Here is an example of two polyhedra which use the same set of 15 vertices but are quite different in appearance. The two polyhedra "nest" together to form a distorted hexagonal prism. It's apparent that the two parts share a set of vertices.

The 15 vertex set comes from a dissection of the dodecahedron. If 4 vertices are added above the 15 vertex set and four are added below the vertex set two different polyhedra are created with regular polygons for faces. Each polyhedron has 19 vertices, 6 pentagonal faces, 3 square faces, and 10 triangular faces. They fit together to form a dodecahedron. All the polyhedra are related to the 3 fold symmetry pair in thing:1380887. OpenSCAD code is included.

MakerBot

MakerBot Replicator (5th Generation)

Doesn't Matter

No

0.3 mm

The distorted hexagonal prism obtained by combining the two polyhedra from the 15 vertex set is the "convex hull" of the vertex set. see - https://en.wikipedia.org/wiki/Convex_hull - If one treats either polyhedron with openSCAD's hull command the same distorted hexagonal prism is output.