Editing Regular icosahedron

{{Short description|Convex polyhedron with 20 triangular faces}}

{{Infobox polyhedron
 | name = Regular icosahedron
 | image = Icosahedron.svg
 | type = [[Deltahedron]],<br />[[Gyroelongated bipyramid]],<br />[[Platonic solid]],<br />[[Regular polyhedron]]
 | faces = 20
 | edges = 30
 | vertices = 12
 | angle = 138.190 (approximately)
 | vertex_config = <math> 12 \times \left(3^5 \right) </math>
 | schläfli = <math> \{3,5\} </math>
 | symmetry = [[icosahedral symmetry]] <math> I_h </math>
 | dual = [[regular dodecahedron]]
 | properties = [[convex set|convex]],<br />[[composite polyhedron|composite]],<br />[[isogonal figure|isogonal]],<br />[[isohedral]],<br />[[isotoxal]]
 | net = Icosahedron flat.svg
}}

The '''regular icosahedron''' (or simply ''icosahedron'') is a [[convex polyhedron]] that can be constructed from [[pentagonal antiprism]] by attaching two [[pentagonal pyramid]]s with [[Regular polygon|regular faces]] to each of its pentagonal faces, or by putting points onto the cube. The resulting polyhedron has 20 [[equilateral triangle]]s as its faces, 30 edges, and 12 vertices. It is an example of a [[Platonic solid]] and of a [[deltahedron]]. The icosahedral graph represents the [[Skeleton (topology)|skeleton]] of a regular icosahedron.

Many polyhedra are constructed from the regular icosahedron. A notable example is the [[stellation]] of regular icosahedron, which consists of 59 polyhedrons. The [[great dodecahedron]], one of the [[Kepler&ndash;Poinsot polyhedra]], is constructed by either stellation or [[faceting]]. Some of the [[Johnson solid]]s can be constructed by removing the pentagonal pyramids. The regular icosahedron's [[dual polyhedron]] is the [[regular dodecahedron]], and their relation has a historical background on the comparison mensuration. It is analogous to a four-dimensional [[polytope]], the [[600-cell]].

Regular icosahedrons can be found in nature; a well-known example is the [[capsid]] in biology. Other applications of the regular icosahedron are the usage of its net in [[cartography]], and the twenty-sided dice that may have been used in ancient times but are now [[d20 System|commonplace in modern]] [[tabletop role-playing games]].

== Construction ==
There are several ways to construct a regular icosahedron:
* The construction started from a [[pentagonal antiprism]] by attaching two [[Pentagonal pyramid|pentagonal pyramids]] with [[Regular polygon|regular faces]] to each of its faces.<ref>{{multiref
 |{{harvnb|Silvester|2001|p=[https://books.google.com/books?id=VtH_QG6scSUC&pg=PA141 140&ndash;141]}}
 |{{harvnb|Cundy|1952}}
}}</ref> Such construction led to the regular icosahedron becoming known for [[composite polyhedron|composite]]; the pyramids are the elementary, meaning they cannot be sliced again into smaller convex polyhedrons. This process of construction is known as the [[gyroelongation]], like other polyhedrons in the family of [[gyroelongated bipyramid]]. {{sfn|Berman|1971}}
[[File:Icosahedron-golden-rectangles.svg|thumb|right|Three mutually perpendicular [[golden ratio]] rectangles, with edges connecting their corners, form a regular icosahedron.]]
* Another way to construct it is by putting two points on each surface of a cube. In each face, draw a segment line between the midpoints of two opposite edges and locate two points with the golden ratio distance from each midpoint. These twelve vertices describe the three mutually perpendicular planes, with edges drawn between each of them.{{sfn|Cromwell|1997|p=[https://archive.org/details/polyhedra0000crom/page/70/mode/1up?view=theater 70]}}
* The regular icosahedron can also be constructed starting from a [[regular octahedron]]. All triangular faces of a regular octahedron are breaking, twisting at a certain angle, and filling up with other equilateral triangles. This process is known as [[Snub (geometry)|snub]], and the regular icosahedron is also known as '''snub octahedron'''.{{sfn|Kappraff|1991|p=[https://books.google.com/books?id=tz76s0ZGFiQC&pg=PA475 475]}}
* One possible system of [[Cartesian coordinate]] for the vertices of a regular icosahedron, given the edge length 2, is: <math display="block"> \left(0, \pm 1, \pm \varphi \right), \left(\pm 1, \pm \varphi, 0 \right), \left(\pm \varphi, 0, \pm 1 \right), </math> where <math>\varphi = (1 + \sqrt{5})/2 </math> denotes the [[golden ratio]].{{sfn|Steeb|Hardy|Tanski|2012|p=[https://books.google.com/books?id=UdI7DQAAQBAJ&pg=PA211 211]}}
By the constructions above, the regular icosahedron is [[Platonic solid]], because it has 20 [[equilateral triangle]]s as it faces. This also results in that regular icosahedron is one of the eight convex [[deltahedron]].<ref>{{multiref
 |{{harvnb|Shavinina|2013|p=[https://books.google.com/books?id=JcPd_JRc4FgC&pg=PA333 333]}}
 |{{harvnb|Cundy|1952}}
}}</ref> It can be unfolded into 44,380 different [[Net (geometry)|net]]s.{{sfn|Dennis|McNair|Woolf|Kauffman|2018|p=[https://books.google.com/books?id=4hVeDwAAQBAJ&pg=PA169 169]}}

