View source for Regular icosahedron

{{sources|date=October 2023}}
{{Short description|Polyhedron with 20 regular triangular faces}}
{{Reg polyhedra db|Reg polyhedron stat table|I}}
[[File:Regular icosahedron.stl|thumb|3D model of a regular icosahedron]]
In [[Geometry|geometry]], a '''regular [[Icosahedron|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 |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 convex [[Polyhedron|polyhedron]] with 20 faces, 30 edges and 12 vertices. It is one of the five [[Platonic solid]]s, and the one with the most faces.

It has five equilateral triangular faces meeting at each vertex. It is represented by its [[Schläfli symbol]] {3,5}, or sometimes by its [[Vertex figure|vertex figure]] as 3.3.3.3.3 or 3<sup>5</sup>. It is the [[Dual polyhedron|dual]] of the [[Regular dodecahedron|regular dodecahedron]], which is represented by {5,3}, having three pentagonal faces around each vertex. In most contexts, the unqualified use of the word "icosahedron" refers specifically to this figure.

A regular icosahedron is a strictly convex [[Deltahedron|deltahedron]] and a [[Johnson solid#Names|gyroelongated]] [[Pentagonal bipyramid|pentagonal bipyramid]] and a biaugmented [[Pentagonal antiprism|pentagonal antiprism]] in any of six orientations.

The name comes {{ety|gre|''εἴκοσι'' (eíkosi)|twenty||''ἕδρα'' (hédra)|seat}}. The plural can be either "icosahedrons" or "icosahedra" ({{IPAc-en|-|d|r|ə}}).

==Dimensions==
[[File:Icosaedro desarrollo.gif|thumb|Net folding into icosahedron]]
If the edge length of a regular icosahedron is <math>a</math>, the [[Radius|radius]] of a circumscribed [[Sphere|sphere]] (one that touches the icosahedron at all vertices) is
<math display=block>r_u = \frac{a}{2} \sqrt{\varphi \sqrt{5}} = \frac{a}{4} \sqrt{10 +2\sqrt{5}} = a\sin\frac{2\pi}{5} \approx 0.951\,056\,5163 \cdot a</math>
and the radius of an inscribed sphere ([[Tangent|tangent]] to each of the icosahedron's faces) is
<math display=block>r_i = \frac{\varphi^2 a}{2 \sqrt{3}} = \frac{\sqrt{3}}{12} \left(3+ \sqrt{5} \right) a \approx 0.755\,761\,3141\cdot a</math>
while the midradius, which touches the middle of each edge, is
<math display=block>r_m = \frac{a \varphi}{2} = \frac{1}{4} \left(1+\sqrt{5}\right) a = a\cos\frac{\pi}{5} \approx 0.809\,016\,99\cdot a</math>
where <math>\varphi</math> is the [[Golden ratio|golden ratio]].

==Area and volume==
The surface area <math>A</math> and the [[Physics:Volume|volume]] <math>V</math> of a regular icosahedron of edge length <math>a</math> are:
<math display=block>A = 5\sqrt{3}a^2 \approx 8.660\,254\,04a^2</math>
<math display=block>V = \frac{5}{12} \left(3+\sqrt{5}\right)a^3 \approx 2.181\,694\,99a^3</math>
The latter is {{math|''F'' {{=}} 20}} times the volume of a general [[Tetrahedron|tetrahedron]] with apex at the center of the
inscribed sphere, where the volume of the tetrahedron is one third times the base area <math>\frac{\sqrt{3}}{4}a^2</math> times its height <math>r_i</math>.

The volume filling factor of the circumscribed sphere is:
<math display=block>f=\frac{V}{\frac43 \pi r_u^3} = \frac{20\left(3+\sqrt{5}\right)}{\left(2\sqrt{5}+10\right)^{\frac32}\pi}\approx 0.605\,461\,3829,</math>
compared to 66.49% for a dodecahedron. A sphere inscribed in an icosahedron will enclose 89.635% of its volume, compared to only 75.47% for a dodecahedron.

The midsphere of an icosahedron will have a volume 1.01664 times the volume of the icosahedron, which is by far the closest similarity in volume of any platonic solid with its midsphere. This arguably makes the icosahedron the "roundest" of the platonic solids.

==Cartesian coordinates==
[[Image:Icosahedron-golden-rectangles.svg|thumb|right|Icosahedron vertices form three orthogonal golden rectangles]]
The vertices of an icosahedron centered at the origin with an edge length of 2 and a circumradius of <math display=inline>\sqrt{\varphi^2 + 1} \approx 1.902</math> are<ref>{{mathworld |title=Icosahedral group |urlname=IcosahedralGroup}}</ref><math display=block>(0, \pm 1, \pm \varphi)</math>
<math display=block>(\pm 1, \pm \varphi, 0)</math>
<math display=block>(\pm \varphi, 0, \pm 1)</math>where <math display="inline">\varphi = \frac{1 + \sqrt{5}}{2}</math> is the [[Golden ratio|golden ratio]]. Taking all permutations of these coordinates (not just cyclic permutations) results in the [[Compound of two icosahedra]].

The vertices of the icosahedron form five sets of three concentric, mutually orthogonal [[Golden rectangle|golden rectangle]]s, whose edges form [[Borromean rings]].

If the original icosahedron has edge length 1, its dual [[Dodecahedron|dodecahedron]] has edge length <math display=inline>\frac{1}{\varphi} = \varphi - 1 = \frac{\sqrt{5} - 1}{2}</math>.

[[File:Icosahedron model.JPG|right|thumb|Model of an icosahedron made with metallic spheres and magnetic connectors]]
The 12 edges of a regular [[Octahedron|octahedron]] can be subdivided in the golden ratio so that the resulting vertices define a regular icosahedron. This is done by first placing vectors along the octahedron's edges such that each face is bounded by a cycle, then similarly subdividing each edge into the golden mean along the direction of its vector. The [[Compound of five octahedra|five octahedra]] defining any given icosahedron form a regular polyhedral compound, while the [[Compound of two icosahedra|two icosahedra]] that can be defined in this way from any given octahedron form a [[Uniform polyhedron compound|uniform polyhedron compound]].

