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Author Topic: Exponentialish extensions of the mandelbrot set.  (Read 3518 times)
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Schlega
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« on: March 21, 2010, 09:35:08 PM »

This was inspired by Paolo's conjecture in the triplex algebra thread, combined with my inability to get anything but a black screen for exp(2*ln(z))+c using the triplex definitions.

Define exponentialish functions by

expish(x,y,z) = exp(x)*(cos(y)*f1(x,y,z), sin(y)*f2(x,y,z), f3(x,y,z)),
where f1(x,y,0) = f2(x,y,0) = 1 and f3(x,y,0) = 0.

Similarly, define logarithmish functions by
logish(x,y,z) = (ln(x^2+y^2) + g1(x,y,z), atan2(x+iy)+g2(x,y,z), g3(x,y,z),
where g1(x,y,0) = g2(x,y,0) = g3(x,y,0) = 0.

Here is what you get for expish(2*logish(z))+c for

f1(x,y,z) = f2(x,y,z) = cos(z), f3=sin(z):
g1 = ln(x^2+y^2+z^2) - ln(x^2+y^2), g2=0, g3 = asin(z/r).
<a href="http://www.youtube.com/v/q9ncR-Hbl4Y&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/q9ncR-Hbl4Y&rel=1&fs=1&hd=1</a>

I played with some other choices, but youtube doesn't seem to like them at the moment. I'll try again later.
« Last Edit: March 21, 2010, 10:04:00 PM by Schlega » Logged
kram1032
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« Reply #1 on: March 21, 2010, 09:40:08 PM »

pretty much 2D-Mset-ish...
But nice smiley
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Schlega
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« Reply #2 on: March 22, 2010, 06:46:47 AM »

The power 2 sets seem to be mostly flat for the functions I've tried:

<a href="http://www.youtube.com/v/Lf7_Bahlk-c&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/Lf7_Bahlk-c&rel=1&fs=1&hd=1</a>

<a href="http://www.youtube.com/v/CugpGf1s2qQ&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/CugpGf1s2qQ&rel=1&fs=1&hd=1</a>

Here's what the higher powers look like:

<a href="http://www.youtube.com/v/-dkrpbiw1xM&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/-dkrpbiw1xM&rel=1&fs=1&hd=1</a>


And just out of curiosity, I tried swapping sin(z) and cos(z). It doesn't reduce to the mandelbrot set on the xy plane, but it does look more familiar as a power 2 mandelbulb:
<a href="http://www.youtube.com/v/TE4Gvjf0W6s&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/TE4Gvjf0W6s&rel=1&fs=1&hd=1</a>
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Schlega
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« Reply #3 on: March 22, 2010, 09:05:56 AM »

This also gives a definition for triplex powers.

z(2,1,0):


z(2,1,1):


z(2,0,1):


z(8,1,0):


z(8,1,1):


z(8,0,1):
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Sockratease
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« Reply #4 on: March 22, 2010, 11:25:19 AM »

I really enjoyed the "higher powers" animation!

It's like watching a time-lapse video of a flower blooming    cheesy

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kram1032
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« Reply #5 on: March 22, 2010, 03:59:01 PM »

very nice animations and still and I have to agree with Sockratease smiley
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Schlega
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« Reply #6 on: March 23, 2010, 07:26:31 AM »

Thanks! Here's an animation of z(8,m,n)where m and n vary between 0 and 8:
<a href="http://www.youtube.com/v/38hLZIILeD4&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/38hLZIILeD4&rel=1&fs=1&hd=1</a>
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kram1032
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« Reply #7 on: March 23, 2010, 03:14:59 PM »

that's pretty nice cheesy
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Timeroot
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« Reply #8 on: March 24, 2010, 02:19:17 AM »

Sweeet!  :thumbsup1

After seeing those stills, I was just about to request a video of parameter space exploration of z^(a,b,c), and there it was when I scrolled down! Pre-request gratification is fun!  smiley
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Schlega
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« Reply #9 on: June 09, 2010, 01:06:15 AM »

Hey, I'm back. Here's what happens when you add a phase shift to the exponential:

<a href="http://www.youtube.com/v/BG-8tTq03io&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/BG-8tTq03io&rel=1&fs=1&hd=1</a>
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jehovajah
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« Reply #10 on: June 09, 2010, 12:03:53 PM »

Hi schlega. I am pretty excited by the form you have developed. The animations are a great tool to understanding what you have done. Again it is pretty cool that ideas sparked from this forum are at the cutting edge of visualisation of relativistic forms. I can't explain myself now but i will later when i have explored the notions of polynomial multiple operators. A simpler idea is the taylor expansion operator or the Fourier polynomial transforms .

What you are showing is something fractal woman said: our computing power has arrived at a place where the computaional error is not significant enough to mask the relationships we are exploring; i think she said we are close to plank length!

Good luck and good fortune in your explorations!
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Schlega
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« Reply #11 on: June 09, 2010, 01:43:53 PM »

Thanks, jehovajah. I must admit I don't know what relativistic forms are, but I like the sound of it.

Good luck in your own exploration as well!
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KRAFTWERK
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« Reply #12 on: June 10, 2010, 09:03:02 AM »

Very nice animations Schlega! Some seem to be totally new versions, but these two looks exactly like our familliar Positive z:



 and cosine mandelbulbs:



Wicked...  afro


* Bild 36.jpg (11.77 KB, 377x299 - viewed 461 times.)

* Bild 34.jpg (10.27 KB, 307x289 - viewed 435 times.)
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kram1032
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« Reply #13 on: June 10, 2010, 12:53:17 PM »

very interesting but when thinking about how the exponential is related to angular functions, it's somehow expected. smiley
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