Regular dodecahedron a 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
Trapezo-rhombic dodecahedron optimal way to stack spheres. As the Voronoi cell of a regular space pattern, it is a plesiohedron. It is the polyhedral dual of the triangular orthobicupola. The "trapezo-rhombic dodecahedron" can be seen as an elongation of another dodecahedron, which can be called a rhombo-triangular dodecahedron, with 6 rhombi (or squares) and 6 triangles. It also has d symmetry and is space-filling. It has 21 edges and 11 vertices. With square faces it can be seen as a cube split across the 3-fold axis, separated with the two halves rotated 180 degrees, and filling the gaps with triangles. When used