View source for Icosahedron

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{{Short description|Polyhedron with 20 faces}}
[[File:Icosahedron.svg|thumb|Convex [[regular icosahedron]]]]
[[File:Icosahedral tensegrity structure.png|thumb|upright|A [[tensegrity]] icosahedron]]

In [[geometry]], an '''icosahedron''' ({{IPAc-en|ˌ|aɪ|k|ɒ|s|ə|ˈ|h|iː|d|r|ən|,_|-|k|ə|-|,_|-|k|oʊ|-}} or {{IPAc-en|aɪ|ˌ|k|ɒ|s|ə|ˈ|h|iː|d|r|ən}}{{refn|{{Citation |last=Jones |first=Daniel |author-link=Daniel Jones (phonetician) |title=English Pronouncing Dictionary |editor=Peter Roach |editor2=James Hartmann |editor3=Jane Setter |place=Cambridge |publisher=Cambridge University Press |orig-year=1917 |year=2003 |isbn=3-12-539683-2}}}}) is a [[polyhedron]] with 20 faces. The name comes {{ety|grc|''εἴκοσι'' (eíkosi)|twenty||''ἕδρα'' (hédra)|seat}}. The plural can be either "icosahedra" ({{IPAc-en|-|d|r|ə}}) or "icosahedrons".

There are infinitely many non-[[similarity (geometry)|similar]] shapes of icosahedra, some of them being more symmetrical than others. The best known is the ([[convex polyhedron|convex]], non-[[stellation|stellated]]) [[regular icosahedron]]—one of the [[Platonic solid]]s—whose faces are 20 [[equilateral triangle]]s.

==Regular icosahedra==
{{multiple image
|image1      = Icosahedron.png
|caption1    = Convex [[regular icosahedron]]
|image2      = Great icosahedron.png
|caption2    = [[Great icosahedron]]
|header      = Two kinds of regular icosahedra
|total_width = 300px
}}
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===
{{Main|Regular icosahedron}}
[[File:golden_rectangle_regular_icosahedron.jpg|thumb|upright|Three interlocking [[golden rectangle]]s inscribed in a con&shy;vex regular icosahedron]]
The convex regular icosahedron is usually referred to simply as the ''regular icosahedron'', one of the five regular [[Platonic solid]]s, 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===
{{main|Great icosahedron}}
[[File:Icosahedron-spinoza.jpg|alt=A detail of Spinoza monument in Amsterdam|thumb|upright|A detail of [[Baruch Spinoza|Spinoza]] monument in [[Amsterdam]]]]
The [[great icosahedron]] is one of the four regular star [[Kepler-Poinsot polyhedra]]. Its [[Schläfli symbol]] is {3, {{sfrac|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]] {{{sfrac|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.
{{Icosahedron stellations}}

==Pyritohedral symmetry==
{| class=wikitable align=right width="320"
!bgcolor=#e7dcc3 colspan=2|Pyritohedral and tetrahedral symmetries
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]s||{{CDD|node_h|3|node_h|4|node}} (pyritohedral) [[File:Uniform polyhedron-43-h01.svg|24px]]<br />{{CDD|node_h|3|node_h|3|node_h}} (tetrahedral) [[File:Uniform polyhedron-33-s012.svg|24px]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||s{3,4}<br />sr{3,3} or <math>s\begin{Bmatrix} 3 \\ 3 \end{Bmatrix}</math>
|-
|bgcolor=#e7dcc3|[[Face (geometry)|Faces]]||20 triangles:<br />8 equilateral<br />12 isosceles
|-
|bgcolor=#e7dcc3|[[Edge (geometry)|Edges]]||30 (6 short + 24 long)
|-
|bgcolor=#e7dcc3|[[Vertex (geometry)|Vertices]]||12
|-
|bgcolor=#e7dcc3|[[List of spherical symmetry groups#Polyhedral symmetry|Symmetry group]]||[[Pyritohedral symmetry|T<sub>h</sub>]], [4,3<sup>+</sup>], (3*2), order 24
|-
|bgcolor=#e7dcc3|[[Point groups in three dimensions#Rotation groups|Rotation group]]||[[Tetrahedral symmetry|T<sub>d</sub>]], [3,3]<sup>+</sup>, (332), order 12
|-
|bgcolor=#e7dcc3|[[Dual polyhedron]]||[[Pyritohedron]]
|-
|bgcolor=#e7dcc3|Properties||[[convex set|convex]]
|-
!colspan=2|[[File:Pseudoicosahedron flat.png|200px]]<br />[[Net (polyhedron)|Net]]
|-
| colspan=2|
{| width=320px align=right
|[[File:Icosahedron in cuboctahedron.png|220px]]
|[[File:Icosahedron in cuboctahedron net.png|100px]]
|-
|colspan=2|A regular icosahedron is topologically identical to a [[cuboctahedron]] with its 6 square faces bisected on diagonals with pyritohedral symmetry. There exists a [[Kinematics of the cuboctahedron|kinematic transformation between cuboctahedron and icosahedron]].
|}

