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
Regular icosahedron Borromean rings. If the original icosahedron has edge length 1, its dual dodecahedron has edge length = = "ϕ" − 1. The 12 edges of a regular octahedron can be subdivided in the golden ratio so that the resulting vertices define a regular icosahedron. This is done by first placing vectors along the octahedron's edges such that each face is bounded by a cycle, then similarly subdividing each edge into the golden mean along the direction of its vector. The five octahedra defining any given icosahedron form a regular polyhedral compound, while the two icosahedra that can be defined in