== Properties ==
=== Mensuration ===
The [[insphere]] of a convex polyhedron is a sphere inside the polyhedron, touching every face. The [[circumsphere]] of a convex polyhedron is a sphere that contains the polyhedron and touches every vertex. The [[midsphere]] of a convex polyhedron is a sphere tangent to every edge. Therefore, given that the edge length <math> a </math> of a regular icosahedron, the radius of insphere (inradius) <math> r_I </math>, the radius of circumsphere (circumradius) <math> r_C </math>, and the radius of midsphere (midradius) <math> r_M </math> are, respectively:<ref>{{multiref
 |{{harvnb|MacLean|2007|p=[https://books.google.com/books?id=vINuAwAAQBAJ&pg=PA44 43&ndash;44]}}
 |{{harvnb|Coxeter|1973}}, Table I(i), pp. 292–293. See column "<math>{}_1\!\mathrm{R}/\ell</math>", where <math>{}_1\!\mathrm{R}</math> is Coxeter's notation for the midradius, also noting that Coxeter uses <math>2\ell</math> as the edge length (see p. 2).
}}</ref>
<math display="block">
 r_I = \frac{\varphi^2 a}{2 \sqrt{3}} \approx 0.756a, \qquad
 r_C = \frac{\sqrt{\varphi^2 + 1}}{2}a \approx 0.951a, \qquad
 r_M = \frac{\varphi}{2}a \approx 0.809a. 
</math>

[[File:Regular icosahedron.stl|thumb|3D model of a regular icosahedron]]
The [[surface area]] of a polyhedron is the sum of the areas of its faces. Therefore, the surface area <math>(A)</math> of a regular icosahedron is twenty times that of each of its equilateral triangle faces. The volume <math>(V)</math> of a regular icosahedron can be obtained as twenty times that of a pyramid whose base is one of its faces and whose apex is the icosahedron's center; or as the sum of two uniform [[pentagonal pyramid]]s and a [[pentagonal antiprism]]. The expressions of both are:<ref>{{multiref
|{{harvnb|MacLean|2007|p=[https://books.google.com/books?id=vINuAwAAQBAJ&pg=PA44 43&ndash;44]}}
|{{harvnb|Berman|1971}}
}}</ref>
<math display="block">
 A = 5\sqrt{3}a^2 \approx 8.660a^2, \qquad
 V = \frac{5 \varphi^2}{6}a^3 \approx 2.182a^3. 
</math>
A problem dating back to the ancient Greeks is determining which of two shapes has a larger volume, an icosahedron inscribed in a sphere, or a dodecahedron inscribed in the same sphere. The problem was solved by [[Hero of Alexandria|Hero]], [[Pappus of Alexandria|Pappus]], and [[Fibonacci]], among others.{{sfn|Herz-Fischler|2013|p=[https://books.google.com/books?id=aYjXZJwLARQC&pg=PA138 138–140]}} [[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.{{sfn|Simmons|2007|p=[https://books.google.com/books?id=3KOst4Mon90C&pg=PA50 50]}} Both volumes have formulas involving the [[golden ratio]], but taken to different powers.{{sfn|Sutton|2002|p=[https://books.google.com/books?id=vgo7bTxDmIsC&pg=PA55 55]}} As it turns out, the icosahedron occupies less of the sphere's volume (60.54%) than the dodecahedron (66.49%).{{efn|1=Numerical values for the volumes of the inscribed Platonic solids may be found in {{harvnb|Buker|Eggleton|1969}}.}}

