Regular dodecahedron Regular dodecahedron A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of twelve regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges, and 160 diagonals (60 face diagonals, 100 space diagonals). It is represented by the Schläfli symbol {5,3}. If the edge length of a regular dodecahedron is "a", the radius of a circumscribed sphere (one that touches the regular dodecahedron at all vertices) is and the radius of an inscribed sphere (tangent to each of the regular dodecahedron's
First stellation of rhombic dodecahedron First stellation of rhombic dodecahedron In geometry, the first stellation of the rhombic dodecahedron is a stellation of the rhombic dodecahedron. This polyhedron is sometimes called "Escher's solid"; it appears in M. C. Escher's works "Waterfall" and in a study for "Stars" (although "Stars" itself features a different shape, the compound of three octahedra). It has 48 faces, all triangles. It has 26 vertices in total: 6 with degree 8, 8 with degree 6, and 12 with degree 4. It has 72 edges in total, giving an Euler characteristic of 26 + 48 − 72 = +2. Its vertices are