Square Midget Tutorial

From Ultrafractal Wiki

Hi folks,

Some of you might enjoy this:

Contents 1 How to find square midgets 1.1 Step 1 1.2 Step 2 1.3 Step 3 1.4 Step 4 1.4.1 Comment From Kerry 1.5 Step 5 1.6 Step 6 1.7 Step 7 - The Square Midget 1.8 Elephant Valley Example 1.9 Update 1.10 A Tweak by Evie 1.11 Tweak Of Evie's Tweak 1.11.1 Toby Said 2 Embedded Squares 2.1 upr 3 See Also

How to find square midgets

Here is a mini-tutorial that describes a method of finding squarish midgets.

Step 1

1. Zoom in to the Elephant Valley on the West Midget. This is the largest Mandelbrot-like figure on the filament that extends to the left out of the main Mandelbrot Set. Elephant Valley is the indentation on the far right of the figure, which is bisected by the filament.

Copyright © 2008 by Toby Marshall

squareMidgetFig.1 {
fractal:
 title="square midget fig.1" width=1000 height=1000 layers=1
 resolution=200 credits="Computer;1/1/2002;Toby;3/17/2008"
layer:
 caption="Background" opacity=100 mergemode=overlay method=multipass
mapping:
 center=-1.748371483625/0.000151181879725 magn=243.75493 angle=-3.313
formula:
 maxiter=10000 percheck=off filename="Standard.ufm"
 entry="Mandelbrot" p_start=0/0 p_power=2/0 p_bailout=4
inside:
 density=.87 transfer=none
outside:
 density=.8 transfer=linear solid=4286722382 filename="Standard.ucl"
 entry="Basic" p_type=Iteration
gradient:
 linked=yes smooth=no rotation=-69 index=139 color=16777215 index=-93
 color=0 index=308 color=0
opacity:
 smooth=no rotation=-69 index=139 opacity=255 index=-93 opacity=255
 index=308 opacity=255
}

Step 2

Now go deeper still. You will see a main filament running down the middle, and long, slightly curved filaments coming out of the figures near the periphery where the inside of the set (solid black) becomes the outside of the set.

Copyright © 2008 by Toby Marshall

squareMidgetFig.2 {
fractal:
 title="square midget fig.2" width=1000 height=1000 layers=1
 resolution=200 credits="Computer;1/1/2002;Toby;3/17/2008"
layer:
 caption="Background" opacity=100 mergemode=overlay method=multipass
mapping:
 center=-1.749646212795/-5.1510483185e-6 magn=5299.0202 angle=-3.313
formula:
 maxiter=10000 percheck=off filename="Standard.ufm"
 entry="Mandelbrot" p_start=0/0 p_power=2/0 p_bailout=4
inside:
 density=.87 transfer=none
outside:
 density=.8 transfer=linear solid=4286722382 filename="Standard.ucl"
 entry="Basic" p_type=Iteration
gradient:
 linked=yes smooth=no rotation=-69 index=139 color=16777215 index=-93
 color=0 index=308 color=0
opacity:
 smooth=no rotation=-69 index=139 opacity=255 index=-93 opacity=255
 index=308 opacity=255
}

Step 3

Zoom in even more and you will see that at the source of each of the long filaments are two spirals that almost touch.

Copyright © 2008 by Toby Marshall

squareMidgetFig.3 {
fractal:
 title="square midget fig.3" width=1000 height=1000 layers=1
 resolution=200 credits="Computer;1/1/2002;Toby;3/17/2008"
layer:
 caption="Background" opacity=100 mergemode=overlay method=multipass
mapping:
 center=-1.749871249811115/0.0000180379897339836 magn=504092.48
 angle=-3.313
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 maxiter=10000 percheck=off filename="Standard.ufm"
 entry="Mandelbrot" p_start=0/0 p_power=2/0 p_bailout=4
inside:
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outside:
 density=.8 transfer=linear solid=4286722382 filename="Standard.ucl"
 entry="Basic" p_type=Iteration
gradient:
 linked=yes smooth=no rotation=-69 index=139 color=16777215 index=-93
 color=0 index=308 color=0
opacity:
 smooth=no rotation=-69 index=139 opacity=255 index=-93 opacity=255
 index=308 opacity=255
}

Step 4

Now zoom into the filament at the point where the two spirals are closest.

