The Twelve Variable Cube Octahedral Cup seems like 'duh' material. I was able to get this to work in Excel by projecting a Tetrahedron across an inverted two dimensional triangle. Then, summated, it is no longer five dimensional. For a long time I couldn't make the leap from this equation to the Cube Octahedron.

This interesting calculation produces results with symmetry chopped in half along a hexagonal equator of a Cube Octahedron, however the larger numbers are still in the center. Now we also have the algebraic equation for this shape.

The geometry is still n-based, starting here at N=1. It is viewable here up to N=5

12^{1} = 12

12^{2} = 144

12^{3} = 1,728

12^{4} = 20,736

12^{5} = 248,832