The [[dihedral angle]] of a regular icosahedron can be calculated by adding the angle of pentagonal pyramids with regular faces and a pentagonal antiprism. The dihedral angle of a pentagonal antiprism and pentagonal pyramid between two adjacent triangular faces is approximately 38.2°. The dihedral angle of a pentagonal antiprism between pentagon-to-triangle is 100.8°, and the dihedral angle of a pentagonal pyramid between the same faces is 37.4°. Therefore, for the regular icosahedron, the dihedral angle between two adjacent triangles, on the edge where the pentagonal pyramid and pentagonal antiprism are attached is 37.4° + 100.8° = 138.2°.<ref>{{multiref
 |{{harvnb|Johnson|1966|loc=See table II, line 4.}}
 |{{harvnb|MacLean|2007|p=[https://books.google.com/books?id=vINuAwAAQBAJ&pg=PA44 43&ndash;44]}}
}}</ref>

=== Symmetry ===
[[File:Sphere symmetry group ih.png|thumb|Illustration of a icosahedral symmetry. The five-fold, three-fold, and two-fold are labeled in blue, red, and magenta respectively. The mirror planes are the cyan [[great circle]].]]
The regular icosahedron has six five-fold rotation axes passing through two opposite vertices, ten three-fold axes rotating a triangular face, and fifteen two-fold axes passing through any of its edges. It has fifteen mirror planes as in a cyan [[great circle]] on the sphere meeting at order <math>\pi/5, \pi/3, \pi/2</math> angles, dividing a sphere into 120 triangles [[fundamental domain]]s. The full symmetry group of the icosahedron (including reflections) is known as the [[full icosahedral symmetry]] <math> \mathrm{I}_\mathrm{h} </math>.<ref>{{multiref
 |{{harvnb|Cann|2012|p=[https://books.google.com/books?id=mDbsN9LvE8gC&pg=PA34 34]}}
 |{{harvnb|Benz|Neumann|2014|p=[https://books.google.com/books?id=YagtBAAAQBAJ&pg=RA1-SA1-PA98 1-98]}}
}}</ref> It is isomorphic to the product of the rotational symmetry group and the [[cyclic group]] of size two, generated by the reflection through the center of the regular icosahedron.{{sfn|Seidel|1991|p=[https://books.google.com/books?id=brziBQAAQBAJ&pg=PA364 364]}} It shares the [[dual polyhedron]] of a regular icosahedron, the regular dodecahedron: a regular icosahedron can be inscribed in a regular dodecahedron by placing its vertices at the face centers of the dodecahedron, and vice versa.<ref>{{multiref
|{{harvnb|Erickson|2011|p=[https://books.google.com/books?id=LgeP62-ZxikC&pg=PA62 62]}}
|{{harvnb|Herrmann|Sally|2013|p=[https://books.google.com/books?id=b2fjR81h6yEC&pg=PA257 257]}}
}}</ref>

