Icosidodecahedron

The icosidodecahedron is a convex polyhedron with twenty triangular faces and twelve pentagonal faces, thirty identical vertices, and sixty edges, formed by rectifying either a regular icosahedron or a regular dodecahedron, a construction that cuts each vertex of the original solid exactly at the midpoints of its edges. It is one of the thirteen Archimedean solids, the convex polyhedra built from more than one kind of regular polygon meeting in an identical arrangement at every vertex, a class first fully catalogued in a surviving written source by the German astronomer Johannes Kepler in his 1619 book Harmonices Mundi, though the solids themselves are traditionally credited to Archimedes in an earlier work that has not survived. At every vertex of the icosidodecahedron, two triangles and two pentagons meet alternately, and every one of its edges lies on the boundary between one triangular face and one pentagonal face, a fully alternating pattern shared by only a small handful of other Archimedean solids. The icosidodecahedron's dual polyhedron, formed by placing a vertex at the center of each of its faces, is the rhombic triacontahedron, a solid whose thirty rhombic faces echo the icosidodecahedron's own thirty vertices.