Deep Zooms

From Ultrafractal Wiki

Toby Marshall, In a Post to the UF Mailing List

I've always loved diving deep, and this is a lot safer than with a scuba rig...I hope no one minds that I've gone a bit extra-curricular, and would like to share some thoughts and observations.

Note: These are all in UF4 Parameters in \MyFormulas\MandalaPossibilities

Toby

Elephant Valley

One of the things that fascinates me about the Mandelbrot set is the fact that the symmetries become increasingly complex the deeper one goes.

I mean, we usually think that things in the macro world are more complex than those in the micro: a human is more complex than a bug, which is more complex than a bacterium which is more complex than a virus. But things are reversed in the Mandelbrot set.

Elephant Midget 1

These first two midgets with their associated structures illustrate this: Number 1 is much closer to the "surface";

elephant_midget_1_toby {
fractal:
title="elephant_midget_1_toby" width=500 height=400 layers=1
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p_start=0/0 p_power=2/0 p_bailout=1e20
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Elephant Midget 2

Number 2 is seven orders of magnitude higher in magnification. They are neighbors in terms of X/Y position, which is why the fractal structures are similar, but note how much more complex the latter is than the former, in terms of layers of symmetry (around the midget before the ring becomes reasonably round) and the multiplication of the symmetrical elements.

elephant_midget_2_toby {
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title="elephant_midget_2_toby" width=500 height=400 layers=1
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Elephant Midget 3

I took the opportunity to go deeper into this valley than I ever have before. Render times eventually become unbearably long: although the magnifications are not particularly high, one needs to set the number of iterations much higher than when one is more shallow. I am not sure of the reason, but this last ufr from Elephant Valley needed 1,000,000 iterations to completely color the outside points, even though the magnification is 100x less than #1, which was satisfied with 5,000. One thing that is fascinating is that the deeper into the valley one descends, the more the spirals "unwind", and the elements and spaces between the elements seem to become more chaotic and irregular. Compare this upr with the previous, deliciously symmetrical ones above.

elephant_midget_3_toby {
fractal:
title="elephant_midget_3_toby" width=500 height=400 layers=1
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}

Seahorse Valley

Some similar observations in Seahorse Valley.

Seahorse Midget 1

Upr #1 and #2 are analogues to the first two in Elephant Valley.

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fractal:
title="seahorse_midget_1_toby" width=500 height=400 layers=1
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Seahorse Midget 2

Number 2 being in a similar position but with much deeper magnification.

seahorse_midget_2_toby {
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Seahorse Midget 3

Moving down the valley, which is actually a river (it is connected, except that the exact middle appears to be simply a point), the wonderful spirals begin to unwind. #3 is quite near the midpoint, again without much magnification but with an extremely high iteration count (10,000,000). The elements repeat but are actually quite different in size--not at all symmetrical in that aspect.

seahorse_midget_3_toby {
fractal:
title="seahorse_midget_3_toby" width=500 height=400 layers=1
resolution=200 credits="Toby;3/3/2008"
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Seahorse Midget 4

Raising the magnification for a smaller midget in the same area (#4), we find more complexity and symmetry of the elements, but again without the tightly coiled, thin-armed spirals characteristic of points higher up the arms (#1 and 2).

seahorse_midget_4_toby {
fractal:
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Seahorse Midget 5

Number 5 is even a bit closer to the midpoint, and again one can see that the spirals are quite relaxed and somewhat linear, if you have a bit of imagination you can see the angels appearing...:

seahorse_midget_5_toby {
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Seahorse Midget 6

Number 6 is about as close to the midpoint as my poor notebook can reasonably get me, and it is getting quite clear that while complex, the spirals have become almost pendulous, again with chaotic content widening and taking up more area.

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Sceptre Valley

I love this spot for the dynamism of the elements. The deeper you go into the coil of a spiral for your midget, the more "wraps" you will have as these arms get longer and longer. That holds true AFAIK throughout the Mandelbrot set.

Sceptre Midget 1

Number 1 is just pretty to my eye.

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Sceptre Midget 2

I like the flowing tails of number 2.

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Sceptre Midget 3

Number 3 is a much smaller cousin to number 2; Note again the similarity of the elements but their increased complexity.

I think it is important to realize the power of our fractal-viewing tools: the magnification of number 3 is 8E24.

Let's put that in physical terms: If the main body of the Mandelbrot set were expanded so that the distance from one end to the other were the distance from the earth to the sun, the midget in this image would be only about twice the diameter of an atomic nucleus, and about 5000 times smaller than the diameter of an atom including the electron orbits.

If that doesn't boggle your mind, nothing will!

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To Infinity and Beyond...

One of my favorite diving spots is out past the main set, straight out the front of the set to the left.

Bud 1

You will find a number of smaller Mandelbrot analogues down that line. #1 is directly on it, just behind the first of the two larger mini-Mandelbrots in its Elephant valley, which the line bisects. You'll find filaments everywhere out here, and it is easy to find midgets sitting on all of them, just waiting to be explored.

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Bud 2

Now we have #2 which is deeper than #1, and just off the center line, so we begin to get a bit of twist to the elements (I've moved the gradient to better shade them).

There is a lovely sobriety to the fractal structures out here:

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Bud 3

And finally, to go out with a bang, I've gone deeper than I've ever gone before anywhere on the set, in a position just near #2. To go back to my earlier analogy, if the length of the Mandelbrot set were the distance from the earth to the sun, the length of this midget would be about 1/100th the diameter of an atomic nucleus.... "(8-o

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