The rotational [[symmetry group]] of the regular icosahedron is [[isomorphic]] to the [[alternating group]] on five letters. This non-[[abelian group|abelian]] [[simple group]] is the only non-trivial [[normal subgroup]] of the [[symmetric group]] on five letters.{{sfn|Gray|2018|p=[https://books.google.com/books?id=gl1oDwAAQBAJ&pg=PA371 371]}} 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,{{sfn|Rotman|1998|p=[https://books.google.com/books?id=0kHhBwAAQBAJ&pg=PA75 74&ndash;75]}} and [[Felix Klein]] wrote a book that made use of the theory of icosahedral symmetries to derive an analytical solution to the general quintic equation.<ref>{{multiref
|{{harvnb|Klein|1884}}. See [[Icosahedral symmetry#Related geometries|related geometries of that symmetry group]] for further history and related symmetries on seven and eleven letters.
|{{harvnb|Klein|1888}}
}}</ref>

The regular icosahedron, as one of the Platonic solids, is a [[regular polyhedron]]. It is [[Isogonal figure|isogonal]], [[isohedral]], and [[isotoxal]]: any two vertices, two faces, and two edges of a regular icosahedron respectively can be transformed by rotations and reflections under its symmetry orbit, which preserves the appearance. Each regular polyhedron has a [[convex hull]] on its edge midpoints; [[icosidodecahedron]] is the convex hull of a regular icosahedron.{{sfn|Senechal|1989|p=[https://books.google.com/books?id=OToVjZW9CKMC&pg=PA12 12]}} Each vertex is surrounded by five equilateral triangles, so the regular icosahedron denotes <math> 3.3.3.3.3 </math> in [[vertex configuration]] or <math> \{3,5\} </math> in [[Schläfli symbol]].{{sfn|Walter|Deloudi|2009|p=[https://books.google.com/books?id=nVx-tu596twC&pg=PA50 50]}}

== Icosahedral graph ==
[[File:Icosahedron graph.svg|thumb|right|Icosahedral graph]]
Every [[Platonic graph]], including the '''icosahedral graph''', is a [[polyhedral graph]]. This means that they are [[Planar graph|planar graphs]], graphs that can be drawn in the plane without crossing its edges; and they are [[K-vertex-connected graph|3-vertex-connected]], meaning that the removal of any two of its vertices leaves a connected subgraph. According to [[Steinitz theorem]], the icosahedral graph endowed with these heretofore properties represents the [[Skeleton (topology)|skeleton]] of a regular icosahedron.{{sfn|Bickle|2020|p=[https://books.google.com/books?id=2sbVDwAAQBAJ&pg=PA147 147]}}

The icosahedral graph has twelve vertices, the same number of vertices as a regular icosahedron. These vertices are connected by five edges from each vertex, making the icosahedral graph [[Regular graph|5-regular]].{{sfn|Fallat|Hogben|2014|loc=Section 46|p=[https://books.google.com/books?id=8hnYCwAAQBAJ&pg=SA46-PA29 29]}} The icosahedral graph is [[Hamiltonian graph|Hamiltonian]], because it has a cycle that can visit each vertex exactly once.{{sfn|Hopkins|2004}} Any subset of four vertices has three connected edges, with one being the central of all of those three, and the icosahedral graph has no [[induced subgraph]], a [[claw-free graph]].{{sfn|Chudnovsky|Seymour|2005}}

The icosahedral graph is a [[graceful graph]], meaning that each vertex can be labeled with an [[integer]] between 0 and 30 inclusive, in such a way that the [[absolute difference]] between the labels of an edge's two vertices is different for every edge.{{sfn|Gallian|1998}}

