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A thought for 3D Mandelbrot - extend standard trig to 3D using complex ratios for a "2D" angle ?

Cos(a) = (y + i*z) / h  ; maybe change sign of i*z
Sin(a) = (x + i*z) / h  ; maybe change sign of i*z

For positive signs of i*z then:

h = sqrt((y+i*z)^2+(x+i*z)^2) = sqrt(x^2+y^2-2*z^2+ 2*z*(x+y)*i)

Of course if z==0 then the above reduces to normal complex numbers and standard trig.

  Do you mean to:

theta= acos [(y + i*z) / h]
phi=    asin [(x + i*z) / h]

  with the regular arccos/arcsin functions?

No, I mean *one* *complex* angle theta such that theta=acos((y+i*z)/h)=asin((x+i*z)/h) where h is sqrt((y+i*z)^2+(x+i*z)^2) where acos() and asin() are the complex version of the inverse trig functions.
I have a suspicion that one of the "+ i*z" should be "- i*z" ;)

(Note that since complex numbers are a "field" then Pythagoras of course holds true in all cases)

Also complex theta = atan2((x+i*z),(y+i*z)).

The idea behind this is that to define a sphere only requires a (real) magnitude and two (real) angles - here this is changed to a magnitude and a single complex angle.
So to square our number (x,y,z) we do the usual i.e. square the magnitude and double the angle, but in this case the angle is complex and doubling it will "rotate" around more than one dimension ;)
Obviously multiplication can be acheived in a similar comparable manner, i.e. the product of the magnitudes and the sum of the (complex) angles.

Another point is that the above gives us a trilplex form than can be extended upward to R4,R5,R6.....Rn using a similar method - unlike the quaternion and hypercomplex methods which only give us Rn where n=2^x (x integer >=2).

OK, missed something obvious as usual - of course after doubling the angle (to give a+i*b) and returning acos(a+i*b) and asin(a+i*b) that will give us x+i*z1 and y+i*z2 ! i.e. two values for z.

I guess a fudge is one solution :) e.g. z = (z1+z2)/2 !

The other obvious idea is go to R4 (e.g. (x,y,z,w)) and use the complex angle as atan2(z+i*w,x+i*y) such that after angle doubling our new R4 value is (acos(a+i*b),asin(a+i*b)), but I'm guessing that's just bi-complex again ?
And of course puts us back to method than only gives Rn where n=2^x (x integer>=2).

Just tried the 4D version i.e. a complex angle plus complex pythagoras - it's most definitely *not* an alternative way of describing standard bicomplex.
It wasn't that interesting until I started playing with the signs - I suggest folks so inclined try this 4D version of z^2+c, obviously just convert the complex calculations into the real equivalents if you don't have complex as a type.

      r = x*x + y*y + z*z + w*w
      complex xy = x + flip(y)
      complex zw = z + flip(w)
      complex ccs = 1.0/sqrt(sqr(conj(xy)) + sqr(conj(zw)))
      complex csn = zw*ccs
      ccs = xy*ccs
      xy = r*(sqr(ccs)-sqr(csn))
      zw = 2.0*r*ccs*csn
      x = real(xy) + cx
      y = imag(xy) + cy
      z = real(zw) + cz
      w = imag(zw) + cw

For DE the standard method for quaternionic z^2+c should work OK.
I just rendered it using my really old UF code (in mmf.ufm) so it's not worth posting one of my own renders at the moment.
It's very reminiscent of the Mandelbar but in 3D, so I think further investigations into variations on the theme may prove fruitful ;)

Well one cross section is a 2d mandy, more or less, the other the mandelbar.

  I messed around with absolute values a bit, got the outline of the good old burning ship (more or less), and some emerging structure near the "flames", but nothing great (at first glance).  But... for the most part, it was one of those stretchy stringy ones (at least the varieties I played with).