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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:

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/R_y²[(x₁y₁+x₂y₂)(1+(x₃y₃)/(r_xr_y)),(x₂y₁-x₁y₂)(1+(x₃y₃)/(r_xr_y)),r_yx₃-r_xy₃]

with

R_y = Sqrt(y1²+y2²+y3²)
r_x = Sqrt(x1²+x2²)
r_y = Sqrt(y1²+y2²)

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:

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

1/R_y² [y1,-y2,-y3]

Actually kind of a weird inverse. But I guess, it's a weird algebra.

Oh, that's the cause of the DE going crazy  :)


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


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

not a good inversion trust me ;)

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?

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. :o

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?

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

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 ;D

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  ;D

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.

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

Done. You can lock now...