Regular icosahedron
Type	Deltahedron, Gyroelongated bipyramid, Platonic solid, Regular polyhedron
Faces	20
Edges	30
Vertices	12
Vertex configuration	${\displaystyle 12\times \left(3^5\right)}$
Schläfli symbol	${\displaystyle \{3,5\}}$
Symmetry group	icosahedral symmetry${\displaystyle {\mathrm{I}}_{\mathrm{h}}}$
Dihedral angle (degrees)	138.190 (approximately)
Dual polyhedron	regular dodecahedron
Properties	convex, composite, isogonal, isohedral, isotoxal
Net

Theregular icosahedron (or simply icosahedron) is aconvex polyhedron that can be constructed frompentagonal antiprism by attaching twopentagonal pyramids withregular faces to each of its pentagonal faces, or by putting points onto the cube. The resulting polyhedron has 20equilateral triangles as its faces, 30 edges, and 12 vertices. It is an example of aPlatonic solid and of adeltahedron. The icosahedral graph represents theskeleton of a regular icosahedron.

Many polyhedra and other related figures are constructed from the regular icosahedron, including its 59stellations. Thegreat dodecahedron, one of theKepler–Poinsot polyhedra, is constructed by either stellation of theregular dodecahedron orfaceting of the icosahedron. Some of theJohnson solids can be constructed by removing the pentagonal pyramids. The regular icosahedron'sdual polyhedron is theregular dodecahedron, and their relation has a historical background in the comparison mensuration. It is analogous to a four-dimensionalpolytope, the600-cell.

Regular icosahedra can be found in nature; a well-known example is thecapsid in biology. Other applications of the regular icosahedron are the usage of its net incartography, and the twenty-sided dice that may have been used in ancient times but are nowcommonplace in moderntabletop role-playing games.

The regular icosahedron is atwenty-sided polyhedron wherein the faces areequilateral triangles. It is one of the eight convexdeltahedra, a polyhedron wherein all of its faces are equilateral triangles.^[1] Variously, it can be constructed as follows:

• Started by attaching twopentagonal pyramids withregular faces to the base of apentagonal antiprism. These components areelementaries—they cannot be disintegrated into smaller convex polyhedra with regular faces again. Replacing bases of a pentagonal antiprism with ten triangular pyramids,^[2] the regular icosahedron is classified as a composite polyhedron, the opposite of an elementary polyhedron.^[3] This construction led to the alternative names calledbicapped pentagonal antiprism,^[4] orgyroelongated pentagonal bipyramid due to its construction process throughgyroelongation—polyhedra construction by attaching two pyramids onto the base of anantiprism.^[5]

• The twelve vertices of a regular icosahedron describe the three mutually perpendiculargolden rectangular planes, whose corners are connected. These rectangular planes can be constructed from a pair of vertices located on the midpoints of the opposite edges on acube's surface, drawing a segment line between those two, and divides the segment line in agolden ratio${\displaystyle \phi =(1+\sqrt{5})/2}$ from its midpoint.^[6] Both the vertices of a regular icosahedron have an edge length of 2 and the three planes can be illustrated throughCartesian coordinate system:^[7]$${\displaystyle \left(0,\pm 1,\pm \phi \right),\left(\pm 1,\pm \phi ,0\right),\left(\pm \phi ,0,\pm 1\right).}$$
• One cansnub aregular octahedron, by separating all of its faces and filling them with more equilateral triangles. As suggested by the process, the regular icosahedron is also known assnub octahedron.^[8]

The regular icosahedron can be unfolded into 43,380 differentnets.^[9] The earliest net appeared inAlbrecht Dürer'sPainter's Manual in 1525.^[10]

Thesurface area of a polyhedron is the sum of the areas of its faces. In the case of a regular icosahedron, its surface area${\displaystyle A}$ is twenty times that of each of its equilateral triangle faces. Its volume${\displaystyle V}$ can be obtained as twenty times that of a pyramid whose base is one of its faces and whose apex is the regular icosahedron's center; or as the sum of the volume of two uniformpentagonal pyramids and apentagonal antiprism. Given that the edge length${\displaystyle a}$ of a regular icosahedron, both expressions are:^[11]$${\displaystyle A=5\sqrt{3}{a}^2\approx 8.660a^2,V=\frac{5{\phi }^2}{6}{a}^3\approx 2.182a^3.}$$

