Regular icosahedron

Regular icosahedron
Type	Deltahedron, Gyroelongated bipyramid, Platonic solid, Regular polyhedron
Faces	20
Edges	30
Vertices	12
Vertex configuration	${\displaystyle 12\times \left(3^{5}\right)}$
Schläfli symbol	${\displaystyle \{3,5\}}$
Symmetry group	icosahedral symmetry ${\displaystyle I_{h}}$
Dihedral angle (degrees)	138.190 (approximately)
Dual polyhedron	regular dodecahedron
Properties	convex, composite, isogonal, isohedral, isotoxal
Net

The regular icosahedron (or simply icosahedron) is a convex polyhedron that can be constructed from pentagonal antiprism by attaching two pentagonal pyramids with regular faces to each of its pentagonal faces, or by putting points onto the cube. The resulting polyhedron has 20 equilateral triangles as its faces, 30 edges, and 12 vertices. It is an example of a Platonic solid and of a deltahedron. The icosahedral graph represents the skeleton of a regular icosahedron.

Many polyhedra are constructed from the regular icosahedron. A notable example is the stellation of regular icosahedron, which consists of 59 polyhedrons. The great dodecahedron, one of the Kepler–Poinsot polyhedra, is constructed by either stellation or faceting. Some of the Johnson solids can be constructed by removing the pentagonal pyramids. The regular icosahedron's dual polyhedron is the regular dodecahedron, and their relation has a historical background on the comparison mensuration. It is analogous to a four-dimensional polytope, the 600-cell.

The appearance of regular icosahedron can be found in nature, such as virus with icosahedral-shaped shells and radiolarians. Other applications of the regular icosahedron are the usage of its net in cartography, twenty-sided dice that may have been used in ancient times and in modern role-playing games.