[[File:Вписанный правильный икосаэдр и четыре плоскости.gif|thumb|right|Regular icosahedron and its [[Circumscribed sphere|circumscribed sphere]]. Vertices of the regular icosahedron lie in four parallel planes, forming in them four [[Equilateral triangle|equilateral triangle]]s; this was proved by Pappus of Alexandria]]

The vertices of an icosahedron centred at the origin with an edge length of <math display=inline>\sqrt{2 \cdot \left(1-\frac{1}{\sqrt{5}} \right)} \approx 1.051</math> and a circumradius of 1 (with four points lying in the x-y plane) are<math display="block">\pm(1,0,0)</math>
<math display="block">\pm \left(\frac{1}{\sqrt{5}},\frac{2}{\sqrt{5}},0 \right)</math>
<math display="block">\pm \left(\frac{1}{\sqrt{5}},\frac{1-\frac{1}{\sqrt{5}}}{2},\pm\sqrt{\frac{1+\frac{1}{\sqrt{5}}}{2}} \right)</math>
<math display="block">\pm \left(\frac{1}{\sqrt{5}},\frac{-1-\frac{1}{\sqrt{5}}}{2},\pm\sqrt{\frac{1-\frac{1}{\sqrt{5}}}{2}} \right)</math>

=== Spherical coordinates ===
The locations of the vertices of a regular icosahedron can be described using spherical coordinates, for instance as latitude and longitude. If two vertices are taken to be at the north and south poles (latitude ±90°), then the other ten vertices are at latitude {{nowrap|± arctan {{sfrac|1|2}}}} = ±26.57°. These ten vertices are at evenly spaced longitudes (36° apart), alternating between north and south latitudes.

This scheme takes advantage of the fact that the regular icosahedron is a pentagonal [[Gyroelongated bipyramid|gyroelongated bipyramid]], with D<sub>5d</sub> [[Dihedral symmetry in three dimensions|dihedral symmetry]]—that is, it is formed of two congruent pentagonal pyramids joined by a pentagonal [[Antiprism|antiprism]].

==Orthogonal projections==
The icosahedron has three special orthogonal projections, centered on a face, an edge and a vertex:
{|class=wikitable width=360
|+ Orthogonal projections
|-
!Centered by
!Face
!Edge
!Vertex
|- align=center
!Coxeter plane
!A<sub>2</sub>
!A<sub>3</sub>
!H<sub>3</sub>
|- align=center
!Graph
|[[File:Icosahedron A2 projection.svg|100px]]
|[[File:Icosahedron graph A3 1.png|100px]]
|[[File:Icosahedron H3 projection.svg|100px]]
|- align=center
!Projective<BR>symmetry
|[6]
|[2]
|[10]
|- align=center
!Graph
|[[File:Icosahedron fnormal.png|100px]]<BR>Face normal
|[[File:Icosahedron graph A3 2.png|100px]]<BR>Edge normal
|[[File:Icosahedron vnormal.png|100px]]<BR>Vertex normal
|}

== As a configuration ==
This [[Platonic solid#As a configuration|configuration matrix]] represents the icosahedron. The rows and columns correspond to vertices, edges, and faces. The diagonal numbers say how many of each element occur in the whole icosahedron. The nondiagonal numbers say how many of the column's element occur in or at the row's element.<ref>Coxeter, Regular Polytopes, sec 1.8 Configurations</ref><ref>Coxeter, Complex Regular Polytopes, p.117</ref>

<math>\begin{bmatrix}\begin{matrix}12 & 5 & 5 \\ 2 & 30 & 2 \\ 3 & 3 & 20 \end{matrix}\end{bmatrix}</math>

Here is the configuration expanded with ''k''-face elements and ''k''-figures. The diagonal element counts are the ratio of the full [[Coxeter group]] H<sub>3</sub>, order 120, divided by the order of the subgroup with mirror removal.
{| class=wikitable
!H<sub>3</sub>||{{CDD|node_1|3|node|5|node}}
! ''k''-face||f<sub>k</sub>||f<sub>0</sub> || f<sub>1</sub>||f<sub>2</sub>||[[Vertex figure|''k''-fig]]
!Notes
|- align=center
|A<sub>2</sub> || {{CDD|node_x|2|node|5|node}} || ( ) 
!f<sub>0</sub>
|| 12 || 5 || 5 ||{5} || H<sub>3</sub>/H<sub>2</sub> = 120/10 = 12
|- align=center
|A<sub>1</sub>A<sub>1</sub> ||{{CDD|node_1|2|node_x|2|node}} || { }
!f<sub>1</sub>
|align=center| 2 || 30 || 2 || { } || H<sub>3</sub>/A<sub>1</sub>A<sub>1</sub> = 120/4 = 30
|- align=center
|H<sub>2</sub> ||{{CDD|node_1|3|node|2|node_x}} ||{3}
!f<sub>2</sub>
|align=center| 3 || 3 || 20 || ( ) || H<sub>3</sub>/A<sub>2</sub> = 120/6 = 20
|}

==Spherical tiling==
The icosahedron can also be represented as a spherical tiling, and projected onto the plane via a [[Stereographic projection|stereographic projection]]. This projection is [[Conformal map|conformal]], preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
{|class=wikitable
|- align=center valign=top
|[[File:Uniform tiling 532-t2.png|160px]]
|[[File:icosahedron stereographic projection.svg|160px]]
|-
![[Orthographic projection]]
!colspan=1|[[Stereographic projection]]
|}