== Related figures ==
The regular icosahedron has a large number of [[stellation]]s, constructed by extending the faces of a regular icosahedron. {{harvtxt|Coxeter|du Val|Flather|Petrie|1938}} in their work, ''[[The Fifty-Nine Icosahedra]]'', identified fifty-nine stellations for the regular icosahedron. The regular icosahedron itself is the zeroth stellation of an icosahedron, and the first stellation has each original face augmented by a low pyramid. The [[final stellation of the icosahedron|final stellation]] includes all of the cells in the icosahedron's stellation diagram, meaning every three intersecting face planes of the icosahedral core intersect either on a vertex of this polyhedron or inside it.<ref>{{multiref
 |{{harvnb|Coxeter|du Val|Flather|Petrie|1938|p=8&ndash;26}}
 |{{harvnb|Coxeter|du Val|Flather|Petrie|1999|p=30–31}}
 |{{harvnb|Wenninger|1971|p=65}}
}}</ref> The [[great dodecahedron]] of [[Kepler–Poinsot polyhedron]] is considered part of subsequent stellation.{{sfn|Wenninger|1971|p=23&ndash;69}}

{{multiple image
 | image1 = Stellation icosahedron A.png
 | image2 = Stellation icosahedron B.png
 | image3 = Stellation icosahedron C.png
 | image4 = Stellation icosahedron Ef1g1.png
 | image5 = Stellation icosahedron G.png
 | image6 = Stellation icosahedron H.png
 | align = center
 | total_width = 800
 | footer = The six stellations of the regular icosahedron according to {{harvtxt|Coxeter|du Val|Flather|Petrie|1938}}: regular icosahedron (zeroth), the first stellation, regular [[compound of five octahedra]], [[excavated dodecahedron]], [[great icosahedron]], and [[final stellation of the icosahedron|final stellation]]. See the [[The Fifty-Nine Icosahedra#Table of the fifty-nine icosahedra|list]] for more.
}}

{{multiple image
 | image1 = Great dodecahedron.png
 | image2 = Truncatedicosahedron.jpg
 | image3 = Triakisicosahedron.jpg
 | image4 = Double diminished icosahedron.png
 | footer = Top left to bottom right: [[great dodecahedron]], [[truncated icosahedron]], [[triakis icosahedron]], and [[edge-contracted icosahedron]]
 | perrow = 2
 | total_width = 300
 | align = right
}}

The [[triakis icosahedron]] is the [[Catalan solid]] constructed by attaching the base of triangular pyramids onto each face of a regular icosahedron, the [[Kleetope]] of an icosahedron.{{sfn|Brigaglia|Palladino|Vaccaro|2018}} The [[truncated icosahedron]] is an [[Archimedean solid]] constructed by truncating the vertices of a regular icosahedron; the resulting polyhedron may be considered as a [[ball (association football)|football]] because of having a pattern of numerous hexagonal and pentagonal faces.<ref>{{multiref
 |{{harvnb|Chancey|O'Brien|1997|p=[https://books.google.com/books?id=wcQIEAAAQBAJ&pg=PA13 13]}}
 |{{harvnb|Kotschick|2006}}
}}</ref>

The [[great dodecahedron]] has other ways to construct from the regular icosahedron. Aside from the stellation, the great dodecahedron can be constructed by [[faceting]] the regular icosahedron, that is, removing the pentagonal faces of the regular icosahedron without removing the vertices or creating a new one; or forming a regular pentagon by each of the five vertices inside of a regular icosahedron, and twelve regular pentagons intersecting each other, making a [[pentagram]] as its [[vertex figure]].<ref>{{multiref
 |{{harvnb|Inchbald|2006}}
 |{{harvnb|Pugh|1976a|p=[https://books.google.com/books?id=IDDxpYQTR7kC&pg=PA85 85]}}
 |{{harvnb|Barnes|2012|p=[https://books.google.com/books?id=7YCUBUd-4BQC&pg=PA46 46]}}
}}</ref>

A [[Johnson solid]] is a polyhedron whose faces are all regular but which is not [[Uniform polyhedron|uniform]]. In other words, they do not include the [[Archimedean solid]]s, the [[Catalan solid]]s, the [[Prism (geometry)|prism]]s, or the [[antiprism]]s. Some Johnson solids can be derived by removing part of a regular icosahedron, a process known as ''diminishment''. They are [[gyroelongated pentagonal pyramid]], [[metabidiminished icosahedron]], and [[tridiminished icosahedron]], which remove one, two, and three pentagonal pyramids from the icosahedron respectively.{{sfn|Berman|1971}}

