Fibonacci Mandelbrot & Julia

Kerry Mitchell
06 December 2000

Basic information

Instead of computing the Mandelbrotset in the standard way:

        zn = zn-1 * zn-1 + c,

the previous 2 values of z are used:

        zn = zn-1 * zn-2 + c.

This is similar to the Fibonaccisequence, wherein each subsequent number is the sum of the previous 2:

        Fn = Fn-1 + Fn-2.

The standard Fibonacci sequence isinitialized with 1, 1, and the Fibonacci Mandelbrot formula initializes2 z variables to the same value before beginning iterating.  Fromthere, the new z is found from the product of the previous 2, and the previousz values are updated.  The Julia variation works similarly, initializingboth z variables to the pixel value.  The "switch" feature togglesfrom Mandelbrot to Julia formula, and back.

Suggested coloring formulas

Anything that works well with the standard Mandelbrot and Julia formulas should work fine with the Fibonacci varieties.
 

Parameters

Sample images

Fibonacci Mandelbrot

fibonacci-mandelbrot { ; copyright Kerry Mitchell 06dec2000
fractal:
  title="Fibonacci Mandelbrot" width=640 height=480 author="Kerry Mitchell"
  created="December 6, 2000" numlayers=1
layer:
  method=multipass caption="Layer 1" visible=yes alpha=no
mapping:
  center=-0.196016295020040017/0.988335576644839503
  magn=51.8280916861211811 angle=346.607221186724228
formula:
  filename="lkm.ufm" entry="fibonacci-mandelbrot" maxiter=1000
  percheck=off p_initialz=0/0 p_bailout=1000 p_order="2" p_weight1=1/0
  p_weight2=1/0 p_weight3=1/0 p_weight4=1/0
inside:
  transfer=none repeat=yes
outside:
  transfer=linear repeat=yes
gradient:
  smooth=yes numnodes=9 index=33 color=127 index=83 color=16638 index=133
  color=65535 index=134 color=327679 index=183 color=12582910 index=228
  color=16711551 index=283 color=16695296 index=333 color=16646144
  index=383 color=4128768
}

Fibonacci Julia

fibonacci-julia { ; copyright Kerry Mitchell 06dec2000
fractal:
  title="Fibonacci Julia" width=480 height=480 author="Kerry Mitchell"
  created="December 6, 2000" numlayers=1
layer:
  caption="Layer 1" visible=yes alpha=no
mapping:
  center=-0.12/0 magn=1.75 angle=270
formula:
  filename="lkm.ufm" entry="fibonacci-julia" maxiter=1000 percheck=off
  p_julparam=0.494204037933724129/0.0472742539328047258 p_bailout=1000
  p_order="3" p_weight1=1/0 p_weight2=1/0 p_weight3=1/0 p_weight4=1/0
inside:
  transfer=none repeat=yes
outside:
  filename="dmj.ucl" entry="dmj-Smooth" transfer=sqrt repeat=yes
  p_power=1.9/0 p_bailout=1000000
gradient:
  smooth=yes numnodes=8 index=14 color=11796660 index=64 color=14319578
  index=114 color=16777215 index=164 color=8355839 index=214 color=255
  index=264 color=127 index=314 color=0 index=364 color=5898330
}