dy/dx <email@example.com> wrote: >1treePetrifiedForestLane wrote: >> >> yes; the consequence of "one irregular facet" are to weird. >> >> "then all of the faces are regular polygons" as being true ... > >It's clear that if all but one of the polygons is *equilateral*, >then they all must be. > >However, I exhibit a counterexample to the claim that if all >but one are *regular* they all must be, if a dihedral angle of >exactly 180 degrees is permitted: > >A rhombus with internal angles of 60 and 120 degrees, four >square faces, and two equilateral triangular faces, forming a >prism. The two triangular faces, combined, form a duplicate of >the rhombic face.
Ok, but if I understand it correctly, your construction yields a polyhedron with two adjacent coplanar faces, so those are not acceptable as _distinct_ faces of a convex polyhedron.