3D Mandelbrot type z^3 + p

[Buddhi]

Hi
I have just render my first 3D Mandelbrot with z in 3-rd power:

z(n+1) = z(n)^3 + p

where z is Quatermion:
z = ra + ib + jc + kd

multiplying rules:
r*i = i; r*j = j; r*k=k;
r*r=r; i*i=-1; j*j=-1; k*k=-1;
i*j = k; j*k=i; k*i=j;

I used 4D Quaternions to make 4D formula for render 3D fractal on 2D screen :-). After very long multiplying on paper (two A4 pages :-) I have got following formula for iterations:

Code:

newx = x*x*x - 3*(x*y*y + 3*x*z*z + 3*x*w*w) - 6*y*z*w + a
newy = -y*y*y + 3*(x*x*y - y*z*z - y*w*w) + 6*z*z*w + b
newz = -z*z*z + 3*(x*x*z - z*y*y - z*w*w) + 6*x*y*w + c
neww = -w*w*w + 3*(x*x*w - w*y*y - w*z*z) + 6*x*y*z + d   (d = 0)

http://www.fractalforums.com/gallery/0/640_19_07_09_9_36_29.jpg
full resolution: http://picasaweb.google.com/buddhi1980/Fraktale#5360064631222029058

[David Makin]

Hi
I have just render my first 3D Mandelbrot with z in 3-rd power:

z(n+1) = z(n)^3 + p

where z is Quatermion:
z = ra + ib + jc + kd

multiplying rules:
r*i = i; r*j = j; r*k=k;
r*r=r; i*i=-1; j*j=-1; k*k=-1;
i*j = k; j*k=i; k*i=j;

I used 4D Quaternions to make 4D formula for render 3D fractal on 2D screen :-). After very long multiplying on paper (two A4 pages :-) I have got following formula for iterations:

Code:

newx = x*x*x - 3*(x*y*y + 3*x*z*z + 3*x*w*w) - 6*y*z*w + a
newy = -y*y*y + 3*(x*x*y - y*z*z - y*w*w) + 6*z*z*w + b
newz = -z*z*z + 3*(x*x*z - z*y*y - z*w*w) + 6*x*y*w + c
neww = -w*w*w + 3*(x*x*w - w*y*y - w*z*z) + 6*x*y*z + d   (d = 0)

full resolution: http://picasaweb.google.com/buddhi1980/Fraktale#5360064631222029058

Nice to see that it seems to "work" at higher powers too
Just a small point - you shouldn't really call it quaternionic because quaternions are a very specific 4D number form not a general term for 4D number systems and this system is quite different from the quaternions

[Buddhi]

Just a small point - you shouldn't really call it quaternionic because quaternions are a very specific 4D number form not a general term for 4D number systems and this system is quite different from the quaternions

I agree with you. I know that multiplication of quaternions is not commutative and in our calculations we use commutative multiplication (ore some custom multiplication matrix). We use number system only SIMILAR to quaternions. But it WORKS!

[cKleinhuis]

this seems to be working really ... i can see the shape of the z^3 shape vertically and horizontally, nice work !

note: could it be that the shape is non-fractal around  the 45 deegree ? this might be the visual outcome of the drawback used formulas  ....

[lycium]

Just a small point - you shouldn't really call it quaternionic because quaternions are a very specific 4D number form not a general term for 4D number systems and this system is quite different from the quaternions

I agree with you. I know that multiplication of quaternions is not commutative and in our calculations we use commutative multiplication (ore some custom multiplication matrix). We use number system only SIMILAR to quaternions. But it WORKS!

by that reasoning we could call cars "dogs" instead because they both have "four thingies on the ground"