Small snub icosicosidodecahedron Small snub icosicosidodecahedron In geometry, the small snub icosicosidodecahedron or snub disicosidodecahedron is a uniform star polyhedron, indexed as U. It has 112 faces (100 triangles and 12 pentagrams), 180 edges, and 60 vertices. Its stellation core is a truncated pentakis dodecahedron. It also called a holosnub icosahedron, ß{3,5}. The 40 non-snub triangular faces form 20 coplanar pairs, forming star hexagons that are not quite regular. Unlike most snub polyhedra, it has reflection symmetries. Its convex hull is a nonuniform truncated icosahedron. Cartesian coordinates for the vertices of a small snub icosicosidodecahedron are all the even permutations of where ϕ
Regular icosahedron and symmetric. The automorphism group has order 120. The vertices can be colored with 4 colors, the edges with 5 colors, and the diameter is 3. The icosahedral graph is Hamiltonian: there is a cycle containing all the vertices. It is also a planar graph. There are 4 related Johnson solids, including pentagonal faces with a subset of the 12 vertices. The similar dissected regular icosahedron has 2 adjacent vertices diminished, leaving two trapezoidal faces, and a bifastigium has 2 opposite sets of vertices removed and 4 trapezoidal faces. The pentagonal antiprism is formed by removing two opposite vertices. The