The regular icosahedron (or simply icosahedron) is a convex polyhedron that can be constructed from a 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 occur both in natural and human-made objects. A notable natural example is the adenovirus, while human applications include cartography (where its net shape is employed) and the twenty-sided dice that may have been used in ancient times but are now commonplace in modern tabletop role-playing games.
Construction The regular icosahedron is a twenty-sided polyhedron wherein the faces are equilateral triangles. It is one of the eight convex deltahedra, a polyhedron wherein all of its faces are equilateral triangles. Variously, it can be constructed as follows:
Started by attaching two pentagonal pyramids with regular faces to the base of a pentagonal antiprism. These components are elementaries—they cannot be disintegrated into smaller convex polyhedra with regular faces again. Replacing bases of a pentagonal antiprism with ten triangular pyramids, the regular icosahedron is classified as a composite polyhedron, the opposite of an elementary polyhedron. This construction led to the alternative names called bicapped pentagonal antiprism, or gyroelongated pentagonal bipyramid due to its construction process through gyroelongation—polyhedra construction by attaching two pyramids onto the base of an antiprism.
The twelve vertices of a regular icosahedron describe the three mutually perpendicular golden rectangular planes, whose corners are connected. These rectangular planes can be constructed from a pair of vertices located on the midpoints of the opposite edges on a cube's surface, drawing a segment line between those two, and divides the segment line in a golden ratio φ = ( 1 + 5 ) / 2 {\displaystyle \varphi =(1+{\sqrt {5}})/2} from its midpoint. Both the vertices of a regular icosahedron have an edge length of 2 and the three planes can be illustrated through Cartesian coordinate system: ( 0 , ± 1 , ± φ ) , ( ± 1 , ± φ , 0 ) , ( ± φ , 0 , ± 1 ) . {\displaystyle \left(0,\pm 1,\pm \varphi \right),\left(\pm 1,\pm \varphi ,0\right),\left(\pm \varphi ,0,\pm 1\right).}
One can snub a regular octahedron, by separating all of its faces and filling them with more equilateral triangles. As suggested by the process, the regular icosahedron is also known as snub octahedron. The regular icosahedron can be unfolded into 43,380 different nets. The earliest net appeared in Albrecht Dürer's Painter's Manual in 1525.
Properties
Surface area and volume
The surface area of a polyhedron is the sum of the areas of its faces. In the case of a regular icosahedron, its surface area A {\displaystyle A} is twenty times that of each of its equilateral triangle faces. Its volume V {\displaystyle V} can be obtained as twenty times that of a pyramid whose base is one of its faces and whose apex is the regular icosahedron's center; or as the sum of the volume of two uniform pentagonal pyramids and a pentagonal antiprism. Given that the edge length a {\displaystyle a} of a regular icosahedron, both expressions are:
A = 5 3 a 2 ≈ 8.660 a 2 , V = 5 φ 2 6 a 3 ≈ 2.182 a 3 . {\displaystyle A=5{\sqrt {3}}a^{2}\approx 8.660a^{2},\qquad V={\frac {5\varphi ^{2}}{6}}a^{3}\approx 2.182a^{3}.}
Relation to the spheres The insphere of a convex polyhedron is a sphere touching every polyhedron's face within. The circumsphere of a convex polyhedron is a sphere that contains the polyhedron and touches every vertex. The midsphere of a convex polyhedron is a sphere tangent to every edge. Given that the edge length a {\displaystyle a} of a regular icosahedron, the radius of insphere (inradius) r I {\displaystyle r_{I}} , the radius of circumsphere (circumradius) r C {\displaystyle r_{C}} , and the radius of midsphere (midradius) r M {\displaystyle r_{M}} are, respectively:
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