distorted minibrot outline:



It's got some holes in it but it's definitely a somewhat-warped minibrot.

(I checked, and all 3 positions marked turned out to harbor minibrots, not just #2.)
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fractalchemist
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« Reply #9 on: January 25, 2013, 11:53:51 AM »

Thanks for this clear explanation  Yes !!
Show you the results later on!
Eveline
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Pauldelbrot
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pderbyshire2
« Reply #10 on: January 25, 2013, 07:56:13 PM »

You're welcome ... and have fun.

(Bah! Someone's monkeying around with something. My images in that post keep disappearing and appearing, and I haven't edited it.)
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