Icosahedron

In

Two kinds of regular icosahedra

Convex regular icosahedron

Great icosahedron

There are two objects, one convex and one nonconvex, that can both be called regular icosahedra. Each has 30 edges and 20 equilateral triangle faces with five meeting at each of its twelve vertices. Both have icosahedral symmetry. The term "regular icosahedron" generally refers to the convex variety, while the nonconvex form is called a great icosahedron.

Convex regular icosahedron

The convex regular icosahedron is usually referred to simply as the regular icosahedron, one of the five regular Platonic solids, and is represented by its Schläfli symbol {3, 5}, containing 20 triangular faces, with 5 faces meeting around each vertex.

Its dual polyhedron is the regular dodecahedron {5, 3} having three regular pentagonal faces around each vertex.

Great icosahedron

The

Its dual polyhedron is the great stellated dodecahedron {5/2, 3}, having three regular star pentagonal faces around each vertex.

Stellated icosahedra

Stellation is the process of extending the faces or edges of a polyhedron until they meet to form a new polyhedron. It is done symmetrically so that the resulting figure retains the overall symmetry of the parent figure.

In their book The Fifty-Nine Icosahedra, Coxeter et al. enumerated 58 such stellations of the regular icosahedron.

Of these, many have a single face in each of the 20 face planes and so are also icosahedra. The great icosahedron is among them.

Other stellations have more than one face in each plane or form compounds of simpler polyhedra. These are not strictly icosahedra, although they are often referred to as such.

Notable stellations of the icosahedron
Regular	Uniform duals			Regular compounds			Regular star	Others
(Convex) icosahedron	Small triambic icosahedron	Medial triambic icosahedron	Great triambic icosahedron	Compound of five octahedra	Compound of five tetrahedra	Compound of ten tetrahedra	Great icosahedron	Excavated dodecahedron	Final stellation
The stellation process on the icosahedron creates a number of related compounds with icosahedral symmetry .

Pyritohedral symmetry

Pyritohedral and tetrahedral symmetries
Coxeter diagrams	(pyritohedral) (tetrahedral)
Schläfli symbol	s{3,4} sr{3,3} or ${\displaystyle s{\begin{Bmatrix}3\\3\end{Bmatrix}}}$
Faces	20 triangles: 8 equilateral 12 isosceles
Edges	30 (6 short + 24 long)
Vertices	12
Symmetry group	Th , [4,3+], (3*2), order 24
Rotation group	Td, [3,3]+, (332), order 12
Dual polyhedron	Pyritohedron
Properties	convex
Net
A regular icosahedron is topologically identical to a cuboctahedron with its 6 square faces bisected on diagonals with pyritohedral symmetry. There exists a kinematic transformation between cuboctahedron and icosahedron.

A regular icosahedron can be distorted or marked up as a lower

Coxeter diagrams: and respectively, each representing the lower symmetry to the regular icosahedron , (*532), [5,3] icosahedral symmetry

The coordinates of the 12 vertices can be defined by the vectors defined by all the possible cyclic permutations and sign-flips of coordinates of the form (2, 1, 0). These coordinates represent the truncated octahedron with alternated vertices deleted.

This construction is called a snub tetrahedron in its regular icosahedron form, generated by the same operations carried out starting with the vector (ϕ, 1, 0), where ϕ is the golden ratio.^[3]

Jessen's icosahedron

In Jessen's icosahedron, sometimes called Jessen's orthogonal icosahedron, the 12 isosceles faces are arranged differently so that the figure is non-convex and has right dihedral angles.

It is scissors congruent to a cube, meaning that it can be sliced into smaller polyhedral pieces that can be rearranged to form a solid cube.

Cuboctahedron

A regular icosahedron is topologically identical to a

Other icosahedra

Rhombic icosahedron

The

Pyramid and prism symmetries

Common icosahedra with pyramid and prism symmetries include:

• 19-sided pyramid (plus 1 base = 20).
• 18-sided prism (plus 2 ends = 20).
• 9-sided antiprism (2 sets of 9 sides + 2 ends = 20).
• 10-sided bipyramid (2 sets of 10 sides = 20).
• 10-sided trapezohedron (2 sets of 10 sides = 20).

Johnson solids

Several Johnson solids are icosahedra:^[5]

J22	J35	J36	J59	J60	J92
Gyroelongated triangular cupola	Elongated triangular orthobicupola	Elongated triangular gyrobicupola	Parabiaugmented dodecahedron	Metabiaugmented dodecahedron	Triangular hebesphenorotunda
16 triangles 3 squares   1 hexagon	8 triangles 12 squares	8 triangles 12 squares	10 triangles   10 pentagons	10 triangles   10 pentagons	13 triangles 3 squares 3 pentagons 1 hexagon

See also

• Truncated icosahedron
• 600-cell
• Icosoku

References

Wikimedia Commons has media related to Icosahedron.