Four icosahedra can meet at a point (Q1106477)

Let \(N_{pq}\) denote the maximal number of congruent copies of a Platonic solid of type \(\{\) p,q\(\}\) which can share a common vertex but are otherwise disjoint. Obviously, for the cube one has \(N_{43}=8\), and in this paper it is proved that \(N_{35}=N_{53}=4\). Additionally, it is conjectured that \(N_{33}=20\) and, with less conviction, that \(N_{34}=7\).