==Other facts==
{{trivia section|date=January 2023}}
*[[File:Icosahedral angle.gif|thumb|arcctg 1/2 ≈ 63,434949°]]An icosahedron has 43,380 distinct [[Net (polyhedron)|nets]].<ref>{{mathworld |urlname=RegularIcosahedron |title=Regular Icosahedron}}</ref>
*To color the icosahedron, such that no two adjacent faces have the same color, requires at least 3 colors.{{efn|1=This is true for all convex polyhedra with triangular faces except for the tetrahedron, by applying [[Brooks' theorem]] to the [[Dual graph|dual graph]] of the polyhedron.}}
*A problem dating back to the ancient Greeks is to determine which of two shapes has larger volume, an icosahedron inscribed in a sphere, or a [[Dodecahedron|dodecahedron]] inscribed in the same sphere. The problem was solved by [[Biography:Hero of Alexandria|Hero]], [[Biography:Pappus of Alexandria|Pappus]], and [[Biography:Fibonacci|Fibonacci]], among others.<ref>{{citation|title=A Mathematical History of the Golden Number|first=Roger|last=Herz-Fischler|publisher=Courier Dover Publications|year=2013|isbn=9780486152325|pages=138–140|url=https://books.google.com/books?id=aYjXZJwLARQC&pg=PA138}}.</ref> [[Biography:Apollonius of Perga|Apollonius of Perga]] discovered the curious result that the ratio of volumes of these two shapes is the same as the ratio of their surface areas.<ref>{{citation|title=Calculus Gems: Brief Lives and Memorable Mathematics|first=George F.|last=Simmons|publisher=Mathematical Association of America|year=2007|isbn=9780883855614|page=50|url=https://books.google.com/books?id=3KOst4Mon90C&pg=PA50}}.</ref> Both volumes have formulas involving the [[Golden ratio|golden ratio]], but taken to different powers.<ref>{{citation|title=Platonic & Archimedean Solids|series=Wooden Books|first=Daud|last=Sutton|publisher=Bloomsbury Publishing USA|year=2002|isbn=9780802713865|page=55|url=https://books.google.com/books?id=vgo7bTxDmIsC&pg=PA55}}.</ref> As it turns out, the icosahedron occupies less of the sphere's volume (60.54%) than the dodecahedron (66.49%).<ref>Numerical values for the volumes of the inscribed Platonic solids may be found in {{citation|title=The Platonic Solids (Solution to problem E2053)|first1=W. E.|last1=Buker|first2=R. B.|last2=Eggleton|journal=[[American Mathematical Monthly]]|volume=76|issue=2|year=1969|page=192|jstor= 2317282|doi=10.2307/2317282}}.</ref>
*Icosahedral angle - the angle between the closest vertices of the icosahedron, relative to the center of the body of the icosahedron (3D), is equal to the diagonal angle of a double and / or half square (= arcctg 1/2 ≈ 63.434949°)<ref>https://upload.wikimedia.org/wikipedia/commons/d/d8/Icosahedral_angle.gif {{bare URL image|date=August 2023}}</ref>


==Construction by a system of equiangular lines==
{| class=wikitable align=right width=320
|- align=center
|[[File:Icosahedron H3 projection.svg|160px]]<BR>Icosahedron<BR>H<sub>3</sub> Coxeter plane
|[[File:6-cube t5 B5.svg|160px]]<BR>[[6-orthoplex|6-orthoplex]]<BR>D<sub>6</sub> Coxeter plane
|-
|colspan=2|This construction can be geometrically seen as the 12 vertices of the [[6-orthoplex|6-orthoplex]] projected to 3 dimensions. This represents a geometric folding of the D<sub>6</sub> to H<sub>3</sub> [[Coxeter group]]s: [[File:Geometric folding Coxeter graph D6 H3.png]]
Seen by these 2D Coxeter plane orthogonal projections, the two overlapping central vertices define the third axis in this mapping. 
|}

The following construction of the icosahedron avoids tedious computations in the [[Algebraic number field|number field]] <math>\Q[\sqrt{5}]</math> necessary in more elementary approaches.

The existence of the icosahedron amounts to the existence of six [[Equiangular lines|equiangular lines]] in <math>\R^3</math>. Indeed, intersecting such a system of equiangular lines with a Euclidean sphere centered at their common intersection yields the twelve vertices of a regular icosahedron as can easily be checked. Conversely, supposing the existence of a regular icosahedron, lines defined by its six pairs of opposite vertices form an equiangular system.

In order to construct such an equiangular system, we start with this 6&nbsp;×&nbsp;6 square [[Matrix (mathematics)|matrix]]:
<math display=block>A=\left(\begin{array}{crrrrr}
0&1&1&1&1&1\\
1&0&1&-1&-1&1\\
1&1&0&1&-1&-1\\
1&-1&1&0&1&-1\\
1&-1&-1&1&0&1\\
1&1&-1&-1&1&0\end{array}\right).</math>

A straightforward computation yields <math>A^2=5I</math> (where <math>I</math> is the 6&nbsp;×&nbsp;6 identity matrix). This implies that <math>A</math> has eigenvalues <math>-\sqrt{5}</math> and <math>\sqrt{5}</math>, both with multiplicity 3 since <math>A</math> is [[Symmetric matrix|symmetric]] and of [[Trace (linear algebra)|trace]] zero.

The matrix <math>A+\sqrt{5}I</math> induces thus a [[Euclidean space|Euclidean structure]] on the [[Quotient space (linear algebra)|quotient space]] <math>\R^6/\operatorname{ker}(A+\sqrt{5}I)</math>, which is [[Isomorphism|isomorphic]] to <math>\R^3</math> since the kernel <math>\operatorname{ker}(A+\sqrt{5}I)</math> of <math>A+\sqrt{5}I</math> has [[Physics:Dimension|dimension]] 3. The image under the [[Projection (linear algebra)|projection]] <math>\pi:\R^6\to\R^6/\operatorname{ker}(A+\sqrt{5}I)</math> of the six coordinate axes in <math>\R^6</math> forms a system of six equiangular lines in <math>\R^3</math> intersecting pairwise at a common acute angle of <math>\arccos 1/\sqrt{5}</math>. Orthogonal projection of the positive and negative basis vectors of <math>\R^6</math> onto the <math>\sqrt{5}</math>-eigenspace of <math>A</math> yields thus the twelve vertices of the icosahedron.

A second straightforward construction of the icosahedron uses [[Group representation|representation theory]] of the [[Alternating group|alternating group]] <math>A_5</math> acting by direct [[Isometry|isometries]] on the icosahedron.

