Regular icosahedron

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.

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:

The regular icosahedron can be unfolded into 43,380 different nets. The earliest net appeared in Albrecht Durer's Painter's Manual in 1525.

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 ${\displaystyle A}$ is twenty times that of each of its equilateral triangle faces. Its volume ${\displaystyle V}$ 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 ${\displaystyle a}$ of a regular icosahedron, both expressions are: $${\displaystyle A=5{\sqrt {3}}a^{2}\approx 8.660a^{2},\qquad V={\frac {5\varphi ^{2}}{6}}a^{3}\approx 2.182a^{3}.}$$

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 ${\displaystyle a}$ of a regular icosahedron, the radius of insphere (inradius) ${\displaystyle r_{I}}$, the radius of circumsphere (circumradius) ${\displaystyle r_{C}}$, and the radius of midsphere (midradius) ${\displaystyle r_{M}}$ are, respectively: $${\displaystyle r_{I}={\frac {\varphi ^{2}a}{2{\sqrt {3}}}}\approx 0.756a,\qquad r_{C}={\frac {\sqrt {\varphi ^{2}+1}}{2}}a\approx 0.951a,\qquad r_{M}={\frac {\varphi }{2}}a\approx 0.809a.}$$

A problem dating back to the ancient Greeks is determining which of two shapes has a larger volume: a regular icosahedron inscribed in a sphere, or a regular dodecahedron inscribed in the same sphere. The problem was solved by Hero, Pappus, and Fibonacci, among others. Apollonius of Perga discovered the curious result that the ratio of volumes of these two shapes is the same as the ratio of their surface areas. Both volumes have formulas involving the golden ratio, but taken to different powers. As it turns out, the regular icosahedron occupies less of the sphere's volume (60.54%) than the regular dodecahedron (66.49%).