A monohedron is a convex polyhedron with all faces mutually congruent but with no other symmetry necessarily needed. So obviously, this is a wide class of polyhedra that includes the Platonic solids and isohedra. An earlier post is What are the known convex polyhedra with congruent faces?
Questions: Are there monohedroa with odd numbers of faces (it is known that isohedra necessarily have even numbers of faces — as stated in the MathWorld article Isohedron)? What are the values for the number of edges on a face for which monohedrons are possible? Will relaxing convexity (of the body, not of the faces) have an impact on the answers to these questions?
Best Answer
The answer to the question in the title is negative in dimension 3: it was shown by Grunbaum that every 3-polytope with congruent facets has an even number of facets. See p. 414 of his book "Convex Polytopes", 2nd edition. The original reference is:
Grunbaum, B. On polyhedra in $\mathbb{E}^3$ having all faces congruent. Bull. Res. Council Israel, 8F (1960), 215-218.