==Symmetry==
[[File:Sphere symmetry group ih.png|thumb|Full [[Icosahedral symmetry|Icosahedral symmetry]] has 15 mirror planes (seen as cyan [[Great circle|great circle]]s on this sphere) meeting at order <math>\pi/5, \pi/3, \pi/2</math> angles, dividing a sphere into 120 triangle [[Fundamental domain|fundamental domain]]s. There are 6 5-fold axes (blue), 10 3-fold axes (red), and 15 2-fold axes (magenta). The vertices of the regular icosahedron exist at the 5-fold rotation axis points.]]
{{main|Icosahedral symmetry}}
The rotational [[Symmetry group|symmetry group]] of the regular icosahedron is isomorphic to the [[Alternating group|alternating group]] on five letters. This non-[[Abelian group|abelian]] [[Simple group|simple group]] is the only non-trivial [[Normal subgroup|normal subgroup]] of the [[Symmetric group|symmetric group]] on five letters. Since the [[Galois group]] of the general quintic equation is isomorphic to the symmetric group on five letters, and this normal subgroup is simple and non-abelian, the general quintic equation does not have a solution in radicals. The proof of the [[Abel–Ruffini theorem]] uses this simple fact, and [[Biography:Felix Klein|Felix Klein]] wrote a book that made use of the theory of icosahedral symmetries to derive an analytical solution to the general quintic equation, {{Harv|Klein|1884}}. See [[Icosahedral symmetry#Related geometries|icosahedral symmetry: related geometries]] for further history, and related symmetries on seven and eleven letters.

The full symmetry group of the icosahedron (including reflections) is known as the full icosahedral group, and is isomorphic to the product of the rotational symmetry group and the group <math>C_2</math> of size two, which is generated by the reflection through the center of the icosahedron.

==Stellations==
The icosahedron has a large number of [[Stellation|stellation]]s. According to specific rules defined in the book ''[[The Fifty-Nine Icosahedra]]'',{{Sfn|Coxeter|du Val|Flather|Petrie|1938}} 59 stellations were identified for the regular icosahedron. The first form is the icosahedron itself. One is a regular [[Kepler–Poinsot polyhedron]]. Three are regular compound polyhedra.

{| class=wikitable
|+ 21 of 59 stellations
|-
|rowspan=3 width=200|[[File:Stellation diagram of icosahedron.svg|200px]]<br>The faces of the icosahedron extended outwards as planes intersect, defining regions in space as shown by this stellation diagram of the intersections in a single plane.
|[[Image:Zeroth stellation of icosahedron.svg|50px]]
|[[Image:First stellation of icosahedron.png|50px]]
|[[Image:Second stellation of icosahedron.png|50px]]
|[[Image:Third stellation of icosahedron.svg|50px]]
|[[Image:Fourth stellation of icosahedron.png|50px]]
|[[Image:Fifth stellation of icosahedron.png|50px]]
|[[Image:Sixth stellation of icosahedron.png|50px]]
|-
|[[Image:Seventh stellation of icosahedron.png|50px]]
|[[Image:Eighth stellation of icosahedron.png|50px]]
|[[Image:Ninth stellation of icosahedron.png|50px]]
|[[Image:Tenth stellation of icosahedron.png|50px]]
|[[Image:Eleventh stellation of icosahedron.png|50px]]
|[[Image:Twelfth stellation of icosahedron.png|50px]]
|[[Image:Thirteenth stellation of icosahedron.png|50px]]
|-
|[[Image:Fourteenth stellation of icosahedron.png|50px]]
|[[Image:Fifteenth stellation of icosahedron.png|50px]]
|[[Image:Sixteenth stellation of icosahedron.png|50px]]
|[[Image:Seventeenth stellation of icosahedron.png|50px]]
|[[Image:First compound stellation of icosahedron.png|50px]]
|[[Image:Second compound stellation of icosahedron.png|50px]]
|[[Image:Third compound stellation of icosahedron.png|50px]]
|}

== Facetings==
The [[Small stellated dodecahedron|small stellated dodecahedron]], [[Great dodecahedron|great dodecahedron]], and [[Great icosahedron|great icosahedron]] are three [[Faceting|faceting]]s of the regular icosahedron. They share the same [[Vertex arrangement|vertex arrangement]]. They all have 30 edges. The regular icosahedron and great dodecahedron share the same edge arrangement but differ in faces (triangles vs pentagons), as do the small stellated dodecahedron and great icosahedron (pentagrams vs triangles).
{| class=wikitable
!Convex
!colspan=3|Regular stars
|-
!'''icosahedron'''
![[Great dodecahedron|great dodecahedron]]
![[Small stellated dodecahedron|small stellated dodecahedron]]
![[Great icosahedron|great icosahedron]]
|- align=center
|[[File:Icosahedron.png|150px]]
|[[File:Great dodecahedron.png|150px]]
|[[File:Small stellated dodecahedron.png|150px]]
|[[File:Great icosahedron.png|150px]]
|}

==Geometric relations==

===Inscribed in other Platonic solids===
The regular icosahedron is the [[Dual polyhedron|dual polyhedron]] of the [[Regular dodecahedron|regular dodecahedron]]. An icosahedron can be inscribed in a dodecahedron by placing its vertices at the face centers of the dodecahedron, and vice versa.

An icosahedron can be inscribed in an [[Octahedron|octahedron]] by placing its 12 vertices on the 12 edges of the octahedron such that they divide each edge into its two golden sections. Because the golden sections are unequal, there are five different ways to do this consistently, so five disjoint icosahedra can be inscribed in each octahedron.{{Sfn|Coxeter|du Val|Flather|Petrie|1938|p=4|ps=; "Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the "golden section". Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[Compound of five octahedra|compound of five octahedra]], which comes under our definition of stellated icosahedron. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated triacontahedron.) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a stella octangula, thus forming a [[Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[Icosahedral symmetry|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''chiral''."}}

An icosahedron of edge length <math display=inline>\frac{1}{\varphi} \approx 0.618</math> can be inscribed in a unit-edge-length cube by placing six of its edges (3 orthogonal opposite pairs) on the square faces of the cube, centered on the face centers and parallel or perpendicular to the square's edges.{{Sfn|Borovik|2006|pp=8-9|loc=§5. How to draw an icosahedron on a blackboard}} Because there are five times as many icosahedron edges as cube faces, there are five ways to do this consistently, so five disjoint icosahedra can be inscribed in each cube. The edge lengths of the cube and the inscribed icosahedron are in the [[Golden ratio|golden ratio]].{{Efn|Reciprocally, the edge length of a cube inscribed in a dodecahedron is in the [[Golden ratio|golden ratio]] to the dodecahedron's edge length. The cube's edges lie in pentagonal face planes of the dodecahedron as [[Pentagon#Regular pentagons|regular pentagon diagonals]], which are always in the golden ratio to the regular pentagon's edge. So when a cube is inscribed in a dodecahedron and an icosahedron is inscribed in the cube, the dodecahedron and icosahedron (which do not share any vertices) have the same edge length.}}

===Relations to the 600-cell and other 4-polytopes===
The icosahedron is the [[Four-dimensional space#Dimensional analogy|dimensional analogue]] of the [[600-cell|600-cell]], a [[Regular 4-polytope#Regular convex 4-polytopes|regular 4-dimensional polytope]]. The 600-cell has icosahedral [[600-cell#Polyhedral sections|cross sections]] of two sizes, and each of its 120 vertices is an [[Icosahedral pyramid|icosahedral pyramid]]; the icosahedron is the [[Vertex figure|vertex figure]] of the 600-cell.

