Breaking Symmetry

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One way of breaking the symmetry of formulas is to apply transformations to z as well as the normal formula calculations.

There's a really old formula pair called "Transformation" in mmf.ufm that allows you to do this. Below is an example modifying z2+c with the matrix values changed from the defaults of 1,0,0,1 - the closer the matrix values are to 1,0,0,1 then the closer to the original fractal - note that even small variations from 1,0,0,1 will produce noticeable effects in areas of higher iteration depth. Note that you can choose the base formula used from a number of different fractal types.

bye Dave 

BreakingSymmetry1a {
fractal:
  title="BreakingSymmetry1a" width=640 height=480 layers=1
  credits="Don Berendsen;11/22/2008;David Makin;11/21/2008"
layer:
  caption="Background" opacity=100 method=multipass
mapping:
  center=-0.5/0 magn=1
formula:
  maxiter=100 percheck=off filename="mmf.ufm" entry="MMFp-TransMandel"
  p_fractal=Standard p_startpix=no p_start=0/0 p_test=mod
  p_bailout=65536.0 p_smallbail=disabled p_bailout1=1E-5 p_smooth=Off
  p_fixHinrich=no p_cfudge=1.0 p_fudge=10.0 p_fudge1=100.0
  p_fudge2=1.0 p_ang=0.0 p_x=0.0 p_y=0.0 p_a=0.8 p_b=-0.1 p_c=0.2
  p_d=1.4 p_sigma=no p_product=no p_selfrot=Disabled p_doscale=Off
  p_scale=2.0/0 p_power=2.0 p_power1=2.0/0 p_ndegree=3 p_nr1=1/0
  p_nr2=-0.5/0.8660254 p_nr3=-0.5/-0.8660254 p_nr4=-1/0
  p_nr5=0.5/0.8660254 p_nr6=0.5/-0.8660254 p_usescale=1.0
  p_usecentre=0/0 p_swscale=1.0 p_swcentre=0/0 f_Fn=acos f_Fn1=cos
  f_Fn2=ident f_Fn3=sin
inside:
  transfer=none
outside:
  transfer=linear filename="Standard.ucl" entry="Smooth" p_power=2/0
  p_bailout=128.0
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
}


If you want to do the same thing using UF5 style formulas then you could use either:

reb5.ufm:Object Formulas or mmf5.ufm:Generic Switch Formula + mmf.ulb Switch Formula Transformations

and in either case PlugIn mmf.ulb:Simple Matrix Transform as the pre-transform or post-transform. 

BreakingSymmetry1b {
::r31HIgn2daVQvJuOQ47Vq/Hiy9CxBCQblv0tvun6qKxKtXeSVuJDgf4YH54Al/97YHHaIkST
  fXQ4Z++mM+zeGPr0sUDTc31XFEY4GBQDfQDstc56glHyzBj+QA5twg98MzG6spRBbA+6NG60
  FRBC2BQXSJW2paIjbKphPqkBPAaQmVCy7JkxxxjjjiWc/jsd8sgntB3ZmUb+JkIo5bDWK+Xb
  g4/3WQnzk3Pfc8ELk4wrvy9lcZZKrwwVSMPZpbXrVVyswAVBLlbOQJRRBYOvRlRzrEGeBrs8
  6ryZFF4GqmOINgmeT0okxIW2apN9Xp0IemDRO7duFiNWFgOdDkulqWtKYFXASWOKR55rSGVt
  KP0iHDo+AN8ngE3FpL3zNpbeqOehBFvW8auKDoHgyjLIUpyto0w0Ga04IbccGAIjeDZUsN5s
  GYZ7YyU0onBqLVMRZKDPp8ms7INcSUcuJjqDRVJUCMcfc8reyqCNsjD7tJYDfNTuGoxj8B0L
  O/QgipbvPqS82dP/8Tveym93ItSL2/oR9G0hnze0uXXXLT1qbXn+lWn9SOVJ3BaMEmMu/PfF
  FMdLdZ53mrTszO4P564qzJXXvtOGvcW5tba0u91a3lSJ88v+srjrC1ek4nlSV2w+GjLUVYef
  ufhl9R/9GCWWWB/do/v+DI1ffoAoP2ou9GDtSZokxR9HA73mEvoXmLzZChFEl8P3k0DkU7vv
  ohj3D7cXlnXIgnRpm/+RIhDJOnerdgMOei+N4gVq6vPPUUjsF7JjxmaRzGKHCFRPdc0obX8Z
  U8F29Vzv0g1GMdW4XwdYa3HgvcJZHwDsQuDr6Cv/PcbV+NIo1NP+aoXqZSXoXuzSH0tqY/aw
  vdxCwzi8x3lGQS7Uc6L2WMDbLavcOstYdXv4PH6LqSTrWCpq8cl0db+XVCxA6E0m/AbF0hyw
  qp7Qa4ND6QUibrOE5ySeGUPknzDKZSlEu+K8A/cXCuEHRo18ONV7jqSF9MzzP1sMOa6HKhSj
  Da51yU/S/gDfEEU7bWgt1xXE20HjzGWor3m1fIOnx3nYtReGsO3gO1ZriC74QNB0pXluU3Nf
  gWZYupSJBcZG8O+CrLO0ZJJJTuN2bd2UvZykJTIxTidz76cRmtox3s5znTudu3RcycvjpTmP
  Pxbdy0JerJEMK+5fbnW4j0NpSz0xxJ47m/ldfsSM
}


