Variations on a theme by gerrit

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Offline Spyke

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« Reply #30 on: May 22, 2019, 10:42:34 PM »
a = 5, center = -2, width = 8

All that is left is the main cardioid. Does is have any change of survival?
Earl Hinrichs, offering free opinions on everything.

Offline Spyke

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« Reply #31 on: May 22, 2019, 10:54:23 PM »
a = 4, center = -1, width = 12

The view is now zoomed out 3x so we can see the full antibrot. And for no reason other than I was getting tired of the previous palette, I switched to a 2D color scheme. The lines are not external angles or anything fancy. The 2D palette has two sets of concentric circles. One set transitions black and white, the other steps through silver / gold / blue colors.

The antibrot has shrunk in size. The right most corner point has become rounded and virtually all of the space-filling chaos is gone. The two corner points on the left are still a sharp angle, and retains the chaos.

Offline Spyke

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« Reply #32 on: May 22, 2019, 11:00:32 PM »
a = 3, center = -1, width = 6

The cardioid is fighting back, and putting a dent in the antibrot. And in the center on the left edge of the former triage, there is a hole and it is getting bigger.


Offline Spyke

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« Reply #33 on: May 22, 2019, 11:05:36 PM »
a = 2.5, center = 1, width = 12

There is definitely a crack in the east wall. It is sucking the air out of the antibrot. And the former cardioid appears to be leaking too.

Offline Spyke

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« Reply #34 on: May 22, 2019, 11:10:17 PM »
a = 2, center = -1, width = 12

Inside and outside become the same.

Offline Spyke

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« Reply #35 on: May 22, 2019, 11:25:01 PM »
a = 1, center = -1, width = 12

The two segments become shorter, and thinner. They are almost like small cracks in the universe. As a approaches 0, they continue to get smaller, more (relatively) separated and more linear. Like any particle and antiparticle pair, when they touch it leads to the total annihilation of both of them.

Offline Spyke

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« Reply #36 on: May 22, 2019, 11:30:09 PM »
a = 0, center = -1, width = 12 (not that it matters.)

Yes, I did just post a black rectangle. Think of it as a fractal interpretation of a John Cage symphony.

When a = 0, this formula become the ultimate super attractor. It degenerates to f(z,c) = 0.

Everything thing is swallowed by a black hole. And so ends this little story.

(or does it...?)

Offline gerrit

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« Reply #37 on: May 23, 2019, 03:41:44 AM »
Cool. Here's my little story based on the simple \( \frac{z^2+i}{z^2+1} + c \), which has some nice surprises.

You can start parameter space images at \( z_0=0 \)  or \( z_0=1+c \), the latter giving the orbit of critical point at infinity. They are different, but clearly made by the same artist, if we think of the formula in that capacity.

See images.

Offline gerrit

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« Reply #38 on: May 23, 2019, 03:44:50 AM »
Both parameter space images are similar and when zooming you encounter similar stuff, arranged slightly different.
I haven't found a way to combine the two critical orbit images in one, but either of them are good as catalog for Julia sets.

Below a zoom in the 0 orbit picture, and the corresponding Julia set.

Offline gerrit

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« Reply #39 on: May 23, 2019, 03:51:26 AM »
A rather complex filled Mandelbrot mini shape appears in the z=0 orbit (not in the z=inf orbit: there it's empty).

The warped mini lies in what appears at first sight to be random speckles. The mini is covered with shapes which we recognize as the filled Julia sets one would get if this was a real Mandelbrot set at those specific interior points. In addition the mini is covered with other minibrots (large bluish one in middle most prominent). Those, when zoomed into, do not have anything in the interior.

Finally, if we magnify one of the random speckles around the warped mini, we see each dot is in fact a full minibrot. Second image shows a random one. I scaled the colormap a bit to make it visible.

That little simple formula does a lot of work!
« Last Edit: May 23, 2019, 04:42:22 AM by gerrit »

Offline gerrit

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« Reply #40 on: May 23, 2019, 08:19:52 AM »
Here is a slight change to the formula we have been playing with:\[ \frac{z^2}{\frac{z^2}{a}+1}+c \] When a = \( \infty \) this is the Mandelbrot set. What happens when a is smaller?

a = 20, center = 0, width = 60

A tiny brot is facing off against the antibrot.
Interestingly if you make "a" bigger and bigger the triangle and mini move apart without limit.
If you do this with the other critical orbit it looks sort of the same but the minbrot has vanished.

I found this image for a= 6.71429-2.23809i. First critical point 0, second \( \infty \).

Offline gerrit

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« Reply #41 on: May 24, 2019, 08:26:13 PM »
\( \frac{z^2}{\frac{z^2}{a}+1}+c \), a = 8.33333 + 2.57143i, \( z_0=0 \).
\( z_0=\infty \) leaves just the noisy part (with empty minis, which are filled with interior filled Julia set shapes when using z0=0).

Offline gerrit

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« Reply #42 on: May 25, 2019, 12:52:29 AM »
Original formula a = 1+3i. Picture makes me squint against the glare, which is some sort of optical illusion, as my monitor is pretty dim.

Offline gerson

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« Reply #43 on: May 25, 2019, 01:28:53 AM »
Last one with Fractal Zoomer.
((z^2+(1+3i))/(z^2+1))+c
Bailout 5
iterations 200

Offline 3DickUlus

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« Reply #44 on: May 25, 2019, 02:15:50 AM »
Really nice work gerrit and spyke  :thumbs:

my math insight is terribly myopic but having fun with this, using Utak3r's orbit trap coloring.... implimented as...
Code: [Select]
z = cDiv(cPow(z, power)+A,cPow(z, power)+B)+C;... where A, B, C and power are user input
Fragmentarium is not a toy, it is a very versatile tool that can be used to make toys ;)

https://en.wikibooks.org/wiki/Fractals/fragmentarium


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