The unit-radius 600-cell has tetrahedral cells of edge length <math display=inline>\frac{1}{\varphi} \approx 0.618</math>, 20 of which meet at each vertex to form an icosahedral pyramid (a 4-pyramid with an icosahedron as its base). Thus the 600-cell contains 120 icosahedra of edge length <math display=inline>\frac{1}{\varphi} \approx 0.618</math>. The 600-cell also contains unit-edge-length cubes and unit-edge-length octahedra as [[600-cell#Cell clusters|interior features]] formed by its unit-length [[600-cell#Golden chords|chords]]. In the unit-radius [[120-cell|120-cell]] (another regular 4-polytope which is both the dual of the 600-cell and a compound of 5 600-cells) we find all three kinds of inscribed icosahedra (in a dodecahedron, in an octahedron, and in a cube).

A semiregular 4-polytope, the [[Snub 24-cell|snub 24-cell]], has icosahedral cells.

===Relations to other uniform polytopes===
The icosahedron is unique among the Platonic solids in possessing a [[Dihedral angle|dihedral angle]] not less than 120°. Its dihedral angle is approximately 138.19°. Thus, just as hexagons have angles not less than 120° and cannot be used as the faces of a convex regular polyhedron because such a construction would not meet the requirement that at least three faces meet at a vertex and leave a positive defect for folding in three dimensions, icosahedra cannot be used as the cells of a convex regular polychoron because, similarly, at least three cells must meet at an edge and leave a positive defect for folding in four dimensions (in general for a convex [[Polytope|polytope]] in ''n'' dimensions, at least three facets must meet at a peak and leave a positive defect for folding in ''n''-space). However, when combined with suitable cells having smaller dihedral angles, icosahedra can be used as cells in semi-regular polychora (for example the [[Snub 24-cell|snub 24-cell]]), just as hexagons can be used as faces in semi-regular polyhedra (for example the [[Truncated icosahedron|truncated icosahedron]]). Finally, non-convex polytopes do not carry the same strict requirements as convex polytopes, and icosahedra are indeed the cells of the [[Icosahedral 120-cell|icosahedral 120-cell]], one of the ten non-convex regular polychora.

There are distortions of the icosahedron that, while no longer regular, are nevertheless vertex-uniform. These are [[Invariant (mathematics)|invariant]] under the same [[Physics:Rotation|rotation]]s as the tetrahedron, and are somewhat analogous to the [[Snub cube|snub cube]] and [[Snub dodecahedron|snub dodecahedron]], including some forms which are [[Chirality (mathematics)|chiral]] and some with T<sub>h</sub>-symmetry, i.e. have different planes of symmetry from the tetrahedron.

An icosahedron can also be called a gyroelongated pentagonal bipyramid. It can be decomposed into a [[Gyroelongated pentagonal pyramid|gyroelongated pentagonal pyramid]] and a [[Pentagonal pyramid|pentagonal pyramid]] or into a [[Pentagonal antiprism|pentagonal antiprism]] and two equal pentagonal pyramids.

===Relation to the 6-cube and rhombic triacontahedron===
[[File:6demicube-odd-icosahedron.png|240px|thumb]]
The icosahedron can be projected to 3D from the 6D [[6-demicube|6-demicube]] using the same basis vectors that form the hull of the [[Rhombic triacontahedron]] from the [[6-cube|6-cube]]. Shown here including the inner 20 vertices which are not connected by the 30 outer hull edges of 6D norm length <math>\sqrt{2}</math>. The inner vertices form a [[Dodecahedron|dodecahedron]].

The 3D projection basis vectors [u,v,w] used are:
<math display=block>
\begin{align}
u &= (1, \varphi, 0, -1, \varphi, 0)\\
v &= (\varphi, 0, 1, \varphi, 0, -1)\\
w &= (0, 1, \varphi, 0, -1, \varphi)\\
\end{align}
</math>

===Icosahedral symmetry in the six-dimensional integer lattice===

Icosahedral symmetry cannot be realized in the three-dimensional integer lattice, but does emerge from the six-dimensional lattice. One set of points having this symmetry is given by the twenty 6-cube vertices with coordinates (1,1,1,-1,-1,-1) and all permutations thereof (twenty points in all). The twenty selected lattice points can be grouped into four sets of five, such as the one below, which are equidistant from both (1,0,0,0,0,0) and (-1,0,0,0,0,0):

(1,1,1,-1,-1,-1)

(1,-1,1,1,-1,-1)

(1,-1,-1,1,1,-1)

(1,-1,-1,-1,1,1)

(1,1,-1,-1,-1,1)

The line through (1,0,0,0,0,0) and (-1,0,0,0,0,0), which may also be considered to pass through the origin and (1,0,0,0,0,0), constitutes one of the fivefold axes. Five other such axes with similar properties are identified by choosing each of the other coordinate axes in turn to have the nonzero coordinate values. The ten threefold axes similarly pass through the origin and permutations of (1,1,1,-1,-1,-1), and the fifteen twofold axes pass through the origin and permutations of (1,-1,0,0,0,0). This six-dimensional lattice symmetry enables the use of six-dimensional Miller indices to describe planes in icosahedral quasicrystals.

==Symmetries==
[[File:Icosahedral subgroup tree.png|thumb|[[Icosahedral symmetry|Icosahedral symmetry]] subgroups]]
There are 3 [[Uniform coloring|uniform coloring]]s of the icosahedron. These colorings can be represented as 11213, 11212, 11111, naming the 5 triangular faces around each vertex by their color.

