Based upon the geometrical properties of regular polyhedrons, the possibility of the existence of certain polyhedral molecules composed of only one element is investigated. A very promising candidate - the regular dodecahedron - is selected as the convenient polyhedral structural pattern and phosphorus as the appropriate element. A series of high-level ab initio calculations is performed on the dodecahedral P20 molecule, including predictions of its thermodynamic and kinetic stability, natural bond orbital analysis, vibrational analysis, and inclusion of some elements into the molecular skeleton. Due to the potential stabilizing agents that may eventually form stable inclusion compounds and the estimated high kinetic stability, the question of the possible existence of P20 is answered in the positive.Key words: ab initio calculations, inclusion compounds, P20, phosphorus clusters, polyhedra.