Analysis finds complex realizations of toroidal polyhedra using k-gons, suggesting further simplification is needed.
If all faces of a toroidal polyhedron are k-gons, what are the possible numbers n of faces? It follows from the Euler characteristic that for such polyhedra k is one of 3, 4, 5, 6. The paper provides an almost complete answer in all these cases, although several specific questions remain. Some of the polyhedra found show that the constructions are rather complicated in some instances; this may provide an incentive for searching simpler realizations.
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Grünbaum et al. (2009) studied this question.
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