The icosahedron can be considered a snub tetrahedron, as [[Snub (geometry)|snubification]] of a regular tetrahedron gives a regular icosahedron having chiral [[Tetrahedral symmetry|tetrahedral symmetry]]. It can also be constructed as an alternated truncated octahedron, having pyritohedral symmetry. The pyritohedral symmetry version is sometimes called a pseudoicosahedron, and is dual to the [[Pyritohedron|pyritohedron]].
{| class=wikitable
|- align=center
!
!Regular
!colspan=2|Uniform
!colspan=3|2-uniform
!3-uniform
|- align=center
!Name
|Regular<BR>icosahedron
|[[Icosahedron#Pyritohedral symmetry|Snub<BR>octahedron]]
|Snub<BR>tetratetrahedron
|[[Gyroelongated bipyramid|Gyroelongated<BR>pentagonal<BR>bipyramid]]
|Triangular<BR>gyrobianticupola
|Snub triangular<BR>antiprism<ref>[http://www.orchidpalms.com/polyhedra/snub-anti-prisms/prisms1.html Snub Anti-Prisms ]</ref>
|Snub square<BR>bipyramid
|- align=center
!Image
|[[Image:Uniform polyhedron-53-t2.png|80px]]
|[[Image:Uniform polyhedron-43-h01.svg|80px]]
|[[Image:Uniform polyhedron-33-s012.png|80px]]
|[[Image:Pentagonal gyroelongated bipyramid.png|80px]]
|[[File:Regular triangular gyrobianticupola.png|80px]]
|[[File:snub_triangular_antiprism.png|80px]]
|[[File:snub_square_bipyramid.png|80px]]
|- align=center valign=top
![[Uniform coloring|Face<BR>coloring]]
|(11111)
|(11212)
|(11213)
|(11122)<BR>(22222)
|(12332)<BR>(23333)
|(11213)<BR>(11212)
|(11424)<BR>(22434)<BR>(33414)
|- align=center
!Coxeter<BR>diagram
|{{CDD|node_1|3|node|5|node}}
|{{CDD|node_h|3|node_h|4|node}}
|{{CDD|node_h|3|node_h|3|node_h}}
|
|
|
|
|- align=center
![[Schläfli symbol|Schläfli<BR>symbol]]
|{3,5}
|s{3,4}
|sr{3,3}
|() <nowiki>||</nowiki> {n} <nowiki>||</nowiki> r{n} <nowiki>||</nowiki> ()
|
|ss{2,6}
|sdt{2,4}
|- align=center
![[Conway polyhedron notation|Conway]]
|I
|HtO
|sT
|k5A5
|
|sY3 = HtA3
|HtdP4
|- align=center
![[List of spherical symmetry groups|Symmetry]]
|I<sub>h</sub><br>[5,3]<br>(*532)
|T<sub>h</sub><br>[3<sup>+</sup>,4]<br>(3*2)
|T<br>[3,3]<sup>+</sup><br>(332)
|D<sub>5d</sub><br>[2<sup>+</sup>,10]<br>(2*5)
|D<sub>3d</sub><br>[2<sup>+</sup>,6]<br>(2*3)
|D<sub>3</sub><br>[3,2]<sup>+</sup><br>(322)
|D<sub>2h</sub><br>[2,2]<br>(*222)
|- align=center
!Symmetry<BR>order
|120
|24
|12
|20
|12
|6
|8
|}

==Uses and natural forms==
[[File:Twin2.jpg|thumb|upright|[[Chemistry:Gold|Gold]] nanoparticle viewed by transmission electron microscopy. ]]
[[File:Gamma-bor.jpg|thumb|upright|Structure of γ-boron.]]

=== Biology ===
Many [[Biology:Virus|virus]]es, e.g. [[Biology:Herpesviridae|herpes virus]], have icosahedral [[Biology:Capsid|shells]].<ref>C. Michael Hogan. 2010. [http://www.eoearth.org/article/Virus?topic=49496 ''Virus''. Encyclopedia of Earth. National Council for Science and the Environment]. eds. S. Draggan and C. Cleveland</ref> Viral structures are built of repeated identical [[Biology:Protein|protein]] subunits known as [[Biology:Capsomere|capsomere]]s, and the icosahedron is the easiest shape to assemble using these subunits. A ''regular'' polyhedron is used because it can be built from a single basic unit protein used over and over again; this saves space in the viral [[Biology:Genome|genome]].

Various bacterial organelles with an icosahedral shape were also found.<ref name=Bobik2007>{{Citation |author=Bobik, T.A. |title=Bacterial Microcompartments |year=2007 |journal=Microbe |volume=2 |pages=25–31 |url=http://www.microbemagazine.org/index.php/01-2007-home/2308-bacterial-microcompartments |publisher=Am. Soc. Microbiol. |url-status=dead |archive-url=https://web.archive.org/web/20130729083846/http://www.microbemagazine.org/index.php/01-2007-home/2308-bacterial-microcompartments |archive-date=2013-07-29 }}</ref> The icosahedral shell encapsulating enzymes and labile intermediates are built of different types of proteins with [[Biology:BMC domain|BMC domain]]s.

In 1904, [[Biography:Ernst Haeckel|Ernst Haeckel]] described a number of species of [[Biology:Radiolaria|Radiolaria]], including ''Circogonia icosahedra'', whose skeleton is shaped like a regular icosahedron. A copy of Haeckel's illustration for this radiolarian appears in the article on regular polyhedra.

=== Chemistry ===
The closo-carboranes are chemical compounds with shape very close to icosahedron. [[Physics:Icosahedral twins|Icosahedral]] [[Physics:Crystal twinning|twinning]] also occurs in crystals, especially [[Physics:Nanoparticle|nanoparticle]]s.

Many [[Physics:Crystal structure of boron-rich metal borides|borides]] and [[Chemistry:Allotropes of boron|allotropes of boron]] contain boron B<sub>12</sub> icosahedron as a basic structure unit.

=== Toys and games ===

[[File:Twenty-sided die (icosahedron) with faces inscribed with Greek letters MET 10.130.1158 001.jpg|thumb|upright|left|Twenty-sided die from [[History:Ptolemaic Kingdom|Ptolemaic Egypt]]]]
[[File:Dice in B&amp;W (square).jpg|thumb|upright|Twenty-sided die]]
Icosahedral dice with twenty sides have been used since ancient times.<ref>Cromwell, Peter R. "Polyhedra" (1997) Page 327.</ref>

In several roleplaying games, such as ''Dungeons & Dragons'', the twenty-sided die (d20 for short) is commonly used in determining success or failure of an action. This die is in the form of a regular icosahedron. It may be numbered from "0" to "9" twice (in which form it usually serves as a ten-sided die, or d10), but most modern versions are labeled from "1" to "20".

