Families of polyhedra (Q372493)

If we cut a polyhedron along some edges, so that we can unfold its boundary in the plane, and if this surface does not overlap anywhere, then the author calls this region in the plane together with its folding lines a net.NEWLINENEWLINEIf the net of a polyhedron \(P\) can be cut into two or more nets of some other polyhedra \(P_i\), \(i=1,\dots,k\), then \(P\) is a parent, and each \(P_i\) is an offspring. If we have a total of two offsprings, which are equal, then they are called twins. A regular-faced polyhedron is a convex polyhedron with regular polygons as faces. The \(f\)-vector of a polyhedron is given by \((f_3(P),f_4(P),\dots)\), where \(f_n\) is the number of faces of \(P\) which are \(n\)-gons.NEWLINENEWLINEThe author lists 11 pairs of twin regular-faced polyhedra, but admits difficulties of proving its completeness. He enumerates all families of regular-faced polyhedra, where the faces are triangles, and proves a relationship between a decomposition of a polyhedron into offsprings and a decomposition of its \(f\)-vectors into the \(f\)-vector of its offsprings. The author closes with some general theorems relating the number of vertices, edges, and faces of a polyhedron, its offsprings and their nets.