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Author Topic: Tricomplex Numbers  (Read 2683 times)
Description: Using Tricomplex Numbers for Fractals
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JVillella
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« on: January 26, 2012, 12:17:49 AM »

Hello.
Does anyone know if using a tricomplex number system when calculating the Mandelbrot set would produce any interesting results? Is there any "proof-of-concept" pictures out there showing this?

Thanks, and have a wonderful day smiley
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cKleinhuis
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« Reply #1 on: January 26, 2012, 12:52:34 AM »

very good question, the answer is: yes! we know, and it does produce interesting results, check out the mandelbulb triplex
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Aexion
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« Reply #2 on: January 26, 2012, 01:21:06 AM »

Hmm..
If I understand well, the Tricomplex isn't the Triplex  (as far as I know, but I can be mistaken)...
Tricomplex:
http://en.wikipedia.org/wiki/Tricomplex_number
http://books.google.com/books?id=vv_m6Dn3tM0C&lpg=PA19&ots=MrTDonAl7S&pg=PA19%7CSection#v=onepage&q&f=false

Triplex:
http://www.fractalforums.com/theory/triplex-algebra/

BTW: If anyone want a lot of number systems to test, just check the index of the book that I have linked.. 
 
« Last Edit: January 26, 2012, 01:39:06 AM by Aexion » Logged

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Syntopia
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« Reply #3 on: January 26, 2012, 09:12:55 AM »

yes, the tricomplex numbers and triplex numbers are very different.

I tried the tricomplex numbers a while ago, and did not find anything interesting (just extruded Mandelbrots/Julias - see attached image).

But I only used the multiplication operator - for tricomplex numbers, there is also an unusual modulus operator, which probably should be used instead of the Euclidean length. I tried this, but couldn't get anything but noise with my analytical DE - but perhaps a brute force approach would work here.

Example pic:



* o.jpg (36.47 KB, 640x499 - viewed 286 times.)
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DarkBeam
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« Reply #4 on: January 26, 2012, 09:15:50 AM »

Oh yes! The tricomplex modulus grin ... It makes appear spikes everywhere, don't ever try it wink
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JVillella
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« Reply #5 on: January 26, 2012, 06:10:13 PM »

I tried using tricomplex numbers with my mandelbrot set and I got the exact same geometry as Syntopia. However, when I paired them with the Julia Set I got nothing. No image at all...

The fact that tricomplex numbers are so boring in comparison to triplex and quaternions, does this tell us anything about the underlying or underpinning mathematics behind these number systems? Slightly off topic, but I was wondering about extending complex numbers to three dimensional space. Is tricomplex numbers it. Is that the way? Or is triplex numbers what we want to be using (Some of the mathematics of triplex numbers look a little "merky" to me and it seems it was created more with the focus of beautiful images instead of mathematical beauty).

What about what William Hamilton said...we need a fourth dimension (k) which we take a slice of (3D). So does that mean true hypercomplex 3D numbers are not possible? If so...why?

And I think the biggest question I have is... If we actually extend complex numbers to 3D space how will we know it is correct. And how will we know what a 3D mandelbrot looks like? What criteria do we use? Shouldn't the way to compute the 3D mandelbrot be synonymous with the 2D way? After all its the same shape. So changing its method of formation would not really be 100% analogous. or... Am I just completely off the mark embarrass

Thanks everyone. I am really looking forward to get my questions answered smiley

Farewells,
Julian V.
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Alef
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« Reply #6 on: January 29, 2012, 02:46:52 PM »

I tried to use tricomplex numbers, square (A, Bi, Cj) and got something barely usable. But it could be, that I had fewm mistakes in calculations.

Maybe must do this aproach: http://www.fractalforums.com/new-theories-and-research/mathematical-theory-for-a-'true'-3d-mandelbrot/

Quote
This page has been deleted. The deletion and move log for the page are provided below for reference.

06:55, 29 January 2012 Lankiveil (talk | contribs) deleted "Tricomplex number" ‎ (Wikipedia:Articles for deletion/Tricomplex number)

EDITED:
I just looked throught formula generating this brot, and certainly algorithm is with bug. There should be two x, y and z variables;)


* Tricomplex_pow2.jpg (18.38 KB, 320x240 - viewed 702 times.)
« Last Edit: February 01, 2012, 08:04:37 PM by Asdam » Logged

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JVillella
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« Reply #7 on: January 31, 2012, 12:22:38 AM »

Interesting. Like Asdam quoted the Wikipedia page on tricomplex numbers was taken down.
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Aexion
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« Reply #8 on: January 31, 2012, 01:25:16 AM »

Here's why the page of the Tricomplex algebra was taken down (its interesting to read):
http://en.wikipedia.org/wiki/Wikipedia:Articles_for_deletion/Tricomplex_number
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« Reply #9 on: February 08, 2012, 07:12:30 PM »

I saved the page on desktop. But experimenting with numbers, it looks that extruded property is generated by ixj=1.

If you do (Ar,Bi,Cj)x(Ar,Bi,Cj), in square power ixj is just single part 2*zy*zz. So if 2*zy*zz are in formula for x, very probably that fractal would be extruded.
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