Icosahedral

Regular icosahedron graph
3-fold symmetry
Vertices	12
Edges	30
Radius	3
Diameter	3
Girth	3
Automorphisms	120 (S5)
Chromatic number	4
Properties	Hamiltonian, regular, symmetric, distance-regular, distance-transitive, 3-vertex-connected, planar graph

In geometry, a regular icosahedron (/ˌaɪkɒsəˈhiːdrən, -kə-, -koʊ-/ or /aɪˌkɒsəˈhiːdrən/) is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and also 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 3⁵. 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 six orientations.