I found some combined burning ships in the cubic burning ship:
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Are these common? Are they special? I've never seen any combined mini-set in a fractal this way.
Yeah, with the abs() functions, the sets become mirrored on itself. You can often find Minis in close proximity to each other at moderate zoom levels. When Panzerboy released his absmandvars plugin for Fractal Extreme, one of my first zooms into the needle of the 2nd order Buffalo produced this little formation, at a "moderate" zoom depth of 2^179!
http://www.fractalforums.com/fractal-exteme/fractal-extreme/msg42568/Despite having needles which is very uncharacteristic of odd-order fractals, the 3rd order BS is still very much third order. It contains rotational symmetries of 2 and 6 sided shapes when you zoom in enough, often producing ornate snowflake like patterns, and during deep zoom sequences, minis appear 50% deeper than your last detour rather than twice as deep like with all 2nd order fractals. At shallow zoom levels this causes the minis to appear larger than in the 2nd order abs() fractals, so much so that they often collide with each other. In the cubic Buffalo, as well as Burning Ship, the minis will often get stacked together at shallow depths.
I took some measurements of the set. The three needles of the Cubic Burning ship terminate exactly √2 (square root) distance from the origin, with the diagonal having absolute coordinates of ±1 on both axes. Even Mandelbrots always have needles of legnth (n-1) root of 2. For instance the westward needle of the 2nd order Mandelbrot is the first root of 2, which is located at (duh) -2. 4th order mandelbrot needle terminates at the negative cube root of 2, so the math checks out. Again despite being an odd powered fractal, the presence of abs commands in the formula parameters will create needles due to the even symmetries the reflections create.
The buffalo fractal (2nd or 3rd order) has two abs() functions applied to the entire formula. This confines the resultant exponent to the upper right quadrant always, so reflections occur across both axes. If one of these reflection planes happens to pass in close proximity of a mini, the mini will be reflected as well. The Burning Ship and Celtic fractals have one plane of symmetry in the 2nd order sets (For clarity, the reflections I'm referring to occur in the orbitals, not the shape of the resultant fractal itself. During Zoom sequences, the Burning Ship reflects objects across the X axis while Celtic reflects objects across the Y axis, ultimately deleting the other half of the pattern. This property can be used constructively to customize the levels of complexity during a zoom sequence, much like repeating patterns can be generated in the base 2 Mandelbrot by selective zooming), but I'm still figuring out the symmetries of the Cubic Burning Ship.
Specifically with the Burning Ship fractal, the abs() command is applied to both the real and imaginary components prior to the exponent. The dual abs() commands confine the initial complex value to the upper right quadrant. In the 2nd order Burning Ship, this results in complex coordinates within the upper half of the complex plane after squaring. As a result, you won't find minis in this area (remember the BS fractal is normally rendered flipped vertically). In the 3rd order Burning Ship, the coordinates can exist in one of three quadrants, lower right being the exception. As a result, minis can exist anywhere in the fractal except the upper left quadrant of the fractal behind the twin masts (assuming the fractal is not rendered flipped or rotated).
The reflections in the The Cubic Burning ship appears to have some sort of quasi 3/4 reflective symmetries which ultimately get divided by symmetry lines into forths and twelths due to 3rd order periodicity. Somehow this 270 degree symmetry also generates a third needle equidistant and diagonally offest by 135 degrees to the other two. Being 3rd order, this ultimately creates rotation symmetry of modulo 2, 6, 18, and so on during a zoom sequence. The masts in the 3rd order Burning ship show similarities to the 2nd order BS being assymetrical along the orthagonal needles, and also shares similar properties to the 2nd order Celtic, being perfectly symmetric along the diagonal needle. As an added bonus, the third order Burning Ship also has less "rats nest" material outside of the mast areas compared to the 2nd order version, with even more options ripe for exploration.
Sadly (and this is only my humble opinion) the 2nd and 3rd order Burning Ship fractals are where the amazing symmetry ends. Starting with 4th order and up, the Burning Ship fractals begin to get chaotic and disorganized, with niether the beautifully ornate masts of the 2nd order nor the amazing symmetries contained within the 3rd order. Also like with any progressively higher Mandelbrot powers, fractals eventually tend to get "blobby". That's why I'm not insisting on arbitrary powers. The attempt to expand into a generic arbitrary powers formulation with Kalles Fraktaler might have been why the previous attempt failed. If you can just get the 3rd order BS working alongside the 2nd order, I'll be a very happy camper!
EDIT: Man, I'm becoming like a spokesperson for this formula, or something...