Toby M:Embedded Squares
From Ultrafractal Wiki
Copyright 2009 Toby Marshall
Sorry to have been out of the loop. The internet connection in Manila was too expensive and too slow...
As I mentioned, and others elaborated upon, you can find a squarish midget by going into one of the embedded Julia sets in the West Midget and zooming into the center of one of the first offset Julia sets on either side of the center.
If you zoom into the center, between the four spirals, you will simply find a midget, surrounded by 4-, then 8-, then 16-fold symmetry, etc.
However if you zoom into the first offset on either side (with a similar pattern to the center) you will find an embedded Julia set between the four spirals. Zooming into the center of that set is where you find the "square" midget, in which the first 4-fold symmetry is squarish, the 8-fold symmetry octagon-like, etc.
BUT
If after zooming into the first offset to find the Julia embedded there, you then again zoom into the first offset in that Julia, and zoom into the center of that, you will find first a square 4-fold symmetry containing another square 4-fold symmetry, after which the standard doubling of symmetry begins.
I'm posting a upr in which I zoomed into the first offset in the Julia sets three times as I zoomed in. It has three embedded "squares" before progressing towards the midget.
Toby
embeddedSquares {
fractal:
title="embedded squares" width=600 height=600 layers=1
resolution=200
credits="Computer;1/21/2009;Eveline Berkman;1/12/2009"
layer:
caption="Layer 1" opacity=100 method=multipass
mapping:
center=-1.74979850112456365862710524948/-0.0000342952540473984089169\
3327040465 magn=7.431054E19 angle=29.9441
formula:
maxiter=10000 filename="Standard.ufm" entry="FastMandel" p_start=0/0
p_bailout=100
inside:
transfer=none solid=4278224750
outside:
density=.12 transfer=linear filename="Standard.ucl" entry="Smooth"
p_power=2/0 p_bailout=1e12
gradient:
comments="Simple grayscale gradient." smooth=no rotation=-136
index=67 color=0 index=103 color=0 index=194 color=16777215
opacity:
smooth=no index=0 opacity=255
}