|}
A ''regular icosahedron'' can be distorted or marked up as a lower [[pyritohedral]] symmetry,<ref name=Koca>{{cite journal|
last1=Koca|first1=Nazife|
last2=Al-Mukhaini|first2=Aida|
last3=Koca|first3=Mehmet|
last4=Al Qanobi|first4=Amal|
date=2016-12-01|
page=139|
title=Symmetry of the Pyritohedron and Lattices|
volume=21|
doi=10.24200/squjs.vol21iss2pp139-149|
journal=Sultan Qaboos University Journal for Science [SQUJS]|issue=2 |
doi-access=free}}</ref><ref name=Baez/> and is called a '''snub octahedron''', '''snub tetratetrahedron''', '''snub tetrahedron''', and '''pseudo-icosahedron'''.<ref>{{cite book
 | last = Kappraff | first = Jay
 | year = 1991
 | title = Connections: The Geometric Bridge Between Art and Science
 | publisher = World Scientific
 | edition = 2nd
 | url = https://books.google.com/books?id=tz76s0ZGFiQC&pg=PA475
 | page = 475
}}</ref> This can be seen as an [[alternation (geometry)|alternated]] [[truncated octahedron]]. If all the triangles are [[equilateral triangle|equilateral]], the symmetry can also be distinguished by colouring the 8 and 12 triangle sets differently.

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

These symmetries offer [[Coxeter diagram]]s: {{CDD|node|4|node_h|3|node_h}} and {{CDD|node_h|3|node_h|3|node_h}} respectively, each representing the lower symmetry to the [[regular icosahedron]] {{CDD|node_1|3|node|5|node}}, (*532), [5,3] [[icosahedral symmetry]] of order 120.

===Cartesian coordinates===
[[File:Truncated octahedron internal rectangles.png|150px|thumb|left|Construction from the vertices of a [[truncated octahedron]], showing internal rectangles.]]
The [[Cartesian 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 [[alternation (geometry)|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]].<ref name=Baez>{{cite web|url=http://math.ucr.edu/home/baez/golden.html|title=Fool's Gold|author=John Baez|date=September 11, 2011}}</ref>

===Jessen's icosahedron===
[[File:Jessen's icosahedron.svg|left|thumb|Jessen's icosahedron]]
{{main|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 angle|right]] [[dihedral angle]]s.

It is [[Hilbert's third problem|scissors congruent]] to a cube, meaning that it can be sliced into smaller polyhedral pieces that can be rearranged to form a solid cube.
{{Clear}}

===Cuboctahedron===
[[File:A3-P5-P3.gif|thumb|Progressions between an [[octahedron]], [[pseudoicosahedron]], and cuboctahedron. The cuboctahedron can flex this way even if its edges (but not its faces) are rigid.]]
{{main|Kinematics of the cuboctahedron}}
A regular icosahedron is topologically identical to a [[cuboctahedron]] with its 6 square faces bisected on diagonals with pyritohedral symmetry. The icosahedra with pyritohedral symmetry constitute an infinite family of polyhedra which include the cuboctahedron, regular icosahedron, Jessen's icosahedron, and [[Orientable double cover|double cover]] octahedron. Cyclical kinematic transformations among the members of this family exist.

==Other icosahedra==
[[File:Rhombic icosahedron.png|thumb|[[Rhombic icosahedron]]]]

===Rhombic icosahedron===
{{Main|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 (geometry)|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 solid]]s are icosahedra:<ref>[http://mathworld.wolfram.com/Icosahedron.html Icosahedron] on Mathworld.</ref>
{| class=wikitable width=720
!J22
!J35
!J36
!J59
!J60
!J92
|- align=center valign=top
|[[File:Gyroelongated triangular cupola.png|120px]]<br />[[Gyroelongated triangular cupola]]
|[[File:Elongated triangular orthobicupola.png|120px]]<br />[[Elongated triangular orthobicupola]]
|[[File:Elongated triangular gyrobicupola.png|120px]]<br />[[Elongated triangular gyrobicupola]]
|[[File:Parabiaugmented dodecahedron.png|120px]]<br />[[Parabiaugmented dodecahedron]]
|[[File:Metabiaugmented dodecahedron.png|120px]]<br />[[Metabiaugmented dodecahedron]]
|[[File:Triangular hebesphenorotunda.png|120px]]<br />[[Triangular hebesphenorotunda]]
|-
|[[File:Johnson solid 22 net.png|120px]]
|[[File:Johnson solid 35 net.png|120px]]
|[[File:Johnson solid 36 net.png|120px]]
|[[File:Johnson solid 59 net.png|120px]]
|[[File:Johnson solid 60 net.png|120px]]
|[[File:Johnson solid 92 net.png|120px]]
|- align=center valign=top
|16 triangles<br />3 squares<br />&nbsp;<br />1 hexagon
|8 triangles<br />12 squares
|8 triangles<br />12 squares
|10 triangles<br />&nbsp;<br />10 pentagons
|10 triangles<br />&nbsp;<br />10 pentagons
|13 triangles<br />3 squares<br />3 pentagons<br />1 hexagon
|}

== See also ==
* [[Truncated icosahedron]]
* [[600-cell]]
* [[Icosoku]]

==References==
{{Commons category}}
{{reflist}}

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[[Category:Geodesic polyhedra]]
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