This is my first post in this forum and first participation to the quest for the MandelGrail.
Recently, I tried to see how changing the coefficients of a quadratic "multiplication" could modify the shape of a Mandelbrot 3D.
Here is the iteration function z^2 + c :
x = v * mx * v + cx
y = v * my * v + cy
z = v * mz * v + cz
where v = (x, y, z) and mx, my and mz are 3x3 symmetrical matrices, with coefficients mxij . So, this can be written :
x = mx11 * x^2 + mx12 * x*y + mx13 * x*z + mx21 * x*y + mx22 * y^2 + …,
Let us start with the « revolution Mandelbrot », defined by mx11=1, mx22=mx33=-1, my12=my21=1 and mz13=mz31=1 (other coefficients = 0).
From there, we can slightly change the mij coeffcients, to see the effects on the shape. I will pass on all the coefficients that distort the shape in an « unfavourable » way (i.e. that clearly move us away from what a « 3D Mandelbrot » should be)
Here are some interesting results :
- changing my22, my23/my32, mz23/mz32, mz33 distorts the shape longitudinaly and creates a lateral « fractal spoke » in directions y and z. Watching from the front of the shape :
- changing my33 kind of « splits » the shape in the yz plane in an interesting way. Actually, when you combine the creation of a fractal spoke (on mz23/mz32) and this splitting (on my33), you get three spokes!
To make them identical and equally distributed around the x axis, the « best » parameters seem to be mz32=mz23=0,5 and my33=1/3 (or their opposites).
From there I tried to create additional spokes but could not do that just by changing the matrices coefficients. Since we want some kind of symmetry around the x axis, I tried to see how the iteration equations looked in the following polar coordinates :
x = r cos ph
y = r cos th sin ph
z = r sin th sin ph
This gives us, in the iteration equations :
x = r^2 cos 2ph + cx
y = r^2 cos th sin 2ph + 1/6 r^2 sin^2 ph (cos 2th - 1) + cy
z = r^2 sin th cos 2ph + 1/2 r^2 sin^2 ph (sin 2th) + cz
The first coefficients are the ones of the « revolution Mandelbrot ».
The second coefficients in y and z equations are similar to a Mandelbrot equation in the yz plane, multiplied by a coefficient (r*sin ph) (i.e. r projected in the yz plane).
This gives the idea to choose a different power for th, and write :
x = r^p1 cos(p1*ph) + cx
y = r^p1 cos(th)sin(p1*ph) + 1/3 r^p1 sin(p1*ph)*cos(p2*th) + cy
z = r^ p1 sin(th)cos(p1*ph) + 1/2 r^p1 sin(p1*ph)*sin(p2*th) + cz
with p1 the power for ph, i.e. along x (p1=2 if you want a shape similar to the classical Mandelbrot)
and p2 the power for th, for the shape in the yz plane. It is as if you « combined » two Mandelbrot figures with different powers, one along x and the other along y and z.
When you increase p2, the number of spokes increases in a way similar to the way the 2D Mandelbrot evolves when you increase its power.
(Below for p2 = 3, 6, 20)
As you can see the p2=20 shape is quite hairy.
And yet, still no sign of lateral spheres. Sorry Twinbee!