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 dodecahedron such a regular dodecahedron is 1 : , or : 1, or (5 + ) : 4. For example, an embedded cube with a volume of 64 (and edge length of 4), will nest within a regular dodecahedron of volume 64 + 32"ϕ" (and edge length of 4"ϕ" − 4). Thus, the difference in volume between the encompassing regular dodecahedron and the enclosed cube is always one half the volume of the cube times "ϕ". From these ratios are derived simple formulas for the volume of a regular dodecahedron with edge length "a" in terms of the golden mean: Golden