Find remaining vertices of cuboctahedron.

[Eric Belcastro]

*note - this is not a homework problem.

I have the locations of three vertices of a regular cuboctahedron with edges of unit length (all vertices are length 1 from the center).

They are (1,0,0), (1/2, sqrt(3)/2, 0), (1/2, sqrt(3)/6, sqrt(2)/2)
or in spherical coordinates (1, 0, pi/2), (1, pi/3, pi/2), (1, pi/6, arctan(sqrt(2)/2)) respectively.

Now I am trying to find the remaining vertices of the cuboctahedron.

I believe I can find some of them easily, but I would like to be 100% certain of all of their accuracy. So I went to use maple's functions for defining and exploring archimedean solids, and to define an archimedean solid, you are only allowed to define one point, the center, and the radius, you can't specify any other vertices or angles.

It is really a simple problem, but I don't quite trust my intuition. I would assume I could just add angles where appropriate that correspond to the symmetries of a cuboctahedron and then convert back to cartesian coordinates. Any ideas of how to do this simply, or in maple?
If it is simple to do in maple, that would be nice, because there are many problems like this that pop up all the time when I am exploring something.

Thank you in advance.

 

[Eric Belcastro]

I figured out how to find them, it was fairly easy. I just used a quaternion rotation matrix and rotated about the known vertices according to the known rotation symmetry of a cuboctahedron and did this a few times, generating new vertices, and all was well. I also wrote one of the coordinates wrong in the original post. The third vertex was (1/2,sqrt(3)/6,sqrt(6)/3.