I have seen definitions of regular polyhedra where it is assumed only that the faces are congruent and regular (equilateral). I have seen also definitions where two conditions are assumed:
- Congruent regular faces.
- Congruent Dihedral angles.
I belive that the condition (1) above implies the condition (2) above.
Is this right?
The following link: A congruence problem for polyhedra
Has the following :
Theorem 1.2 (Cauchy, 1839). Two convex polyhedra with corresponding congruent and similarly situated faces have equal corresponding dihedral angles.
I believe this is the line of search that I would go for.
Final Update: At this point the answers with counter-examples are good. I wanted to point out to the book in this link Polyhedra
A set of counter-examples is given in Figure 2.18. But most important is the following theorem:
Theorem: Let $P$ be a convex polyhedron whose faces are congruent regular polygons. Then the following statements about $P$ are equivalent:
- The vertices of $P$ all lie on a sphere.
- All the dihedral angles of $P$ are equal.
- All the vertex figures are regular polygons.
- All the solid angles are congruent.
- All the vertices are surrounded by the same number of faces.
Again, thanks to all who helped.