# Platonic Solids

Volume r 3 s 3 s 3 s 3 2 s 3 4 s 3 A recurring theme in the metric properties of the Platonic Solids is the occurrence of the irrational numbers phi , and the square roots 2, 3, and 5. They are surprisingly elegant when expressed as infinitely continued fractions Their decimal expansions to twelve places, together with that of are 1.618033988750 2 1.414213562373 3 1.732050807569 5 2.236067977500 3.141592653590 The table below gives volumes and surface areas for a sphere radius r, and Platonic Solids edge length s. Also included are the proportional pathways joining each vertex to every other in the Platonic Solids. each embracIng every other 54 exPanSIonS formul Sphere Tetrahedron Octahedron Cube Icosahedron Dodecahedron Surface Area 4 r 2 3 s 2 23 s 2 6 s 2 53 s 2 325105 s 2 Number of Pathways, Length na 6 edges, s 12 edges, s 3 axial diagonals, 2 s 12 edges, s 12 face diagonals inscribed tetrahedra, 2 s 4 axial diagonals, 3 s 30 edges, s 30 face diagonals, s 6 axial diagonals, 21 s 30 edges, s 60 face diagonals inscribed cubes, s 60 interior diagonals inscr. tetrahedra, 2 s 30 interior diagonals, 2 s 10 axial diagonals, 3 s 55