# Icosahedron

Convex regular icosahedron
A tensegrity icosahedron

In geometry, an icosahedron (/ˌkɒsəˈhdrən, -kə-, -k-/ or /ˌkɒsəˈhdrən/[1]) is a polyhedron with 20 faces. The name comes from Ancient Greek εἴκοσι (eíkosi) 'twenty' and from Ancient Greek ἕδρα (hédra) ' seat'. The plural can be either "icosahedra" (/-drə/) or "icosahedrons".

There are infinitely many non-similar shapes of icosahedra, some of them being more symmetrical than others. The best known is the (convex, non-stellated) regular icosahedron—one of the Platonic solids—whose faces are 20 equilateral triangles.

## 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

A detail of Spinoza monument in Amsterdam

The great icosahedron is one of the four regular star Kepler-Poinsot polyhedra. Its Schläfli symbol is {3,5/2}. Like the convex form, it also has 20 equilateral triangle faces, but its vertex figure is a pentagram rather than a pentagon, leading to geometrically intersecting faces. The intersections of the triangles do not represent new edges.

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.

(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
Coxeter diagrams (pyritohedral)
(tetrahedral)
Schläfli symbols{3,4}
sr{3,3} or ${\displaystyle s{\begin{Bmatrix}3\\3\end{Bmatrix}}}$
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
 A regular icosahedron is topologically identical to a cuboctahedron with its 6 square faces bisected on diagonals with pyritohedral symmetry.

A regular icosahedron can be distorted or marked up as a lower pyritohedral symmetry,[2] and is called a snub octahedron, snub tetratetrahedron, snub tetrahedron, and pseudo-icosahedron. 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), [3+,4], 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.

These symmetries offer Coxeter diagrams: and respectively, each representing the lower symmetry to the regular icosahedron , (*532), [5,3] icosahedral symmetry of order 120.

### Cartesian coordinates

Construction from the vertices of a truncated octahedron, showing internal rectangles.

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.[2]

### Jessen's icosahedron

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.

## Other icosahedra

Rhombic icosahedron

### 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.

### 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:[3]

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

• 600-cell
• Icosoku

## References

1. ^ Jones, Daniel (2003) [1917], Peter Roach; James Hartmann; Jane Setter (eds.), English Pronouncing Dictionary, Cambridge: Cambridge University Press, ISBN 3-12-539683-2
2. ^ a b John Baez (September 11, 2011). "Fool's Gold".
3. ^ Icosahedron on Mathworld.

Icosahedron.png

Image d'un icosaèdre, un des solides de Platon
Icosahedron in cuboctahedron net.png
Author/Creator: Tomruen, Licence: CC BY-SA 4.0
A topological en:regular icosahedron in net of a en:Cuboctahedron by bisecting the square faces.
Parabiaugmented dodecahedron.png
Author/Creator: unknown, Licence: CC-BY-SA-3.0
Seventeenth stellation of icosahedron.png
Author/Creator: Jim2k, Licence: CC BY-SA 3.0
Семнадцатая звёздчатая форма икосаэдра.
Icosahedral tensegrity structure.png
Author/Creator: QuarterNotes, Licence: CC BY-SA 4.0
Uniform polyhedron-43-h01.svg
* KaleidoTile, Topology and Geometry Software, Jeff Weeks Icosahedron hand-colored as alternated truncated octahedron.
First compound stellation of icosahedron.png
Author/Creator: The original uploader was Tomruen at English Wikipedia., Licence: CC BY-SA 3.0
First compound stellation of icosahedron
Triangular hebesphenorotunda.png
Author/Creator: unknown, Licence: CC-BY-SA-3.0
Metabiaugmented dodecahedron.png
Author/Creator: unknown, Licence: CC-BY-SA-3.0
Icosahedron-spinoza.jpg
Author/Creator: DFK2021 (Dmitry Feichtner-Kozlov), Licence: CC BY-SA 4.0
Деталь памятника Спинозе в Амстердаме
Third stellation of icosahedron.svg
Author/Creator: Mrmw, Licence: CC0
Third stellation of icosahedron
Elongated triangular orthobicupola.png
Author/Creator: unknown, Licence: CC-BY-SA-3.0
Sixteenth stellation of icosahedron.png
Author/Creator: Jim2k, Licence: CC BY-SA 3.0
Шестнадцатая звёздчатая форма икосаэдра.
Zeroth stellation of icosahedron.png
Author/Creator: The original uploader was Tomruen at English Wikipedia., Licence: CC BY-SA 3.0
Zeroth stellation of icosahedron
Icosahedron.svg
Author/Creator: DTR, Licence: CC-BY-SA-3.0
Icosahedron.
Third compound stellation of icosahedron.png
Author/Creator: Tearfate, Licence: CC BY-SA 3.0
Third compound stellation of icosahedron
Stellation diagram of icosahedron.svg
Author/Creator: Parcly Taxel, Licence: FAL
The stellation diagram of the regular icosahedron, with the original facet marked. Vector version of Zeroth stellation of icosahedron facets.png by Tomruen. Code to generate this SVG file is available here.
Gyroelongated triangular cupola.png
Author/Creator: unknown, Licence: CC-BY-SA-3.0
Icosahedron in cuboctahedron.png
Author/Creator: Tomruen, Licence: CC BY-SA 4.0
A en:regular icosahedron in en:cuboctahedron with square faces bisected on diagonals with en:pyritohedral symmetry.
Elongated triangular gyrobicupola.png
Author/Creator: unknown, Licence: CC-BY-SA-3.0
Ninth stellation of icosahedron.png
Author/Creator: The original uploader was Tomruen at English Wikipedia., Licence: CC BY-SA 3.0
Ninth stellation of icosahedron Could be a medial triambic icosahedron or a great triambic icosahedron.