Regular icosahedron Regular icosahedron In geometry, a regular icosahedron ( or ) is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most sides. It has five equilateral triangular faces meeting at each vertex. It is represented by its Schläfli symbol {3,5}, or sometimes by its vertex figure as 3.3.3.3.3 or 3. It is the dual of the dodecahedron, which is represented by {5,3}, having three pentagonal faces around each vertex. A regular icosahedron is a gyroelongated pentagonal bipyramid and a biaugmented pentagonal antiprism in any of
Deltoidal hexecontahedron two angles, between a short and a long edge each, are both 86.974°. The dihedral angle between all faces is 154.12°. Topologically, the "deltoidal hexecontahedron" is identical to the nonconvex rhombic hexecontahedron. The deltoidal hexecontahedron can be derived from a dodecahedron (or icosahedron) by pushing the face centers, edge centers and vertices out to different radii from the body center. The radii are chosen so that the resulting shape has planar kite faces each such that vertices go to degree-3 corners, faces to degree-five corners, and edge centers to degree-four points. The "deltoidal hexecontahedron" has 3 symmetry positions located on