Regular Icosahedron

The regular icosahedron, often simply called "the" icosahedron, is the regular polyhedron and Platonic solid having 12 polyhedron vertices, 30 polyhedron edges, and 20 equivalent equilateral triangle faces, . It is illustrated above together with a wireframe version and a net that can be used for its construction.

Precisely or approximately icosahedral objects were also made as dice in Greco-Roman Egypt. Surviving examples dated from roughly 200 BC to 400 include a carefully cut serpentinite die labeled with the Greek numerals 1 through 20 and a limestone die whose 20 faces bear divine names in Demotic script (Minas-Nerpel 2007, Platz-Horster 2017).

Third-century BC pottery sherds from Elephantine Island preserve evidence of Greek work on the construction of the regular icosahedron. Text on sherd P12609 closely follows proposition XIII.16 of the Elements. The related sherd P11999, illustrated above, bears Greek text and a figure inscribed in a circle. Together these fragments document the study of advanced geometry on an everyday writing material (Mau and Müller 1962, pp. 1-7).

The regular icosahedron is also the uniform polyhedron with Maeder index 22 (Maeder 1997), Wenninger index 4 (Wenninger 1989), Coxeter index 25 (Coxeter et al. 1954), and Har'El index 27 (Har'El 1993). It is described by the Schläfli symbol and Wythoff symbol . Coxeter et al. (1999) have enumerated 58 icosahedron stellations satisfying Miller's rules (giving a total of 59 solids when the icosahedron itself is included).

A number of symmetric projections of the regular icosahedron are illustrated above.

The regular icosahedron is implemented in the Wolfram Language as Icosahedron[] or UniformPolyhedron["Icosahedron"], and precomputed properties are available as PolyhedronData["Icosahedron"].

Two icosahedra constructed in origami are illustrated above (Gurkewitz and Arnstein 1995, p. 53). This construction uses 30 triangle edge modules, each made from a single sheet of origami paper.

Two icosahedra appear as polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43).

There are 43380 distinct nets for the icosahedron, the same number as for the dodecahedron (Hippenmeyer 1979, Buekenhout and Parker 1998, Bouzette and Vandamme n.d.).

The icosahedron has the icosahedral group of symmetries. The connectivity of the vertices is given by the icosahedral graph.

The regular icosahedron is the convex hull of the first and second dodecahedron stellations, great dodecahedron, great icosahedron, 11th icosahedron stellation, and small stellated dodecahedron.

The dual polyhedron of a regular icosahedron is a regular dodecahedron. The centers of the faces of an icosahedron form a dodecahedron, and vice versa, illustrated above (Steinhaus 1999, pp. 199-201).

In particular, fifteen golden rectangles span the interior of the icosahedron. These rectangles use the 30 edges, each of which pairs up with its opposite edge to form a golden rectangle.

When each of the 20 faces receives a distinct color, there are colorings up to rotation, since the rotational icosahedral group has group order 60 and only its identity element fixes such a coloring. More general questions of polyhedron coloring can be addressed using the Pólya enumeration theorem.

Certain sets of eight centers of the faces of an icosahedron form the vertices of a cube. The five such cubes form the cube 5-compound, which is the basis for Jessen's orthogonal icosahedron.

A plane through the center and perpendicular to a axis of an icosahedron cuts the solid in a regular decagonal cross section (Holden 1991, pp. 24-25).

The long diagonals of the faces of the rhombic triacontahedron give the edges of an icosahedron (Steinhaus 1999, pp. 209-210).

The following table gives polyhedra which can be constructed by augmentation of an icosahedron of unit edge length by pyramids of signed heights , where is its inradius.

A construction for an icosahedron with side length places the end vertices at and the central vertices around two staggered circles of radii and heights . By a suitable rotation, the polyhedron vertices of an icosahedron of side length 2 can also be placed at , , and , where is the golden ratio. These points divide the polyhedron edges of an octahedron into segments with lengths in the ratio . Another orientation of a unit-edge icosahedron places two opposite triangular faces in an orientation parallel to the -plane. In this orientation, the distance from the top plane to the triangle of vertices below it is , equal to the circumradius of a face. The circumradius of is given by

To derive the volume of an icosahedron having edge length , consider the orientation so that two polyhedron vertices are oriented on top and bottom. The vertical distance between the top and bottom pentagonal pyramids is then given by

where

is the height of an equilateral triangle. Let and be the circumradius and apothem of the regular pentagon of side . Its sagitta is

giving

Substituting these expressions for and into the equation for gives

which is identical to the radius of a pentagon of side . The circumradius is then

where

is half the total height of a pentagonal dipyramid. Therefore,

Taking the square root gives the circumradius

The inradius is

The square of the midradius is

so

The dihedral angle is

and the Dehn invariant of a regular icosahedron of unit edge length is

(OEIS A377698), where the first expression uses the basis of Conway et al. (1999, p. 331).

The area of one face is the area of an equilateral triangle

The volume can be computed by taking 20 pyramids of height

Apollonius showed that for an icosahedron and a dodecahedron with the same inradius,

where is the volume and the surface area, with the actual ratio being