[[File:Jessen icosahedron with snub icosahedron.png|thumb|Regular icosahedron and its non-convex variant, which differs from Jessen's icosahedron in having different vertex positions and non-right-angled dihedrals]]
Another related shape can be derived by keeping the vertices of a regular icosahedron in their original positions and replacing certain pairs of equilateral triangles with pairs of isosceles triangles. The resulting polyhedron has the non-convex version of the regular icosahedron. Nonetheless, it is occasionally incorrectly known as [[Jessen's icosahedron]] because of the similar visual, of having the same combinatorial structure and symmetry as Jessen's icosahedron;{{efn|1=Incorrect descriptions of Jessen's icosahedron as having the same vertex positions as a regular icosahedron include:
{{harvnb|Wells|1991|p=161}}; [https://web.archive.org/web/20211018124709/https://mathworld.wolfram.com/JessensOrthogonalIcosahedron.html Jessen's Orthogonal Icosahedron on MathWorld] (old version, subsequently fixed).}} the difference is that the non-convex one does not form a tensegrity structure and does not have right-angled dihedrals.{{sfn|Pugh|1976b|p=[https://books.google.com/books?id=McEOfJu3NQAC&pg=PA11 11], [https://books.google.com/books?id=McEOfJu3NQAC&pg=PA26 26]}}

Apart from the construction above, the regular icosahedron can be inscribed in a regular octahedron by placing its twelve vertices on the twelve edges of the octahedron such that they divide each edge in the [[golden section]]. Because the resulting segments 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}} Another relation between the two is that they are part of the progressive transformation from the [[cuboctahedron]]'s rigid struts and flexible vertices, known as [[jitterbug transformation]].{{sfn|Verheyen|1989}}

The [[edge-contracted icosahedron]] has a surface like a regular icosahedron but with [[coplanar|some faces lie in the same plane]].{{sfn|Tsuruta|2024|p=[https://books.google.com/books?id=kLckEQAAQBAJ&pg=PA112 112]}}

The regular icosahedron is [[Four-dimensional space#Dimensional analogy|analogous]] to the [[600-cell]], a [[Regular 4-polytope#Regular convex 4-polytopes|regular 4-dimensional polytope]].{{sfn|Barnes|2012|p=[https://books.google.com/books?id=7YCUBUd-4BQC&pg=PA79 79]}} This polytope has six hundred regular tetrahedra as its [[Cell (geometry)|cell]]s.{{sfn|Stillwell|2005|p=[http://books.google.com/books?id=I1QPQic_PxwC&pg=PA173 173]}}

== Applications ==
{{multiple image
 | total_width = 300
 | image1 = Twenty-sided die (icosahedron) with faces inscribed with Greek letters MET 10.130.1158 001.jpg
 | caption1 = Twenty-sided dice from Ptolemaic of Egypt, inscribed with Greek letters at the faces
 | image2 = ScatDice.JPG
 | caption2 = The ''Scattergories'' twenty-sided die, excluding the six letters Q, U, V, X, Y, and Z
}}
Dice are the most common objects using different polyhedra, one of them being the regular icosahedron. The twenty-sided die was found in many ancient times. One example is the die from the Ptolemaic of Egypt, which  later used Greek letters inscribed on the faces in the period of Greece and Rome.<ref>{{multiref
 |{{harvnb|Smith|1958|p=[https://books.google.com/books?id=uTytJGnTf1kC&pg=PA295 295]}}
 |{{harvnb|Minas-Nerpel|2007}}
}}</ref>
Another example was found in the treasure of [[Tipu Sultan]], which was made out of gold and with numbers written on each face.{{sfn|Cromwell|1997|p=[https://archive.org/details/polyhedra0000crom/page/4/mode/1up?view=theater 4]}} In several [[roleplaying game]]s, such as ''[[Dungeons & Dragons]]'', the twenty-sided die (labeled as [[Dice#Polyhedral dice|d20]]) is commonly used in determining success or failure of an action. It may be numbered from "0" to "9" twice, in which form it usually serves as a ten-sided die ([[Dice#Non-cubical dice|d10]]); most modern versions are labeled from "1" to "20".<ref>{{cite web |url=https://www.gmdice.com/pages/dungeons-dragons-dice |title=Dungeons & Dragons Dice |website=gmdice.com |access-date=August 20, 2019}}</ref> ''[[Scattergories]]'' is another board game in which the player names the category entires on a card within a given set time. The naming of such categories is initially with the letters contained in every twenty-sided dice.{{sfn|Flanagan|Gregory|2015|p=[https://books.google.com/books?id=hViuEAAAQBAJ&pg=PA85 85]}}

