Ideal 3D mandelbrot?

[Calcyman]

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.

[kram1032]

did you actually try out to just plot simple real graphs of that?

Fractals tend to amplify any accuracy problems (-> butterfly effect) and while it most likely is already overkill to render, 10 steps on that cosine-series are still quite limited to very small values...
At least that's what my testing gave... (I just tried that series manually a few days back )

However, even with those possible inaccuracies, it might be an interesting set to see

[Tglad]

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.

[cKleinhuis]

hi there, first of all: you can use LaTex inside your posting here! there is a [tex] tag build in

and this looks like polynomal representation?
add a nice factor before, try to render it, but do not really know the derivative for it,
where the heck was the definition for deriving triplex addition of potentiation ?

....

[BradC]

where the heck was the definition for deriving triplex addition of potentiation ?

What's "potentiation"?

[kram1032]

Tglad: Interesting...

Very noisy but beautiful
I wonder what some antialiasing could do with it...

certain parts look like some kinds of 3D-Lévy-Curves....

Now I wonder... How would sin and cos in Triplex (and/or the Quaternion Triplex) look like?
both functions shouldn't suffer as much from the limitations as log does, at least if you look at the definitions of the series.... - they don't seem to hide any special problems...

[twinbee]

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.

[Calcyman]

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

Thanks for rendering it. I particularly like the 3rd image, which resembles an ordinary Mandelbulb.

The 4th image resembles the Seahorse valley, in that it contains an infinite set of spirals. It might be interesting to probe that area of the fractal, to see whether it exhibits the same features as its 2D counterpart.

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

These fractals are clearly more sophisticated than 2D Mandelbrot-Julia fractals, which are either completely connected or consist of Cantor dust of zero measure.

where the heck was the definition for deriving triplex addition of potentiation ?

The Maclaurin expansion is designed for real numbers; triplex numbers don't possess well-defined derivatives. Using the expansion is a way to 'cheat' and define any function in triplex algebra.

[mrrgu]

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

[David Makin]

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

Care to tell us how to get cos(q) (triplex) without using the expansion ?

[kram1032]

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....

[BradC]

I don't think knowing cos helps sum the original series because you can't factor the out because multiplication isn't associative. For example, in general, so

   

But it would still be interesting to know what the sin & cos series sum to. I have Mathematica; I'll try to post the results later today.

[kram1032]

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....

[BradC]

a general quaternionic transform variant to triplex algebra was proposed too, recently...

The one in this thread http://www.fractalforums.com/theory/the-real-math-of-the-mandelbulb/?

[mrrgu]

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

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

Care to tell us how to get cos(q) (triplex) without using the expansion ?