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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).
http://www.youtube.com/watch?v=q9ncR-Hbl4Y

I played with some other choices, but youtube doesn't seem to like them at the moment. I'll try again later.

pretty much 2D-Mset-ish...
But nice :)

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

http://www.youtube.com/watch?v=Lf7_Bahlk-c

http://www.youtube.com/watch?v=CugpGf1s2qQ

Here's what the higher powers look like:

http://www.youtube.com/watch?v=-dkrpbiw1xM


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:
http://www.youtube.com/watch?v=TE4Gvjf0W6s

This also gives a definition for triplex powers.

z^(2,1,0):
(http://i624.photobucket.com/albums/tt323/schlega1/pow2-1-0.png)

z^(2,1,1):
(http://i624.photobucket.com/albums/tt323/schlega1/pow2-1-1.png)

z^(2,0,1):
(http://i624.photobucket.com/albums/tt323/schlega1/pow2-0-1.png)

z^(8,1,0):
(http://i624.photobucket.com/albums/tt323/schlega1/pow8-1-0.png)

z^(8,1,1):
(http://i624.photobucket.com/albums/tt323/schlega1/pow8-1-1.png)

z^(8,0,1):
(http://i624.photobucket.com/albums/tt323/schlega1/pow8-0-1.png)

I really enjoyed the "higher powers" animation!

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

very nice animations and still and I have to agree with Sockratease :)

Thanks! Here's an animation of z^(8,m,n)where m and n vary between 0 and 8:
http://www.youtube.com/watch?v=38hLZIILeD4

that's pretty nice :D

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!  :)

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

http://www.youtube.com/watch?v=BG-8tTq03io

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!

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!

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

(http://www.fractalforums.com/index.php?action=dlattach;topic=2081.0;attach=1023;image)

 and cosine mandelbulbs:
(http://www.fractalforums.com/index.php?action=dlattach;topic=2081.0;attach=2087;image)


Wicked...  O0

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