let X,Y,Z be (x1,x2) (y1,y2) (z1,z2) respectively
then this boils down to solving these 6 equations
(x11)^2+x2^2=1
y1^2+y2^2=1
z1^2+(z21)^2=1
(x1y1)^2+(x2y2)^2=1
(y1z1)^2+(y2z2)^2=1
(x1z1)^2+(x2z2)^2=1
Tried for an analytical solution, still working it out, but for now I have Mathematica's solutions which it found these 12 (in order x,y,z)
1 (1/2,sqrt(3)/2) (1,0) (0,0)
2 (1/2,sqrt(3)/2) (1/2,sqrt(3)/2) (0,0)
3 (1/2,sqrt(3)/2) (1,0) (0,0)
4 (1/2,sqrt(3)/2) (1/2,sqrt(3)/2) (0,0)
5 (0,0) (0,1) (sqrt(3),1/2)
6 (0,0) (0,1) (sqrt(3),1/2)
7 (0,0) (sqrt(3)/2,1/2) (sqrt(3)/2,1/2)
8 (0,0) (sqrt(3)/2,1/2) (sqrt(3)/2,1/2)
9 (1,1) (1,0) (1/2,(2sqrt(3))/2)
10 (1,1) (1,0) (1/2,(2+sqrt(3))/2)
11 ((2sqrt(3))/2,1/2) (1,0) (1,1)
12 ((2+sqrt(3))/2,1/2) (1,0) (1,1)

Posted by Daniel
on 20090909 14:58:36 