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 ${\displaystyle \mathrm {I} _{\mathrm {h} }}$
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 and other related figures are constructed from the regular icosahedron, including its 59 stellations. The great dodecahedron, one of the Kepler–Poinsot polyhedra, is constructed by either stellation of the regular dodecahedron or faceting of the icosahedron. 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 in the comparison mensuration. It is analogous to a four-dimensional polytope, the 600-cell.

Regular icosahedra can be found in nature; a well-known example is the capsid in biology. Other applications of the regular icosahedron are the usage of its net in cartography, and the twenty-sided dice that may have been used in ancient times but are now commonplace in modern tabletop role-playing games.