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
Regular dodecahedron ratio rectangles of ratio ("ϕ" + 1) : 1 and "ϕ" : 1 also fit perfectly within a regular dodecahedron. In proportion to this golden rectangle, an enclosed cube's edge is "ϕ", when the long length of the rectangle is "ϕ" + 1 (or "ϕ") and the short length is 1 (the edge shared with the regular dodecahedron). In addition, the center of each face of the regular dodecahedron form three intersecting golden rectangles. It can be projected to 3D from the 6-dimensional 6-demicube using the same basis vectors that form the hull of the rhombic triacontahedron from the 6-cube.