Copyright © 2008 by Toby Marshall

squareMidgetFig.4 {
fractal:
 title="square midget fig.4" width=1000 height=1000 layers=1
 resolution=200 credits="Computer;1/1/2002;Toby;3/17/2008"
layer:
 caption="Background" opacity=100 mergemode=overlay method=multipass
mapping:
 center=-1.749870775833644/0.000017438272291224235 magn=12602312
 angle=-3.313
formula:
 maxiter=10000 percheck=off filename="Standard.ufm"
 entry="Mandelbrot" p_start=0/0 p_power=2/0 p_bailout=4
inside:
 density=.87 transfer=none
outside:
 density=.8 transfer=linear solid=4286722382 filename="Standard.ucl"
 entry="Basic" p_type=Iteration
gradient:
 linked=yes smooth=no rotation=-69 index=139 color=16777215 index=-93
 color=0 index=308 color=0
opacity:
 smooth=no rotation=-69 index=139 opacity=255 index=-93 opacity=255
 index=308 opacity=255
}

You'll see that the filament has many small "islands" spaced across its length. Zoom in and frame one of those.

Comment From Kerry

Kerry Mitchell wrote:

"These islands are embedded Julia sets. Their shape is the shape of the Julia set that is taken from the same place on the main set, relative to where this point is to the West Midget. Here's the corresponding Julia set; you can see that it has the same essential structure (without the filaments, of course):" Embedded Julia Set

Copyright © 2008 by Toby Marshall

Step 5

squareMidgetFig.5 {
fractal:
 title="square midget fig.5" width=1000 height=1000 layers=1
 resolution=200 credits="Computer;1/1/2002;Toby;3/17/2008"
layer:
 caption="Background" opacity=100 mergemode=overlay method=multipass
mapping:
 center=-1.749870777747956968/0.000017403543960428476265
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formula:
 maxiter=10000 percheck=off filename="Standard.ufm"
 entry="Mandelbrot" p_start=0/0 p_power=2/0 p_bailout=4
inside:
 density=.87 transfer=none
outside:
 density=.8 transfer=linear solid=4286722382 filename="Standard.ucl"
 entry="Basic" p_type=Iteration
gradient:
 linked=yes smooth=no rotation=-69 index=139 color=16777215 index=-93
 color=0 index=308 color=0
opacity:
 smooth=no rotation=-69 index=139 opacity=255 index=-93 opacity=255
 index=308 opacity=255
}

Step 6

If you look right in the center you will see a tiny island at the intersection of the crossing of two filaments. Halfway from the center on either side toward the edges on the long axis you will see two similar little islands on crosses. Zoom into one of those.

Copyright © 2008 by Toby Marshall

squareMidgetFig.6 {
fractal:
 title="square midget fig.6" width=1000 height=1000 layers=1
 resolution=200 credits="Computer;1/1/2002;Toby;3/17/2008"
layer:
 caption="Background" opacity=100 mergemode=overlay method=multipass
mapping:
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formula:
 maxiter=10000 percheck=off filename="Standard.ufm"
 entry="Mandelbrot" p_start=0/0 p_power=2/0 p_bailout=4
inside:
 density=.87 transfer=none
outside:
 density=.8 transfer=linear solid=4286722382 filename="Standard.ucl"
 entry="Basic" p_type=Iteration
gradient:
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 color=0 index=308 color=0
opacity:
 smooth=no rotation=-69 index=139 opacity=255 index=-93 opacity=255
 index=308 opacity=255
}

Step 7 - The Square Midget

In the center, you will find a cute, square midget :-)

Copyright © 2008 by Toby Marshall

squareMidgetFig.7 {
fractal:
 title="square midget fig.7" width=1000 height=1000 layers=1
 resolution=200 credits="Computer;1/1/2002;Toby;3/17/2008"
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mapping:
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formula:
 maxiter=10000 percheck=off filename="Standard.ufm"
 entry="Mandelbrot" p_start=0/0 p_power=2/0 p_bailout=4
inside:
 density=.87 transfer=none
outside:
 density=.8 transfer=linear solid=4286722382 filename="Standard.ucl"
 entry="Basic" p_type=Iteration
gradient:
 linked=yes smooth=no rotation=-69 index=139 color=16777215 index=-93
 color=0 index=308 color=0
opacity:
 smooth=no rotation=-69 index=139 opacity=255 index=-93 opacity=255
 index=308 opacity=255
}

Elephant Valley Example

This works all down Elephant Valley of the West Midget--the deeper you go the more complex the structure will be, but it will always be square (or at least squarish). Look around as see what you can find ;-)

Here, from a similar "cross" within the island with the square midget at the center (fig.6) is a square within a square with a midget (in an octagon) at the center...

