Name for regular polytope vertex embedding

Question

What I mean by that is using some vertices of a regular polytope to construct another, such as a tetrahedron in a hexahedron. I've thought about this topic, but I do not know its name, if it has any.

Answer 1

You are looking for facetings. A facet of a poltope is the span of any subset of the vertex set of a polytope (as such some afine subspace) intersected with the given polytope - or, alternatively, the convex hull of those vertices. The most prominent facets of a d-dimensional polytope surely are its d-1-dimensional facets. Those are all the boundary faces, but also several (vertex defined) cross-sections. A faceting (polytope) then is any dihedral polytope, which can be obtained from such (prominent) facets. - As such faceting is the dual concept of stellation. Both concepts will produce several non-convex polytopes from a given convex starting polytope.

--- rk

Answer 2

To add to Richard's answer, faceting (or facetting) is the reciprocal process of stellation. If you begin with polyhedron $P$ and stellate it to say $S$, then dualise the stellation in the conventional way, by reciprocation in a concentric sphere, the result $S^*$ will be a facetting of $P^*$, the dual of $P$.

So, for example, the facettings of the regular dodecahedron are duals of the stellated icosahedra.

Coxeter briefly defines faceting and its reciprocal relation to stellation in his classic Regular Polytopes.

James Bridge studied some facetings of the regular dodecahedron: Bridge, N. J.; "Facetting the dodecahedron", Acta Crystallographica, A30, 1974, pp. 548-552. (pdf)

I looked more systematically at the process involved and some simpler cases: Inchbald, G.; "Facetting Diagrams", The Mathematical Gazette 86, July 2002, p.p. 208-215. (Web page).