Theinsphere of a convex polyhedron is a sphere touching every polyhedron's face within. Thecircumsphere of a convex polyhedron is a sphere that contains the polyhedron and touches every vertex. Themidsphere of a convex polyhedron is a sphere tangent to every edge. Given that the edge length${\displaystyle a}$ of a regular icosahedron, the radius of insphere (inradius)${\displaystyle r_I}$, the radius of circumsphere (circumradius)${\displaystyle r_C}$, and the radius of midsphere (midradius)${\displaystyle r_M}$ are, respectively:^[12]$${\displaystyle r_I=\frac{{\phi }^{2}a}{2\sqrt{3}}\approx 0.756a,r_C=\frac{\sqrt{{\phi }^2+1}}{2}a\approx 0.951a,r_M=\frac{\phi }{2}a\approx 0.809a.}$$

A problem dating back to the ancient Greeks is determining which of two shapes has a larger volume: a regular icosahedron inscribed in a sphere, or a regular dodecahedron inscribed in the same sphere. The problem was solved byHero,Pappus, andFibonacci, among others.^[13]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.^[14] Both volumes have formulas involving thegolden ratio, but taken to different powers.^[15] As it turns out, the regular icosahedron occupies less of the sphere's volume (60.54%) than the regular dodecahedron (66.49%).^[a]

Thedihedral angle of a regular icosahedron is${\displaystyle 2\arcsin (\phi /\sqrt{3})\approx {138.19}^{\circ }}$, obtained 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°.^[16]

The regular icosahedron has three types ofclosed geodesics. These are paths on its surface that are locally straight: they avoid the polyhedron's vertices, follow line segments across the faces that they cross, and formcomplementary angles on the two incident faces of each edge that they cross. The first geodesic forms aregular decagon perpendicular to the longest diagonal and has the length${\displaystyle 5}$. The other two geodesics are non-planar, with lengths${\displaystyle 3\sqrt{3}\approx 5.196}$ and${\displaystyle 2\sqrt{7}\approx 5.292}$.^[17]

The regular icosahedron has the thirty-one axes ofrotational symmetry (that is, rotating around an axis that results in an identical appearance). There are six axes passing through two opposite vertices, ten axes rotating a triangular face, and fifteen axes passing through any of its edges. Respectively, these axes are five-fold rotational symmetry (0°, 72°, 144°, 216°, and 288°), three-fold rotational symmetry (0°, 120°, and 240°), and two-fold rotational symmetry (0° and 180°).^[18] The regular icosahedron also has fifteen mirror planes that can be represented asgreat circles on a sphere. It divides the surface of a sphere into 120 trianglesfundamental domains; these triangles are calledMobius triangles. Both reflections and rotational symmetries are theisometries—transformations in order to maintain the appearance—which forms thefull icosahedral symmetry${\displaystyle {\mathrm{I}}_{\mathrm{h}}}$ of order 120.^[19] This symmetry group isisomorphic to the product of the rotational symmetry group and thecyclic group of size two, generated by the reflection through the center of the regular icosahedron.^[20]

The rotationalsymmetry group of the regular icosahedron is isomorphic to thealternating group on five letters. This non-abeliansimple group is the only non-trivialnormal subgroup of thesymmetric group on five letters.^[21] Since theGalois group of the generalquintic 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 theAbel–Ruffini theorem uses this simple fact,^[22] andFelix Klein wrote a book that made use of the theory of icosahedral symmetries to derive an analytical solution to the general quintic equation.^[23]

The regular icosahedron isisogonal,isohedral, andisotoxal: any two vertices, two faces, and two edges of a regular icosahedron can be transformed by rotations and reflections under its symmetry orbit, which preserves the appearance. Each regular polyhedron has aconvex hull on its edge midpoints;icosidodecahedron is the convex hull of a regular icosahedron.^[24] Each vertex is surrounded by five equilateral triangles, so the regular icosahedron denotes${\displaystyle \mathrm{3.3.3.3.3}}$ invertex configuration or${\displaystyle \{3,5\}}$ inSchläfli symbol.^[25]

A twenty-sided die from Ptolemaic Egypt, inscribed with Greek letters on the faces

TheScattergories twenty-sided die, excluding the six letters Q, U, V, X, Y, and Z

Dice are among the most common objects that utilize different polyhedra, one of which is the regular icosahedron. The twenty-sided die was found in ancient times. One example is the die fromPtolemaic Egypt, which was later used with Greek letters inscribed on the faces in the period of Greece and Rome.^[26]Another example was found in the treasure ofTipu Sultan, which was made out of gold and with numbers written on each face.^[27]