{{multiple image
 | total_width = 350
 | align = right
 | image1 = Circogonia icosahedra.jpg
 | caption1 = The [[radiolarian]] ''Circogonia icosahedra''
 | image2 = Dymaxion projection.png
 | caption2 = [[Dymaxion map]], created by the net of a regular icosahedron
}}
The regular icosahedron may also appear in many fields of science as follows:
* In [[virology]], [[Herpesviridae|herpes virus]] have icosahedral [[capsid|shells]], especially well-known in [[adenovirus]].{{sfn|Gallardo|Pérez-Illana|Martín-González|San Martín|2021}} The outer protein shell of [[HIV]] is enclosed in a regular icosahedron, as is the head of a typical [[myovirus]].{{sfn|Strauss|Strauss|2008|p=35&ndash;62}} Several species of [[radiolarian]]s discovered by [[Ernst Haeckel]], described its shells as the like-shaped various regular polyhedra; one of which is ''Circogonia icosahedra'', whose skeleton is shaped like a regular icosahedron.<ref>{{multiref
 |{{harvnb|Haeckel|1904}}
 |{{harvnb|Cromwell|1997|p=[https://archive.org/details/polyhedra0000crom/page/6/mode/1up?view=theater 6]}}
}}</ref>
* In chemistry, the [[closo cluster|closo]]-[[carboranes]] are [[Chemical compound|compounds]] with a shape resembling the regular icosahedron.{{sfn|Spokoyny|2013}} The [[crystal twinning]] with [[icosahedral twins|icosahedral shapes]] also occurs in crystals, especially [[nanoparticle]]s.{{sfn|Hofmeister|2004}} Many [[Crystal structure of boron-rich metal borides|borides]] and [[allotropes of boron]] such as [[Allotropes of boron#α-rhombohedral boron|α-]] and [[Allotropes of boron#β-rhombohedral boron|β-rhombohedral]] contain boron B<sub>12</sub> icosahedron as a basic structure unit.{{sfn|Dronskowski|Kikkawa|Stein|2017|p=[https://books.google.com/books?id=e0VBDwAAQBAJ&pg=PA436 435&ndash;436]}}
* In cartography, [[R. Buckminster Fuller]] used the net of a regular icosahedron to create a map known as [[Dymaxion map]], by subdividing the net into triangles, followed by calculating the grid on the Earth's surface, and transferring the results from the sphere to the polyhedron. This projection was created during the time that Fuller realized that the [[Greenland]] is smaller than [[South America]].{{sfn|Cromwell|1997|p=[https://archive.org/details/polyhedra0000crom/page/7/mode/1up?view=theater 7]}}
* In the [[Thomson problem]], concerning the minimum-energy configuration of <math>n</math> charged particles on a sphere, and for the [[Tammes problem]] of constructing a [[spherical code]] maximizing the smallest distance among the points, the minimum solution known for <math>n = 12</math> places the points at the vertices of a regular icosahedron, [[Circumscribed sphere|inscribed in a sphere]]. This configuration is proven optimal for the Tammes problem, but a rigorous solution to this instance of the Thomson problem is unknown.{{sfn|Whyte|1952}}

