Abstract
This chapter considers closed three-membered metal cycles of one or several chemical elements surrounded by ligands connected to them. It has been proven that the widespread opinion in the literature about the formation of ligands by atoms in some cases of the semi-correct polyhedron of the anti-cube-octahedron is wrong. Geometrical analysis of the interpenetration of the coordinates of ligand atoms around each of the metal atoms of a closed chain showed that this leads to a different class of special three-dimensional irregular polyhedrons for different clusters. In all cases of homo-element and hetero-element closed metal chains, the cycle itself, located in a certain plane, creates a cross section of the cluster, dividing the cluster into two parts. Each of the parts of a cluster has dimension 4.