Copyright © 2008 by Toby Marshall

squareMidgetFig.8 {
fractal:
 title="square midget fig.8" width=1000 height=1000 layers=1
 resolution=200 credits="Computer;1/1/2002;Toby;3/17/2008"
layer:
 caption="Background" opacity=100 mergemode=overlay method=multipass
mapping:
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 05 magn=3.4478292E16 angle=-9.0236
formula:
 maxiter=10000 percheck=off filename="Standard.ufm"
 entry="Mandelbrot" p_start=0/0 p_power=2/0 p_bailout=4
inside:
 density=.87 transfer=none
outside:
 density=.8 transfer=linear solid=4286722382 filename="Standard.ucl"
 entry="Basic" p_type=Iteration
gradient:
 linked=yes smooth=no rotation=-69 index=139 color=16777215 index=-93
 color=0 index=308 color=0
opacity:
 smooth=no rotation=-69 index=139 opacity=255 index=-93 opacity=255
 index=308 opacity=255
}

Have fun and happy exploring!

Update

I've done a bit more exploring and found out something interesting. Forget the square midget tutorial. It appears that you can find a "squarish" midget in any of the embedded Julia sets in the West Midget, if you zoom into the same point.

The deal is: You always have an outer "ring" with eight spiral arms. If you zoom directly into the center of the embedded Julia (directly between the four spirals) you will find a midget in which this ring has the spiral arms all of equal size. If you move to the next similar meeting of four spirals and zoom into the middle of that, the ring around the midget will have four arms bigger than the other four, forming a somewhat "squarish" figure. If you continue moving out, but looking in the center wherever those four spirals come together, the four larger spiral arms in the ring around the midget keep getting longer and longer.

Because of the structure of the spiral arms, the squarest midgets are to be found in Elephant Valley, but you can find something "squarish" in any valley at the same point in any embedded Julia.

Toby

A Tweak by Evie

Another squarish seahorse, colored by multicolored thin orbit traps.

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Tweak Of Evie's Tweak

I just changed a few of Evie's selected coloring parameters and lowered the maximum interations from 10000 to 1000.

I'm finding a lot of interesting images as I zoom deeper.

Right now, as I write this, I've zoomed to

Layer1Location {
location:
  center=0.384687744880823643355/-0.1092016829478302542255
  magn=1.676729E12 angle=19.0261
}

and see lots of possibilities for further enhancement.

Bob Margolis

squarishSeahorse1_Evie+bobm {
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Toby Said

You gave me an idea. I went to Seahorse Valley to look for the same kind of squarish midget and found it. It appears that the squarish subset appears at the "choke point" of the two main spirals in any valley. Check this out:

squarishSeahorse_Evie+tma {
fractal:
 title="squarish seahorse_Evie+tma" width=600 height=600 layers=1
 resolution=200
 credits="Computer;1/13/2009;Eveline Berkman;1/12/2009"
layer:
 caption="Layer 1" opacity=100 method=multipass
mapping:
 center=-1.7685153505906796828125/0.00092266055203952201852342
 magn=6.0968649E12 angle=-18.8351
formula:
 maxiter=10000 filename="Standard.ufm" entry="FastMandel" p_start=0/0
 p_bailout=100
inside:
 transfer=none solid=4278224750
outside:
 density=.1 transfer=linear filename="Standard.ucl" entry="Smooth"
 p_power=2/0 p_bailout=1e12
gradient:
 comments="Simple grayscale gradient." smooth=no rotation=-100
 index=9 color=0 index=23 color=0 index=105 color=0 index=236
 color=15790320
opacity:
 smooth=no index=0 opacity=255
}

Embedded Squares

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.

Copyright © 2009 by Toby Marshall

upr

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
}

See Also

• Category:Midget example