{{multiple image
 | image1 = Kepler Icosahedron Water.jpg
 | caption1 = Sketch of a regular icosahedron by Johannes Kepler
 | image2 = Mysterium Cosmographicum solar system model.jpg
 | caption2 = [[Johannes Kepler|Kepler's]] Platonic solid model of the [[Solar System]]
 | align = right
 | total_width = 300
}}
As mentioned above, the regular icosahedron is one of the five [[Platonic solid|Platonic solids]]. The regular polyhedra have been known since antiquity, but are named after [[Plato]] who, in his [[Timaeus (dialogue)|''Timaeus'']] dialogue, identified these with the five [[Classical elements|elements]], whose elementary units were attributed these shapes: [[Fire (classical element)|fire]] (tetrahedron), [[Air (classical element) | air]] (octahedron), [[Water (classical element)|water]] (icosahedron), [[Earth (classical element)|earth]] (cube) and the shape of the universe as a whole (dodecahedron). [[Euclid]]'s [[Euclid's Elements|''Elements'']] defined the Platonic solids and solved the problem of finding the ratio of the circumscribed sphere's diameter to the edge length.{{sfn|Heath|1908|p=262, 478, 480}} Following their identification with the elements by Plato, [[Johannes Kepler]] in his ''[[Harmonices Mundi]]'' sketched each of them, in particular, the regular icosahedron.{{sfn|Cromwell|1997|p=[https://archive.org/details/polyhedra0000crom/page/55 55]}} In his ''[[Mysterium Cosmographicum]]'', he also proposed a model of the [[Solar System]] based on the placement of Platonic solids in a concentric sequence of increasing radius of the inscribed and circumscribed spheres whose radii gave the distance of the six known planets from the common center. The ordering of the solids, from innermost to outermost, consisted of: [[regular octahedron]], regular icosahedron, [[regular dodecahedron]], [[regular tetrahedron]], and [[cube]].{{sfn|Livio|2003|p=147}}

== References ==
=== Footnotes ===
{{notelist|group="lower-alpha"}}

=== Notes ===
{{reflist|30em}}

=== Bibliographies ===
{{refbegin|30em}}
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}}
* {{cite book
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* {{cite journal
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 | year = 2024
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 | contribution = Polyhedral Realization as Deltahedra Using Subgroup Isomorphism Test
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 | volume = 1
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}}
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 | issue = 1–3
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 | mr = 0994201
 | pages = 203–250
 | title = The complete set of Jitterbug transformers and the analysis of their motion
 | volume = 17
 | year = 1989
}}
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 | last1 = Walter | first1 = Steurer
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 | year = 2009
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 | series = Springer Series in Materials Science
 | volume = 126
 | url = https://books.google.com/books?id=nVx-tu596twC&pg=PA50
 | page = 50
 | isbn = 978-3-642-01898-5
 | doi = 10.1007/978-3-642-01899-2
}}
* {{cite book
 | last = Wells | first = David
 | location = London
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 | publisher = Penguin
 | title = The Penguin Dictionary of Curious and Interesting Geometry
 | year = 1991
}}
* {{cite book
 | last = Wenninger | first = Magnus J. 
 | year = 1971
 | title = Polyhedron Models
 | publisher = [[Cambridge University Press]]
}}
* {{cite journal
 | last = Whyte | first = L. L.
 | doi = 10.1080/00029890.1952.11988207
 | journal = [[The American Mathematical Monthly]]
 | jstor = 2306764
 | mr = 50303
 | pages = 606–611
 | title = Unique arrangements of points on a sphere
 | volume = 59
 | year = 1952| issue = 9
}}

{{refend}}

== External links ==
{{commons category|Icosahedron}}
{{Wikisource1911Enc|Icosahedron}}
{{Wiktionary|icosahedron}}
*{{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}}
{{Icosahedron stellations}}

[[Category:Composite polyhedron]]
[[Category:Deltahedra]]
[[Category:Platonic solids]]
[[Category:Regular polyhedra]]