An icosahedron is used in the board game ''Scattergories'' to choose a letter of the alphabet. Six letters are omitted (Q, U, V, X, Y, and Z).

In the ''Nintendo 64'' game ''[[Software:Kirby 64: The Crystal Shards|Kirby 64]]'', the boss Miracle Matter is a regular icosahedron.

Inside a Magic 8-Ball, various answers to yes–no questions are inscribed on a regular icosahedron.

=== Tensegrity ===

The octahedron has been widely studied in the field of [[Engineering:Tensegrity|tensegrity]]. Due to its spherical symmetry and high strength to mass ratio, the shape became a good candidate for deployable tensegrity space structures such as NASA's SuperBALL.<ref>{{Cite journal |last1=Sabelhaus |first1=Andrew P. |last2=Bruce |first2=Jonathan |last3=Caluwaerts |first3=Ken |last4=Chen |first4=Yangxin |last5=Lu |first5=Dizhou |last6=Liu |first6=Yuejia |last7=Agogino |first7=Adrian K. |last8=SunSpiral |first8=Vytas |last9=Agogino |first9=Alice M. |date=2014-07-15 |title=Hardware Design and Testing of SUPERball, A Modular Tensegrity Robot |url=https://ntrs.nasa.gov/citations/20140011157 |language=en}}</ref> The robot is composed of rods, cables and actuators of different scales and is currently in development between NASA Ames Research Center's Intelligent Robotics Group and the Dynamic Tensegrity Robotics Lab (DTRL). Its undeployed configuration is highly compact, hence being ideal for fitting within the space-constraints of rocket fairings.<ref>{{Cite web |date=2015-03-02 |title=NASA's Super Ball Bot Could Be the Best Design for Planetary Exploration |url=https://spectrum.ieee.org/nasa-super-ball-bot |access-date=2022-12-22 |website=IEEE Spectrum |language=en}}</ref>

The Icosahedron in tensegrity is composed of six struts and twenty-four cables that connect twelve nodes. One self-stress state is present within the combination achieved through the use of cellular morphogenesis.<ref>{{Cite journal |last1=Aloui |first1=Omar |last2=Flores |first2=Jessica |last3=Orden |first3=David |last4=Rhode-Barbarigos |first4=Landolf |date=2019-04-01 |title=Cellular morphogenesis of three-dimensional tensegrity structures |url=https://www.sciencedirect.com/science/article/pii/S0045782518305814 |journal=Computer Methods in Applied Mechanics and Engineering |language=en |volume=346 |pages=85–108 |doi=10.1016/j.cma.2018.10.048 |issn=0045-7825|arxiv=1902.09953 |bibcode=2019CMAME.346...85A |s2cid=67856423 }}</ref>

=== Others ===

R. Buckminster Fuller and Japanese cartographer Shoji Sadao<ref>{{Cite web |url=http://library.stanford.edu/depts/spc/exhibits/fullersadao.html |title=Fuller and Sadao: Partners in Design |date=September 19, 2006 |access-date=2010-01-26 |url-status=dead |archive-url=https://web.archive.org/web/20100816141849/http://library.stanford.edu/depts/spc/exhibits/fullersadao.html |archive-date=August 16, 2010 }}</ref> designed a world map in the form of an unfolded icosahedron, called the Fuller projection, whose maximum [[Physics:Distortion#Map projections|distortion]] is only 2%. 

The American [[Physics:Electronic music|electronic music]] duo ODESZA use a regular icosahedron as their logo.


==Icosahedral graph==
{{Infobox graph
 | name = Regular icosahedron graph
 | image = Icosahedron graph.svg|image_size=160px
 | image_caption = 3-fold symmetry
 | namesake = 
 | vertices = 12
 | edges = 30
 | automorphisms = 120 ([[Alternating group|A<sub>5</sub>]] [[Direct product of groups|×]] Z<sub>2</sub>)
 | radius = 3
 | diameter = 3
 | girth = 3
 | chromatic_number = 4
 | chromatic_index = 
 | fractional_chromatic_index = 
 | properties = Hamiltonian, [[Regular graph|regular]], [[Symmetric graph|symmetric]], [[Distance-regular graph|distance-regular]], [[Distance-transitive graph|distance-transitive]], [[K-vertex-connected graph|3-vertex-connected]], [[Planar graph|planar graph]]
}}
The [[N-skeleton|skeleton]] of the icosahedron (the vertices and edges) forms a [[Graph (discrete mathematics)|graph]].  It is one of 5 [[Platonic graph]]s, each a skeleton of its [[Platonic solid]].

The high degree of symmetry of the polygon is replicated in the properties of this graph, which is [[Distance-transitive graph|distance-transitive]] and [[Symmetric graph|symmetric]]. The [[Graph automorphism|automorphism group]] has order 120. The vertices can be [[Graph coloring|colored]] with 4 colors, the edges with 5 colors, and the diameter is 3.<ref>{{MathWorld |urlname=IcosahedralGraph |title=Icosahedral Graph}}</ref>

The icosahedral graph is Hamiltonian: there is a cycle containing all the vertices.  It is also a [[Planar graph|planar graph]].

