Editing Regular icosahedron (section)

{{Short description|Convex polyhedron with 20 triangular faces}}

{{Infobox polyhedron
 | name = Regular icosahedron
 | image = Icosahedron.svg
 | type = [[Deltahedron]],<br />[[Gyroelongated bipyramid]],<br />[[Platonic solid]],<br />[[Regular polyhedron]]
 | faces = 20
 | edges = 30
 | vertices = 12
 | angle = 138.190 (approximately)
 | vertex_config = <math> 12 \times \left(3^5 \right) </math>
 | schläfli = <math> \{3,5\} </math>
 | symmetry = [[icosahedral symmetry]] <math> I_h </math>
 | dual = [[regular dodecahedron]]
 | properties = [[convex set|convex]],<br />[[composite polyhedron|composite]],<br />[[isogonal figure|isogonal]],<br />[[isohedral]],<br />[[isotoxal]]
 | net = Icosahedron flat.svg
}}

The '''regular icosahedron''' (or simply ''icosahedron'') is a [[convex polyhedron]] that can be constructed from [[pentagonal antiprism]] by attaching two [[pentagonal pyramid]]s with [[Regular polygon|regular faces]] to each of its pentagonal faces, or by putting points onto the cube. The resulting polyhedron has 20 [[equilateral triangle]]s 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 (topology)|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&ndash;Poinsot polyhedra]], is constructed by either stellation or [[faceting]]. Some of the [[Johnson solid]]s 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]].

Regular icosahedrons 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 [[d20 System|commonplace in modern]] [[tabletop role-playing games]].