Thank you so much for doing this. I just did a quick look at the formulas, and these fractals actually look more like what you'd expect from a 3rd order fractal than HPDZ's cubic ship. I had another look at HPDZ's webpage, and I think this information will be quite helpful.

Introduction

The "Burning Ship" fractal is a spectacularly beautiful variation on the Mandelbrot set invented by Michael Michelitsch and Otto E. Rössler way back in 1992, consisting of a small modification to the Mandelbrot set's formula (see below).

A seemingly small change in the formula makes a striking difference in the appearance of the fractal!! Images of this thing are incredibly beautiful and radically different from any "analytic" fractals (my term). The non-analytic part is the absolute value, which takes away a lot of the curviness and turns it into angles and lines.

This fractal is absolutely seductive. Every time I zoom into a new area, I want to render it at some insane resolution with 25X noise reduction. It just compels enormity of implementation, it is so beautiful.

Technical Details

The formula for the Burning Ship fractal is very similar to the formula for the Mandelbrot set. Instead of iterating

z = z2 + c

which expands to

z = [Re(z) + i Im(z)]2 + c

which generates the Mandelbrot set, the Burning Ship fractal is obtained by iterating

z = [|Re(z)| + i |Im(z)|]2 + c

The difference, in case it's not clear from the 1-pixel wide lines in the above equation, is the absolute values of the real and imaginary components are used in the Burning Ship fractal.

A small detail to note: Traditionally, the Burning Ship fractal is drawn with at least the y-axis inverted (i.e. negative numbers at the top), so that it looks like a ship. Often, the x-axis is reversed as well, so the ship appears to be heading to the left rather than to the right. All images here have only the y-axis reversed.

Source:

http://www.hpdz.net/StillImages/BurningShip.htmI spent some time reading over the documents on HPDZ's website the other night, specifically the Cubic Burning Ship, and at first it wasn't clear to me whether or not the abs() function should be applied before or after taking the square. You seemed to apply the abs() function after the square, while HPDZ applied the abs() function before.

For your square burning ship plugin, you used:

Z_{1} = real(Z^2) + abs(imag(Z^2))i + C

whereas HPDZ used:

Z_{1} = (abs(real(Z))+ abs(imag(Z))i)^2 + C

HPDZ chose to flip the formula vertically; you chose to rotate the formula 180 degrees by subtracting C rather than adding. For the purpose of analysis, I will be ignoring flips/rotations. Examining the Burning Ship formula used by you, the abs() function is applied to the imaginary part of Z^2. This isolates Z^2 to complex quadrants I and IV on the number plane, keeping Z^2 above the x-axis. Examining the formula used by HPDZ, the abs() function is applied to both the imaginary and real parts of Z

_{0}, prior to the squaring. This restricts Z

_{0} to quadrant I on the complex plane, however after the squaring operation is completed, Z^2 can exist either in quadrants I or IV. So, without using complex algebra to develop a rigorous proof, it seems plausible that your implementation of the Burning Ship and HPDZ's implementation of the Burning Ship may very well be congruent to each other, based upon the possible range of values for Z^2.

That being said, we have developed two very different approaches to creating a Cubic equivalent of the Burning Ship. Having seen the results of the render tests, it seems likely that HPDZ applied the abs() function prior to cubing Z

_{0}, whereas with our approach, the abs() function was applied after the cube. Applying the abs() function to both the imaginary and real parts of Z

_{0} will place the final result for Z^3 in quadrants I, IV, or III. Performing the abs() function only to the imaginary part afterwords will place the final result for Z^3 in quadrants I or IV. On a positive note, we have created a new 3rd order plugin that actually looks and feels even more like a third order fractal than the original formulas. The Cubic Buffalo fractal seems especially promising! These are packed with fractal details just like the 2nd order formulas. I'm not entirely sure how to apply the 3rd order Celtic, but I'm pretty sure now that the 3rd order Burning Ship HPDZ used is:

Z_{1} = (abs(real(Z))+ abs(imag(Z))i)^3 + C

or at least that seems highly plausible.

I have attached sample renders showcasing the differences between UF5 and Panzerboy's Cubic FX plugin. Both fractal types (sans rotation) are the identical for 2nd order. UF5 doesn't have a plugin named "Buffalo" in the public formulas library so I don't have one to compare to, but the "Flying Squirrel" mistake and the cubic Buffalo variant in Panzerboy's plugin look especially promising.

Update: I had t return t- school today for spring quarter, but not before I started a new zoom movie on my 8-core rendering machine. The Cubic Buffalo has two needles at right angles to each other, and the intricate details within the fractal have sweeping lines and right angles much like the Square Burning Ship and Buffalo fractals, which transform into beautiful kaleidoscope-like hexagon formations. Stunning! Expect a new Youtube upload sometime this coming weekend, for the Cubic Buffalo in glorious HD, well provided the render finishes, that is! Thanks again, PanzerBoy...

I also uploaded two videos the other day, as an example of what one can do with Panzerboy's original fxabsmandvar plugin:

http://www.fractalforums.com/movies-showcase-%28rate-my-movie%29/aztec-ruins/