{| class=wikitable
|+ Orthogonal projection
|[[File:Icosahedron A2 projection.svg|160px]]
|}
{{Clear}}

== Diminished regular icosahedra==
There are 4 related [[Johnson solid]]s, including pentagonal faces with a subset of the 12 vertices. The similar dissected regular icosahedron has 2 adjacent vertices diminished, leaving two trapezoidal faces, and a bifastigium has 2 opposite sets of vertices removed and 4 trapezoidal faces. The pentagonal antiprism is formed by removing two opposite vertices.
{| class=wikitable
!Form
![[Pentagonal pyramid|J2]]
!Bifastigium
![[Tridiminished icosahedron|J63]]
![[Metabidiminished icosahedron|J62]]
!Dissected<BR>icosahedron
![[Pentagonal antiprism|s{2,10}]]
![[Gyroelongated pentagonal pyramid|J11]]
|-
!Vertices
!6 of 12
!8 of 12
!9 of 12
!colspan=3|10 of 12
!11 of 12
|- align=center
![[List of finite spherical symmetry groups|Symmetry]]
|''C''<sub>5v</sub>, [5], (*55)<BR>order 10
| D<sub>2h</sub>, [2,2], *222<BR>order 8
|''C''<sub>3v</sub>, [3], (*33)<BR>order 6
|colspan=2|''C''<sub>2v</sub>, [2], (*22)<BR>order 4
|D<sub>5d</sub>, [2<sup>+</sup>,10], (2*5)<BR>order 20
|''C''<sub>5v</sub>, [5], (*55)<BR>order 10
|-
!Image
|[[File:Pentagonal pyramid.png|80px]]
|[[File:4-diminished_icosahedron.png|80px]]
|[[File:Tridiminished icosahedron.png|80px]]
|[[File:Metabidiminished icosahedron.png|80px]]
|[[File:Dissected regular icosahedron.png|80px]]
|[[File:Pentagonal antiprism.png|80px]]
|[[File:Gyroelongated pentagonal pyramid.png|80px]]
|}

==Related polyhedra and polytopes==

The icosahedron can be transformed by a [[Truncation (geometry)|truncation]] sequence into its [[Dual polyhedron|dual]], the dodecahedron:

As a snub tetrahedron, and alternation of a truncated octahedron it also exists in the tetrahedral and octahedral symmetry families:
{{Tetrahedron family}}

This polyhedron is topologically related as a part of sequence of regular polyhedra with [[Schläfli symbol]]s {3,''n''}, continuing into the [[Hyperbolic space|hyperbolic plane]].

The regular icosahedron, seen as a ''snub tetrahedron'', is a member of a sequence of [[Snub (geometry)|snubbed]] polyhedra and tilings with vertex figure (3.3.3.3.''n'') and Coxeter–Dynkin diagram {{CDD|node_h|n|node_h|3|node_h}}. These figures and their duals have (''n''32) rotational [[Orbifold notation|symmetry]], being in the Euclidean plane for <math>n=6</math>, and hyperbolic plane for any higher <math>n</math>. The series can be considered to begin with <math>n=2</math>, with one set of faces degenerated into [[Digon|digon]]s.



The icosahedron can tessellate hyperbolic space in the order-3 icosahedral honeycomb, with 3 icosahedra around each edge, 12 icosahedra around each vertex, with [[Schläfli symbol]] {3,5,3}. It is one of four regular tessellations in the hyperbolic 3-space.
{| class=wikitable width=240
|[[File:Hyperb icosahedral hc.png|240px]]<br>It is shown here as an edge framework in a [[Poincaré disk model]], with one icosahedron visible in the center.
|}

==See also==

*[[Great icosahedron]]
*[[Earth:Geodesic grid|Geodesic grid]]s use an iteratively bisected icosahedron to generate grids on a sphere
*[[Physics:Icosahedral twins|Icosahedral twins]]
*Infinite skew polyhedron
*[[Jessen's icosahedron]]
*[[Regular polyhedron]]
*[[Truncated icosahedron]]

==Notes==
{{notelist}}

==Citations==
<!--See Wikipedia:Footnotes for instructions.-->
{{reflist}}

==References==
{{refbegin}}
* {{Citation |last=Klein |first=Felix |title=Lectures on the ikosahedron and the solution of equations of the fifth degree |year=1888 |publisher=Courier Corporation |url=http://digital.library.cornell.edu/cgi/t/text/text-idx?c=math;cc=math;view=toc;subview=short;idno=03070001|isbn=978-0-486-49528-6|postscript=, Dover edition}}, translated from {{cite book |last =Klein |first =Felix |title =Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade |url =https://archive.org/details/vorlesungenberd00kleigoog |publisher =Teubner |year =1884 }}
* {{Cite book | last1=Coxeter | first1=H.S.M. |  last2=du Val | first2=Patrick |  last3=Flather | first3=H.T. | last4=Petrie | author4-link=W:John Flinders Petrie | first4=J.F. | year=1938 | title=The Fifty-Nine Icosahedra | publisher=University of Toronto Studies (Mathematical Series) | volume=6 }}
* {{Cite book|title=The Coxeter Legacy|chapter=Coxeter Theory: The Cognitive Aspects|last=Borovik|first=Alexandre|year=2006|publisher=American Mathematical Society|place=Providence, Rhode Island|editor1-last=Davis|editor1-first=Chandler|editor2-last=Ellers|editor2-first=Erich|pages=17–43|isbn=978-0821837221|url=https://www.academia.edu/26091464}}
{{refend}}

==External links==
*{{KlitzingPolytopes|polyhedra.htm|3D convex uniform polyhedra|x3o5o – ike}}
*{{cite web|first=Michael|last=Hartley| url=http://www.dr-mikes-math-games-for-kids.com/polyhedral-nets.html?net=PKYcLAHcKk9lKCGWdoeJHvnDe7jSJyKXIghPYlKcV5PxkeWqGqwmeoyXUilmvzkXsQnxEoduPHUYFqOf20B8EwLz8CufLOUuc4N5&name=Icosahedron#applet |title=Dr Mike's Math Games for Kids}}
*[http://www.kjmaclean.com/Geometry/GeometryHome.html K.J.M. MacLean, A Geometric Analysis of the Five Platonic Solids and Other Semi-Regular Polyhedra]
*[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra
*[https://web.archive.org/web/20040922084928/http://www.tulane.edu/~dmsander/WWW/335/335Structure.html Tulane.edu] A discussion of viral structure and the icosahedron
*[https://www.flickr.com/photos/pascalin/sets/72157594234292561/ Origami Polyhedra] – Models made with Modular Origami
*[https://www.youtube.com/watch?v=r6JQWCi_lNM Video of icosahedral mirror sculpture]
*[https://archive.today/20121224154747/http://web.uct.ac.za/depts/mmi/stannard/virarch.html] Principle of virus architecture

{{Polyhedra}}
{{Polyhedron navigator}}
{{Polytopes}}

[[Category:Deltahedra]]
[[Category:Planar graphs]]
[[Category:Platonic solids]]

{{Sourceattribution|Regular icosahedron}}