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Author Topic: Re: Triplex 1/z  (Read 2059 times)
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Kali
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« on: March 26, 2012, 07:05:23 PM »

I need some help on how to implement z-1 (or 1/z) using triplex, I'm trying to render a formula and it requires this operation (z+1/z+c).

In complex, 1/z = conjugate(z)/|z|2 - I tried in 3D with conjugate(z)=z.x-z.y-z.z but the result is more like a quaternion (the "revolved brot" one)

I'll appreciate any clues on how to do it with tricomplex system.

Kali

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DarkBeam
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Fragments of the fractal -like the tip of it


« Reply #1 on: March 26, 2012, 07:09:32 PM »

I need some help on how to implement z-1 (or 1/z) using triplex, I'm trying to render a formula and it requires this operation (z+1/z+c).

In complex, 1/z = conjugate(z)/|z|2 - I tried in 3D with conjugate(z)=z.x-z.y-z.z but the result is more like a quaternion (the "revolved brot" one)

I'll appreciate any clues on how to do it with tricomplex system.

Kali

Me too, but I think it's impossible keeping it coherent with W-N standard theory howdy
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kram1032
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« Reply #2 on: March 26, 2012, 07:48:51 PM »

Divisions have been defined...
According to bugmans's original formulation:

x/y=[x1,x2,x3]/[y1,y2,y3] = [x1,x2,x3]*[y1,y2,y3]-1 = 1/Ry2[(x1y1+x2y2)(1+(x3y3)/(rxry)),(x2y1-x1y2)(1+(x3y3)/(rxry)),ryx3-rxy3]

with

Ry = Sqrt(y12+y22+y32)
rx = Sqrt(x12+x22)
ry = Sqrt(y12+y22)

By putting x=(1,0,0), you should get exactly what you want.
« Last Edit: March 26, 2012, 09:19:39 PM by kram1032 » Logged
Kali
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« Reply #3 on: March 26, 2012, 09:00:59 PM »

By putting x=(1,0,0), you should get exactly what you want.

Thanks.

If I get it right it should simplify to: 1/Ry2[y1,-y2/(rx.ry),-rx.y3)]

Also rx=Sqrt(1)

I tried it with the formula I mentioned (z+1/z+c with initial z=1) and something that could be interesting emerged... but now I have to deal with the DE angry
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kram1032
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« Reply #4 on: March 26, 2012, 09:21:25 PM »

well... not quite, I did a small mistake with the ()s...
It would be:

1/Ry2 [y1,-y2,-y3]

Actually kind of a weird inverse. But I guess, it's a weird algebra.
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Kali
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« Reply #5 on: March 26, 2012, 10:12:35 PM »

well... not quite, I did a small mistake with the ()s...

Oh, that's the cause of the DE going crazy  smiley

It would be:

1/Ry2 [y1,-y2,-y3]

So... it's the same as I first posted:

In complex, 1/z = conjugate(z)/|z|2 - I tried in 3D with conjugate(z)=z.x-z.y-z.z but the result is more like a quaternion (the "revolved brot" one)

conjugate(z)/|z|2 = 1/|z|2 [z.x,-z.y,-z.z]  grin


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DarkBeam
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« Reply #6 on: March 26, 2012, 11:46:38 PM »

not a good inversion trust me wink
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kram1032
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« Reply #7 on: March 27, 2012, 09:38:53 AM »

Well, it's the inversion that would make sense...
You could aribitarily define something else but whatever you do, it wouldn't work as expected, most likely...

If you just take the pseudo-inverse of a standard vector (x,y,z) , what you get is 1/r (x,y,z)... Not very exciting either but maybe that works better for you?
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DarkBeam
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Fragments of the fractal -like the tip of it


« Reply #8 on: March 27, 2012, 10:12:59 AM »

That's ok, it's coherent with the theory but the resulting fractals are not very good. That's what I meant with "impossible"

Probably the best thing should be to study a new number set that works good with neg powers. shocked
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Kali
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« Reply #9 on: March 27, 2012, 10:39:06 AM »

I was trying to do a 3D version of this brot (z+1/z+c), only works with initial z=1

See attached image... I was curious on how it could look as a Mandelbulb but no good results yet... with the previous methods only a revolved brot around the x-axis,

some more ideas anyone?



* Fractal1.jpg (55.61 KB, 800x469 - viewed 341 times.)
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DarkBeam
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Fragments of the fractal -like the tip of it


« Reply #10 on: March 27, 2012, 10:45:21 AM »

That is Talis formula, already implemented in Mb "ages" ago, not good in 3D wink
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Kali
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« Reply #11 on: March 27, 2012, 11:14:14 AM »

That is Talis formula, already implemented in Mb "ages" ago, not good in 3D wink

Oh, I didn't know it was called Talis, and I don't even recall where I first saw this formula, but I think it was on fracmonk's megathread.

But I won't give up just because your "ages-ago" MB3D implementation is not good grin

Btw, maybe is a DE problem? in Fragmentarium, in order to render the "revolved brot" I mentioned, I must set the default mandelbulb DE to 0,5 pow and scale it by 4. Don't ask me why, you know, I'm more an intuitive "brute-force" fractalist rather than a mathematician  grin
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cKleinhuis
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formerly known as 'Trifox'


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« Reply #12 on: March 27, 2012, 12:38:56 PM »

is this really related to triplex math ? if not please consider opening up a new thread...
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Kali
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« Reply #13 on: March 27, 2012, 07:41:17 PM »

is this really related to triplex math ? if not please consider opening up a new thread...

It was at first, but it didn't work, and now we are a bit off-topic I guess... will open a new thread for related posts, sorry Chris.
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cKleinhuis
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formerly known as 'Trifox'


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« Reply #14 on: March 27, 2012, 10:05:45 PM »

i or you can split this off, i would like to lock this thread because it has really valuable information in it ...!
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