Deep Zoom Julias
From Ultrafractal Wiki
Hi all.
As has been noted by Toby, Kerry and Morgen et al, zooming into midgets can be a fascinating pass time. One can get a feel for how simple combinatorial rules leads to an infinity of complicated variations of form.
Contents |
20090114aStart
I'm gonna start an exploration on an elephant valley style Julia off of a random midget and show how the presence of square midgets arises:
Copyright © 2008 by Stephen Francis Johnson
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Evolution1
The Julia has 2 fold rotation symmetry and if you go zoom deeper you get a structure with 4 then 8 then 16 rotation symmetry. Here is the 8 fold structure; a circle:
Copyright © 2008 by Stephen Francis Johnson
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Evolution2
If we head off to the second biggest midget from our staring point we see it is distorted and if we zoom in again we get the square. We have 4 extra "C scrolls" that are pushing out the corners:
Copyright © 2008 by Stephen Francis Johnson
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Evolution3
We will now go out toward the edge of our starting structure and see how the structure mutates further, we now have 8 extra "C scrolls" which are causing our corners to stick out:
Copyright © 2008 by Stephen Francis Johnson
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Evolution12
We can carry on taking steps out toward the edge of our staring point, adding extra "C scrolls" which leads to further distorted structures. Here is the "cross shape" which has formed with 12 steps toward the edge of start. There are 12x4=48 extra C scrolls:
Copyright © 2008 by Stephen Francis Johnson
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So a circle has evolved into a square and then into a cross shape with ever lengthening arms!
Evolution12Spiral
In the above sequence we have selected midgets by taking a path toward the edge of our starting structure. Here is what happens if we go down a spiral. Again this is twelve steps from the central midget. Instead of forming a cross the structure "twists"around:
Copyright © 2008 by Stephen Francis Johnson
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Evolution13Spiral3Turns
If we go around three more turns of the spiral and select a midget there, the structure "twists" around the center the corresponding number of times:
Copyright © 2008 by Stephen Francis Johnson
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Evolution14SpiralOval
More complex structures are obtained by doing sequences of jumps between midgets as you zoom. Here is a shape which I obtained by zooming into my twelve step spiral midget via a path of "sub midgets", building on the structure with each step (This zoom is at 1022 - may take a while to render):
Copyright © 2008 by Stephen Francis Johnson
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Evolution15SpiralSquare
Zooming into the center of this produces a more convoluted square! (mag 1022):
Copyright © 2008 by Stephen Francis Johnson
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The same priniples can be applied to any structure in the Mandelbrot set and related fractals. The following link shows variations that I found in a Julia type structure (this time embedded in a Nova Mandelbrot) see the following link:
I hope this has been informative.
Stephen
