# Icosahedron

In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes from Greek είκοσι (eíkosi), meaning "twenty", and εδρα (hédra), meaning "seat". The plural can be either "icosahedra" or "icosahedrons" (-).

The most notable icosahedra have twenty triangular faces, with five meeting at each vertex. These include the convex regular icosahedron, one of the five highly symmetrical regular Platonic solids.

There are many other icosahedra.

## Icosahedral symmetry

### Regular icosahedron

Main article: Regular icosahedron

A regular icosahedron has 20 regular triangle faces with five meeting at each of its twelve vertices. The term is usually applied to the convex form, which is one of the five regular Platonic solids.

It is represented by its Schläfli symbol {3,5}, and sometimes by its vertex figure as 3.3.3.3.3 or 35. Its dual is the regular dodecahedron {5,3}, having three regular pentagonal faces around each vertex.

### Great icosahedron

Main article: Great icosahedron

The great icosahedron is one of the four regular star Kepler-Poinsot polyhedra.

### Stellated icosahedra

Stellation is the process of extending the faces or edges of a polyhedron until they meet to form a new polyhedron. In their book The fifty nine icosahedra, Coxeter et. al. enumerated 58 such stellations of the regular icosahedron, all having icosahedral symmetry.

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.

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

## Pyritohedral symmetry

Pyritohedral and tetrahedral symmetries

Four views of an icosahedron with tetrahedral symmetry, with eight equilateral triangles (red and yellow), and 12 blue isosceles triangles. Yellow and red triangles are the same color in pyritohedral symmetry.
Coxeter diagrams (pyritohedral)
(tetrahedral)
Schläfli symbols{3,4}
sr{3,3} or $s\begin{Bmatrix} 3 \\ 3 \end{Bmatrix}$
Faces20 triangles:
8 equilateral
12 isosceles
Edges30 (6 short + 24 long)
Vertices12
Symmetry groupTh, [4,3+], (3*2), order 24
Rotation groupTd, [3,3]+, (332), order 12
Dual polyhedronPyritohedron
Propertiesconvex

Net
Construction from the vertices of a truncated octahedron, showing internal rectangles with edge length ratios of 2:1.

A regular icosahedron can be constructed with pyritohedral symmetry, and is called a snub octahedron or snub tetratetrahedron or snub tetrahedron. this can be seen as an alternated truncated octahedron. If all the triangles are equilateral, the symmetry can also be distinguished by colouring the 8 and 12 triangle sets differently.

Pyritohedral symmetry has the symbol (3*2), [4,3+], with order 24. Tetrahedral symmetry has the symbol (*332), [3,3]+, with order 12. These lower symmetries allow geometric distortions from 20 equilateral triangular faces, instead having 8 equilateral triangles and 12 congruent isosceles triangles.

### Cartesian coordinates

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.[1]

### Jessen's icosahedron

Main article: Jessen's icosahedron

In Jessen's icosahedron, sometimes called Jessen's orthogonal icosahedron, the 12 isosceles faces are arranged differently such that the figure is non-convex. It 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.

## Other symmetries

### Rhombic icosahedron

Main article: Rhombic icosahedron

The rhombic icosahedron is a zonohedron made up of 20 congruent rhombs. It can be derived from the rhombic triacontahedron by removing 10 middle faces. Even though all the faces are congruent, the rhombic icosahedron is not face-transitive.

### Tetrahedral colouring

20 triangles can also be arranged with tetrahedral symmetry (332), [3,3]+, seen as the 8 triangles marked (colored) in alternating pairs of four, with order 12. These symmetries offer Coxeter diagrams: and respectfully, each representing the lower symmetry to the regular icosahedron , (*532), [5,3] icosahedral symmetry of order 120.

### 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:[2]

J22J35J36J59J60J92

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

## References

1. ^ John Baez (September 11, 2011). "Fool's Gold".
2. ^ Icosahedron on Mathworld.