Name for regular polytope vertex embedding 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.

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.

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