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The classic 3D Mandelbulb has the equation:



where q denotes the triplex variable, in the same way that z is used for complex numbers.

As you are already aware, choosing a low order for the polynomial results in a 'whipped cream' effect, whereas a very high value results in an almost spherical fractal. Effectively, deciding on an exponent is a compromise.

A way forward is to combine multiple powers of q. For example, the Order 8 Mandelbulb and Order 12 Mandelbulb could be combined as follows:



To achieve infinite detail, an infinite number of Mandelbulbs should be combined. However, this has the disadvantage that it takes an infinite amount of time to calculate. Moreover, the series could diverge as more terms are added, so the result is not well-defined.


Fortunately, I have developed a solution: to make successive powers of q increasingly negligable. An easy way to do this is to use the factorial function:






If you haven't already noticed, this is equivalent to the Maclaurin expansion for the expression q^8 cos q. In other words, this infinite series has a fundamental mathematical basis, and is thus only slightly more complicated than the existing 'polynomial' Mandelbulbs.

I do not have sufficient tools to render this fractal, so would be very grateful to see an image of it. Only the first few terms of the infinite series need to be considered, since they rapidly decrease in size: as you can see, the term in q^16 is divided by 40320, and further terms will have minimal effect.

This might be the closest thing to the elusive 'True 3D Mandelbrot'. It has bilateral symmetry, same as the 2D Mandelbrot, which is rather reassuring.

And here's what it looks like (maybe someone else can do a better job with the lighting model)

First 2 only go up to the 1/24 (though adding the 1/720 makes little difference)
3rd is a zoom, using the 1/720 term
I'm not sure if the expansion still applies on a power 2 mandelbulb, but the 4th pic is what the crest looks like (also using 1/720 term), it doesn't look as pretty when seen from other angles.

I like the look, though as you can see, it is unconnected in places.

What's "potentiation"?

Calcyman, welcome to the forum!

WOW!! This really looks as though it might have some potential. Dare I say it has a good chance of looking somewhat better than the 'classic' Mandelbulb?

I'm pretty sure it's not the holy grail (not even with using multiple or even infinite powers together), but the glimpses of ideal3 and ideal4 are fascinating.

What happens if you use your expression q^8 cos q directly and not the expansion ? Would that not be ideal?

Could someone plug the sin/cos series into mathematica to find solutions?
For sin and cos it should be easier than for log as the series are a lot clearer with less complex behaviour....

a general quaternionic transform variant to triplex algebra was proposed too, recently...
Did anyone yet try to use that?
In the paper, it was concluded that the quaternion variant (expressing the mandelbulb formula as a quaternionic transform) is both comutative and associative... If that's true, it would be interesting to see what happens if you plug that into mathematica... Maybe the results are more correct, then....

Oops
Perhaps numerically, solving the differential equation q'' +q = 0,  q(0) = (1,0,0)  and plugin triplexes instead of ordinary numbers...