Don't think this is going to work as a straight out number system.. rather it's a system of rotations being calculated algebraically, rather than trigonometrically, which might not be too useful....
A new number system based on the 3d fractal formula I posted in the
"search for the holy grail" thread that generates images for z^2, such as this one:
In this system components are divided into planar and linear components as follows:
x, y, and z are the linear components.
w, v, and u are the planar components.
w is the planar component that relates to the linear component x.
w is the planar magnitude of the y-z plane: w= sqrt(y^2 + z^2)
v is the planar component that relates to the linear component y.
v is the planar magnitude of the x-z plane: w= sqrt(x^2 + z^2)
u is the planar component that relates to the linear component z.
u is the planar magnitude of the x-y plane: w= sqrt(x^2 + y^2)
The planar triplex z is noted in this way:
z
1= [ (x
1,w
1) ; (y
1,v
1) ; (z
1,u
1) ]
z
2= [ (x
2,w
2) ; (y
2,v
2) ; (z
2,u
2) ]
Addition and subtraction follow the same basic format, here is an addition:
z
1 + z
2 = [(x
1+x
2,w
1+w
2) ; (y
1+y
2,v
1+v
2) ; (z
1+z
2,u
1+u
2)
Multiplication and division follow slightly different basic formats, here is a multiplication:
First, we take the magnitude of the system:
|q| = |z1 & z2 *| = sqrt(x1*x2 + w1*w2 + y1*y2 + v1*v2 + z1*z2 + u1*u2)
Updated magnitude information will be here later today.
Then we calculate new components:
x
1*2 = |
q| * (x
1*x
2 - w
1*w
2) / (x
1*x
2 + w
1*w
2)
w
1*2 = |
q| * (x
2*w
1 + x
1*w
2) / (x
1*x
2 + w
1*w
2)
y
1*2 = |
q| * (y
1*y
2 - v
1*v
2) / (y
1*y
2 + v
1*v
2)
v
1*2 = |
q| * (y
2*v
1 + y
1*v
2) / (y
1*y
2 + v
1*v
2)
z
1*2 = |
q| * (z
1*z
2 - u
1*u
2) / (z
1*z
2 + u
1*u
2)
u
1*2 = |
q| * (z
2*u
1 + z
1*u
2) / (z
1*z
2 + u
1*u
2)
Division will come later. Movie night...
Here is an image of a z^13 quadrant of the new formula (all positives for planes, I like the positive quadrant, can set planes according to corresponding xyz signs in order to have 8 different quadrants, but I am still exploring the positive):
I guess I should include the corresponding z^2 positive quadrant: