a modular Mandelbulb
A piece from the geometric view. A modular power 8 Mandelbulb consisting of 3 power 2 Mandelbulbs. Just a gimmick but with some nice features. And - just to mention - this system has eluded algebraists for 15 years.
I must admit, that I do not fully understand, why this system works. This was a shot in the dark, based on a vague idea, that turned out to be true.
The formula coders of the Mandelbulber team have noticed that system, so I am courious, what they come up wirth.
Further experiments show, that several Mandelbulb formula variations can be configured to a modular Mandelbulb, at least two of them show a proper power 8 M-set without any angular change. That means, that the angular alteration described on following pages, is maybe necessary in the used system, but is not elementary for the overlay.
In Mandelbulb 3d I can’t implement that system, because I can’t hinder a module to add global c. But the solo module can be built.
The M-set
The objective is simple. We build a power 8 Mandelbulb from three superimposed power two Mandelbulbs. One of them contains a right angle in an internal rotation (beta). Global c is added one time after the third power 2 Mandelbulb.Result is something very similar to a power 8 Mandelbulb.
top view
Similar we can say, because it has defined properties, that match the expectations of a power 8 Mandelbulb.
Mandelbulbs of power n have in top view n-1 “antennas”. Remember: The 2-dimensional Mandelbrot set has n-1=2-1=1 antenna. The modular Mandelbulb has seven antennas. That’s why we can be sure, that this is really similar to a power 8 Mandelbulb. By altering the power of Mandelbulb#1 you can change the power of the whole Mandelbulb. Some examples.
power 2.25, power2, power 2 creates a 2,25*2*2=power 9 Mandelbulb (counting in top view)
power 3, power 2, power 2 creates a 3*2*2=power 12 Mandelbulb
So at this point we can claim, that this seems to be a fundamental mathematic correlation in the modular Mandelbulb construction.
modular and native
Similar but not equal we must say, because the native power 8 Mandelbulb of the same formula base looks sightly different than the modular construction.
The parameter is somewhat optimized, so you can change between modular and native display with just a few clicks.
the modules
edit > I’ve got it more simple with only one module - the 90° 270° version. I will update ASAP
These are the two modules. Without a formal evaluation, one can determine in the experimental proof that a rotation of 90° for beta rotates the underlying Mandelbrot set by 180°. The Origin is the rotation center and is marked by a little orange dot.
ortho plantage
As an artist, my first interest is of course the artistic quality, the possibility of making something unique out of it. And the first thing that came to my mind was adding c in different places. Here c is added after the first slot and not the third. Result is a shape, quite unique for Mandelbulbs. And as the distance between two add c operations stays three slots, the “bulbish” character of the shape remains intact.
