[Atomic-S (OP)]

It is well known in solid geometry that there exist five, and only five, possible shapes of spherically compatible regular polyhedra. This of course refers to non-reentrant structures (those having but a single surface between the center and the outside). If, however, we allow re-entrant structures, the situation changes significantly. At first sight it appears that an infinitude of polyhedra become possible, however this is quickly whittled down by other considerations. Nonetheless, I am aware of no general formula which tells us what is and what is not a possible re-entrant regular polyhedron.  For further explanation see http://polyhedra.atspace.com . Does anyone know the answer?