bye Dave

You might add a transformation to the mapping tab. Two good ones are Twist and Spiral in lp.uxf. Note: the default values are too strong for just breaking symmetry. Here's a simple example: 



Fractal2 {
fractal:
 title="Fractal2" width=640 height=480 layers=1
 credits="James Blue;11/21/2008"
layer:
 caption="Background" opacity=100 method=multipass
mapping:
 center=0/0 magn=1 transforms=1
transform:
 filename="lp.uxf" entry="Spiral" p_center=0.0/0.8 p_twist=0.25
formula:
 maxiter=250 filename="Standard.ufm" entry="Julia" p_seed=0.4/0.0
 p_power=2/0 p_bailout=128
inside:
 transfer=none
outside:
 transfer=linear filename="Standard.ucl" entry="Smooth" p_power=2/0
 p_bailout=128.0
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
}

Jim Blue

Hi,

If you want to do the same thing using UF5 style formulas then you could use either:

reb5.ufm:Object Formulas or mmf5.ufm:Generic Switch Formula + mmf.ulb Switch Formula Transformations

and in either case plug-in mmf.ulb:Simple Matrix Transform as the pre-transform or post-transform.

bye Dave

Txs David for this interesting approach but what do you mean with the matrix transform as pre transform or post transform...Can't find that option Thanks for support :)

Joe Ds

Hi,

See UPR below.

bye Dave 

Fractal1 {
::pwTGwin2daVwONOMQ07Ix/QUuTacSTbBkvAslTdFSg0esyk426tO2ROOt0/+dsjTJNNUCrEq
 CPz7NZ8zeGPrUkUNhf31X55pZaOF7Pv2Ey3bPLTvBPZco3GKb9GNe8sQPO5AVViRGGpKaGTX
 i9fisjl5tgslJuHhGFB/FGO7+5gfqit17VO7vbpqci4+pjiiNej8v+Kbssf7UShmJFY/HIpb
 XrkViMfPZBJlpPgRhhe5U9GZGOvirZFkyyrvKnUUwErrpTFaqCfTYQyIALZtwkgrkKAPxiIn
 8BzAxErCqKdDNdLWual3KGnKI5wGPPfVSQ1qcfDeIgqDY/npCYDk+6emOdz86457VssYZuMj
 iPQLPuAhFS7iSNRpxhjCNxxagSzw3gCiMJnxAJbHRkCGdMAdpiwLTJg+7MZ2RK6JRx6GFUHi
 qSaJlA7jjf1TWVoo7Y09mEshviIWTxRBuA6EnH5gYa37BV83vbxi5LPZz+GQr0g9PKQvpK/z
 ZHsb56aZqWd760t04sXypSxOqCChOj5+nvjCkul2s8Hz1K2ZHcncdc15krr3WHjXOrc21Na3
 +at7SpEc+Xf21xVhcPQ8rSpKTYfnw4yKIvP3P3w+o/eDBJLrg9Bt/v+DA13OUQxP1ou9GDlU
 qxoRh9HAz3GFNrXmvmT4cDIM6X3k0DkUzvvooHvH25uKLvgTXASN7jjQ8HScO9W7AZc8E9Hw
 BqUV/cegoGaK2TGBN1CnMUOIMge8owgbn9VUcF29Vz/qGqNIqM/vh7w0uPBf5SyOgHYhcHW1
 Fe/PcbV+NIo1NP+eoXqZSXoXuzSH0tqY/ewvfxCwzi8x3lGQSbVc8LmWMDbLaucOstYdXvov
 G6LySdrWCpy8cpweb+3Vc+A6E0m/AbF0hywqp7Qa4ND6QUAbrOEZiSWGte0NrHQyESB96rgD
 8zdxZCYEhWz70UtHUly7ZmnnVkMGY6RJXqgBtcaZqbpbwhPDCo9NLg26wLCb6jxZDL01bz6P
 Fnz47Ss2IPDWnbQn6sVRhZcomAa1rSbqbnPQJ1E7UpIPmIj+B8CrNO4JJJJx3G5sOZszMKOO
 GFFHZCTtL0kZN+mMd6U0tTdOiSm6cMOe60En14xxOrJIIKu5fbnWwj0NpSz0xRJw7m/D2PlH
 RD==
}
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