Necks of regular polyhedra

A neck of a convex polyhedron P is a part of P where a circle can be attached to hold P. A neck is called balanced if the neck divides the number of vertices in half, otherwise, it is called unbalanced. Among other things, we prove that (1) a regular icosahedron has exactly three types of necks, two types are balanced necks and the other type is unbalanced. (2) There is a holding circle of icosahedron that can move between a balanced neck and an unbalanced neck with crossing over a vertex. (3) The set of diameters of all holding circles of a regular icosahedron is disconnected. © 2013 Springer Basel.