Differing Mandelbulbs?

[Dranorter]

Hi there, I started following the 'search for the 3D Mandelbulb' when twinbee's first article was posted (I think a Google Alert brought me to it!). Just yesterday I encountered the slashdot story of 'success', though of course this new object is found lacking in several respects.

I read through twinbee's new article and much of twinbee's thread yesterday and today. There is a discrepancy in some of the images which bothers me.

One of the main things about this object which is different from the hoped-for object (as shown in some sketches of it) and different from the 2D higher-power Mandelbrot sets is the lack of dendritic arms coming off the object which have further buds upon them. Here is an example on a sixth power Mandelbrot set:



What bothers me is that there are some images of the Mandelbulb in which this behavior appears to occur:




However, most images clearly don't do this.



There are other differences between the objects in these pictures, but I'm aware that there are some slightly different methods running around for producing them. What bothers me is that just looking at these pictures it seems to me there is a possibility that some of them are topologically spherical while others are not.

However, all of these objects exhibit stretched-looking areas, and I agree with the sentiment expressed somewhere around http://www.fractalforums.com/3d-fractal-generation/true-3d-mandlebrot-type-fractal/525/ that maybe there is a simple variation of this which will show a more 'genuine' feeling shape.

In any case am I imagining things with the differences between the above mandelbulbs???

(By the way, I have been interested in fractals a long time, but my 'holy grail' fractal is a bit different... I want to show the Mandelbrot Set equivalent to a universal turing machine. XD)

[Dranorter]

Here is another 3D image of the budding I'm talking about, though I suppose there aren't really great 'dendrites'. But I never thought high-power Mandelbrots had much variation in them anyway...



Created using Dmitry Brant's program: http://www.fractalforums.com/mandelbulb-implementation/opengl-mandelbulb-implementation/

[twinbee]

Hi there,

The reason for the differences you see is because I use a simple extension of spherical coords thusly:

{x,y,z}^n = r^n { sin(theta*n) * cos(phi*n) , sin(theta*n) * sin(phi*n) , cos(theta*n) }
...where: r = sqrt(x^2 + y^2 + z^2), theta = atan2( sqrt(x^2+y^2), z ), phi = atan2(y,x)

Whilst Paul uses a slightly different version - an extension of the original quadratic bulb (with offsets):
{x,y,z}^n = r^n { cos(theta)*cos(phi), sin(theta) * cos(phi), -sin(phi) }
...where: r = sqrt(x^2+y^2+z^2), theta = n*atan2(y,x), phi = n*asin(z/r)

[bugman]

Hi there,

The reason for the differences you see is because I use a simple extension of spherical coords thusly:

{x,y,z}^n = r^n { sin(theta*n) * cos(phi*n) , sin(theta*n) * sin(phi*n) , cos(theta*n) }
...where: r = sqrt(x^2 + y^2 + z^2), theta = atan2( sqrt(x^2+y^2), z ), phi = atan2(y,x)

Whilst Paul uses a slightly different version - an extension of the original quadratic bulb (with offsets):
{x,y,z}^n = r^n { cos(theta)*cos(phi), sin(theta) * cos(phi), -sin(phi) }
...where: r = sqrt(x^2+y^2+z^2), theta = n*atan2(y,x), phi = n*asin(z/r)

This formula appears to be equivilent to the cosine formula that Garth Thornton was using. Here is a link to the non-trigonometric expansions for the cosine formula:
http://www.fractalforums.com/theory/non-trigonometric-expansions-for-cosine-formula/

[twinbee]

Dranorter,

Just to also add (after reading your post again), that the more dendrite-like tentacles in Paul's colorful pic is due to a higher iteration count. At higher iterations, things look more fragmented. But the fragmentation isn't as cool as one might expect (it doesn't produce more Mandelbrot-like variety).

For a vivid example of a super-high iteration count, see here.

[bugman]

It looks like there are actually 3 popular variations. I have tried to summarize the different formulas here:
http://www.fractalforums.com/theory/summary-of-3d-mandelbrot-set-formulas/