3d complex triplex absolute modifications (correct 2d M-set cross-sections)

[M Benesi]

  So for this series, I was pondering the fact that we don't carry the signs of the magnitude for the imaginary component of the first part of the complex triplex:
complex_1 = x + sqrt ( y^2 + z^2) i  <---- no sign

  Now, this really doesn't effect the calculations of the 2d set, as can be seen visually (2d Mandelbrot is reflected over the x axis) or mathematically (I wrote a maxima formula to check the variations).  However, for the 3d set we end up with a variation when the y pixel component changes sign (for all even z^n: n= 2,4,6,8..... a positive y pixel component yields a different magnitude result from a negative y pixel component).

  So this got me thinking (again) about ways to correct the issue (absolute value anyone?), but I'd already done the absolute value of the pixel components and made a "corrected" 3d Mandelbrot (that was adequately detailed for z^2), so I decided to experiment with taking the absolute value of specific components (a lot like the Burning Ship fractal).

  I ended up making several interesting discoveries, including the fact that the regular 3d complex triplex gains TONS of details when you correctly apply +/- and absolute values to the various components.  Many absolute value/sign combinations are discontinuous, only certain ones work (the hints in my new formula set (under complex compound corrected version) tell you the main combos of values/signs that I found to be continuous).  

  Anyways, here are a few of the images I made recently.... I didn't even include the 7th order zoom series, which shows some really cool details...

4th order:

5th order:

5th order:

6th order:

6th order

8th order (full size here)

8th order, bluffs near the tower:

8th order spinners (full size here):


2nd order straight on large (view full size here):