Faceting
A faceting of a polytope is another polytope that has a subset of the same vertices, in the same locations. Some authors require facetings to have the same symmetry as the original polytope, and a faceting may be required to have all of the vertices of the original polytope.
If the faceting of a polytope has a subset of the original's ridges too (that is, it doesn't create any new ones or divide up any old ones), it can be called a ridgefaceting. A ridgefaceting of a polyhedron is called an edgefaceting.
New polyhedra may be discovered by faceting. The first noble kipiscoidal icositetrahedron was discovered by faceting the snub cube, and the snub hexecontatetrasnubsnub disoctachoron is a faceting of the disnub disicositetrachoron under a different symmetry.
Terminology edit
If a polytope has all the same vertices of another polytope, some refer to them as being "in the same army."
The regiment of each uniform polytope contains facetings of that uniform polytope. All members of a regiment have the same vertices and edges.
Examples edit

The fully symmetric facetings of the cuboctahedron:

the octahemioctahedron,

and the cubohemioctahedron.
The latter two include hexagonal faces.
Altogether, these make up the regiment of the cuboctahedron.
External links edit
 Klitzing, Richard. "RidgeFacetings".