Higher dimensional Mandelbulbs

[s31415]

Hi,

I'm not sure if someone already emphasized this, but the trick for "squaring" a vector in 3d easily generalizes to higher dimensions. So we can define Mandelbulbs in any dimension. To be more specific, in any dimension d, you have spherical coordinates r, phi_1, phi_2, phi_(d-1). Denoting the cartesian coordinates by x_1, ..., x_d and defining y_i = sqrt(x_1^2 + ... + x_i^2), the coordinate change reads:
r = y_d = sqrt(x_1^2 + x_2^2 + ... + x_d^2)
phi_i = Atan(x_(i+1)/y_i)   [EDIT: I corrected a mistake, it's x_(i+1), not x_i.]
From here, we can square r and double each phi_i. Then to go back to Cartesian coordinates, we apply:
x_d = r sin(phi_(d-1))
x_(d-1) = r cos(phi_(d-1)) sin(phi_(d-2))
...
x_2 = r cos(phi_(d-1)) cos(phi_(d-2)) ... sin(phi_1)
x_1 = r cos(phi_(d-1)) cos(phi_(d-2)) ... cos(phi_1)
Just as with the usual Mandelbulb, we can worry or not worry about the angles going out of their usual range after the doubling.

These Mandelbulbs have a very nice feature. Take the Mandelbulb in dimension d and consider the d' dimensional slice obtained by setting the last (d-d') coordinates to zero. Then you can check that in this slice, you will obtain the d' dimensional Mandelbub!

This leads me to a question: Did anybody tried to look at 2d slices of the 3d Mandelbulb? If you slice through the equator, you get the usual M-set. What about other slices?

Also, what about Julia sets ?

Sam

[cKleinhuis]

if the formula you propose is totally "general" it would, show for dimension = 2 the normal mandelbrot formula ?
 

[s31415]

Yes, indeed. For dimension 2 you have only two cartesian coordinates x_1 (corresponding to the real part of the complex number) and x_2 (corresponding to the imaginary part). The polar coordinates are r, the modulus of the complex number, and phi, its argument. Now you can see the process of squaring the complex number as taking the complex number
z = (r, phi) -> (r^2, 2phi) = z^2.

In the general formula, if you set all x_i to zero for i > 2, then all the phi_i are zero except phi_1, so you recover exactly the operation of squaring a complex number. Therefore the usual M-set lies in all these Mandelbulbs, in the x_1 x_2 plane.

Of course, in the post above, I only explain how to "square" a vector. To implement the usual Mandelbrot algorithm you should really iterate v -> v^2 + c, where ^2 is the operation described above, and the original vector.

Sam

[cKleinhuis]

this is a very good view on the formulas, and though the mandelbrot is contained, it is really getting interesting,
although we will never "see" those higher dimension mandelbulbs, but for sure it is a wealthy lot of forms and
shapes that could be seen in those, and we spend the last 30 years with searching around in the simple 2d mandelbrot

i like the idea of the "number system", it should be proved that it is a mathematical body ( + ( inverse - ) and * ( inverse / ), with a neutral element for both operations ...

this is a lot nicer than the usual quaternion way of extending fractal formulas like our beloved , i once made a formula
evolver ( mutatorkammer ), and i am now thinking of extending it to 3rd dimension using this formula ... although it seems to be
quite hard to achieve the "remaining" operations, like squaring with non integer powers, or sine and cosine values for this system, but
i have to admit, it all makes sense .. the best thing about the mandelbulb is its easy formulation:

"use the polar coordinates and extend them for an additional rotation axis, and proceed like before ..." excellent and easy !

anyway, i will love to convert existing simple fractal formulas, like barnsley, burning ship .... to view them in 3d, i just hadnt had time to
implement a gpu renderer for viewing those ...

 

[bib]

This leads me to a question: Did anybody tried to look at 2d slices of the 3d Mandelbulb? If you slice through the equator, you get the usual M-set. What about other slices?

Other slices parrallel to what you call the equator are less interesting. However, the slice perpendicular to the equator shows shapes that can't be seen (I think) in the 2D M-set. e.g :
http://www.fractalforums.com/gallery/1/492_16_11_09_11_33_06.jpg
http://www.fractalforums.com/gallery/1/492_16_11_09_11_21_29.jpg

This video shows different slices perpendicular to the equator in the 2nd order Mandelbulb:
<a href="http://www.youtube.com/v/XhVXC_0arIs&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/XhVXC_0arIs&rel=1&fs=1&hd=1</a>
It's easy to imagine the whipped cream.

[dougfractal]

I've done quite a bit of work on this

4D Evolution
<a href="http://www.youtube.com/v/eS7qCfttmBk&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/eS7qCfttmBk&rel=1&fs=1&hd=1</a>

4D Parallel Universe
<a href="http://www.youtube.com/v/D7j-Fj7JozQ&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/D7j-Fj7JozQ&rel=1&fs=1&hd=1</a>

A quick glimpse into 5 Dimensions
<a href="http://www.youtube.com/v/BassAa0Y3Ho&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/BassAa0Y3Ho&rel=1&fs=1&hd=1</a>