I am looking for basic information about the following idea:
(I) Consider a square. By collapsing two adjacent vertices, we obtain a triangle.
(II) Consider a three-dimensional cube. By collapsing a single face to a line, we obtain a prism with triangular base. Next, by collapsing a line longitudinally, we obtain a pyramix with square base. Next, by contracting two adjacent vertices of the square base, we obtain a tetrahedron.
How can this can be generalized to higher dimensions? It is obvious that we can transform a hypercube to a simplex by a sequence of contractions. But not all contractions of faces produce a polyhedron (with flat faces), and many contractions produce the same object up to congruence. Since the idea is so basic, I hope somebody else has already written about it.
Hence I look for information, in particular about the following questions:
- Which contractions actually produce polyhedra, and which class of polyhedra is actually produced?
- Since different contractions produce congruent polyhedra, how can we relate (sequences of contractions) and congruence relations?
- Do these operations have an analogon on the symmetry groups of the polyhedra?
Of course, it would be best to have a book reference on this topic.