Compounding Tweaked

Mandelbrot & Julia

by Kerry Mitchell, 23 October 2000

Basic information

Variations on the Mandelbrot and Julia sets in which either z or c is tweaked each iteration.  The tweaks compound over the orbit, creating some very different details and an outline that resembles the standard set.  Four types of tweaks can be applied to either z or c, from the list of built-in functions.  The "switch" feature toggles from Mandelbrot to Julia formula, and back.

Suggested coloring formulas

There can be a lot of structure (or noise) created with these formulas, so stick with basic iteration outside coloring until you get the hang of things.  For inside lake areas, anything that works well with the standard Mandelbrot and Julia formulas should work fine.

Parameters

Hint

The switch feature will not necessarily show a Julia set that has the same shape as that region of the Mandelbrot set.  This is because the tweaked Mandelbrot set usually doesn't start with the critical point of the tweaked function.  The descrepancy gets worse as the amount of tweaking increases.

Sample images


CompoundingSample1 { ; copyright Kerry Mitchell 23oct2000
;
; sample for "Compounding Tweaked Mandelbrot" formula
;
fractal:
  title="Compounding Sample 1" width=600 height=480 author="Kerry Mitchell"
  created="October 23, 2000" numlayers=1
layer:
  method=multipass caption="Layer 1" visible=yes alpha=no
mapping:
  center=-0.7/0 magn=1.3 angle=0
formula:
  filename="lkm.ufm" entry="compounding-tweaked-mandelbrot" maxiter=1000
  percheck=off p_manparam=0/0 p_bailout=1000 p_nexp=2/0
  p_tweaktype="c; fn(c*z)" p_tweakage=0.01/0 f_tweakfunction=cotan
inside:
  transfer=none repeat=yes
outside:
  filename="dmj.ucl" entry="dmj-Smooth" transfer=log repeat=yes p_power=2/0
  p_bailout=1000
gradient:
  smooth=yes numnodes=4 index=0 color=0 index=100 color=78079 index=200
  color=131071 index=300 color=16711680
}

CompoundingSample2 { ; copyright Kerry Mitchell 23oct2000
;
; sample for "Compounding Tweaked Julia" formula
;
fractal:
  title="Compounding Sample 2" width=600 height=300 author="Kerry Mitchell"
  created="October 23, 2000" numlayers=1
layer:
  method=multipass caption="Layer 1" visible=yes alpha=no
mapping:
  center=0/0 magn=1.8 angle=0
formula:
  filename="lkm.ufm" entry="compounding-tweaked-julia" maxiter=1000
  percheck=off p_julparam=-1/0 p_nexp=2/0 p_bailout=1000
  p_tweaktype="c; fn(z)" p_tweakage=-0.01/0 f_tweakfunction=recip
inside:
  transfer=none repeat=yes
outside:
  filename="dmj.ucl" entry="dmj-Smooth" transfer=sqrt repeat=yes
  p_power=2/0 p_bailout=1000
gradient:
  smooth=yes numnodes=4 index=99 color=16711680 index=199 color=131071
  index=299 color=78079 index=399 color=0
}