In severalroleplaying games, such asDungeons & Dragons, the twenty-sided die (labeled asd20) 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 (d10); most modern versions are labeled from "1" to "20".^[28]Scattergories is another board game in which the player names the category entries on a card within a given set time. The naming of such categories is initially with the letters contained in every twenty-sided dice.^[29]

TheradiolarianCircogonia icosahedra

Dymaxion map, created by the net of a regular icosahedron

Invirology,herpes virus have icosahedralshells, especially well-known inadenovirus.^[30] The outer protein shell ofHIV is enclosed in a regular icosahedron, as is the head of a typicalmyovirus.^[31] Several species ofradiolarians discovered byErnst Haeckel, described their shells as like-shaped various regular polyhedra; one of which isCircogonia icosahedra, whose skeleton is shaped like a regular icosahedron.^[32]

In chemistry, thecloso-carboranes arecompounds with a shape resembling the regular icosahedron.^[33] Thecrystal twinning withicosahedral shapes also occurs in crystals, especiallynanoparticles.^[34] Manyborides andallotropes of boron such asα- andβ-rhombohedral contain boron B₁₂ icosahedron as a basic structure unit.^[35]

In cartography,R. Buckminster Fuller used the net of a regular icosahedron to create a map known asDymaxion 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 thatGreenland is smaller thanSouth America.^[36]

In theThomson problem, concerning the minimum-energy configuration of${\displaystyle n}$ charged particles on a sphere, and for theTammes problem of constructing aspherical code maximizing the smallest distance among the points, the minimum solution known for${\displaystyle n=12}$ places the points at the vertices of a regular icosahedron,inscribed in a sphere. This configuration is proven optimal for the Tammes problem, and also for the Thomas problem.^[37]

In tensegrity, the regular icosahedron 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 cellularmorphogenesis.^[38]

Sketch of a regular icosahedron by Johannes Kepler

Kepler's Platonic solid model of theSolar System

The regular icosahedron is one of the fivePlatonic solids. The regular polyhedra have been known since antiquity, but are named afterPlato who, in hisTimaeus dialogue, identified these with the fiveelements, whose elementary units were attributed these shapes:fire (tetrahedron),air (octahedron),water (icosahedron),earth (cube) and the shape of the universe as a whole (dodecahedron).Euclid'sElements defined the Platonic solids and solved the problem of finding the ratio of the circumscribed sphere's diameter to the edge length.^[39]

Following their identification with the elements by Plato,Johannes Kepler in hisHarmonices Mundi sketched each of them, in particular, the regular icosahedron.^[40] In hisMysterium Cosmographicum, he also proposed a model of theSolar 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, andcube.^[41]

Leonardo da Vinci'sDivina proportione drew both regular icosahedron and dodecahedron.^[42]

EveryPlatonic graph, including theicosahedral graph, is apolyhedral graph: theycan be drawn in the plane without crossing its edges, andthe removal of any two of its vertices leaves a connected subgraph. According toSteinitz's theorem, the icosahedral graph endowed with these heretofore properties represents theskeleton of a regular icosahedron.^[43]

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 graph5-regular.^[44] The icosahedral graph isHamiltonian, because it has a cycle that can visit each vertex exactly once.^[45] Any subset of four vertices has three connected edges, with one being the central of all of those three, and the icosahedral graph has noinduced subgraph, aclaw-free graph.^[46]

The icosahedral graph is agraceful graph, meaning that each vertex can be labeled with aninteger between 0 and 30 inclusive, in such a way that theabsolute difference between the labels of an edge's two vertices is different for every edge.^[47]

The regular icosahedron can be represented as aconfiguration matrix, amatrix in which the rows and columns correspond to the elements of a polyhedron as the vertices, edges, and faces. Thediagonal of a matrix denotes the number of each element that appears in a polyhedron, whereas the non-diagonal of a matrix denotes the number of the column's elements that occur in or at the row's element. The following matrix is:^[48]

The regular icosahedron is thedual polyhedron of theregular dodecahedron. A regular icosahedron can be inscribed in a regular dodecahedron by placing its vertices at the face centers of the regular dodecahedron, and vice versa. The regular dodecahedron has icosahedral symmetry.^[49]

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 thegolden 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.^[50] Another relation between the two is that they are part of the progressive transformation from thecuboctahedron's rigid struts and flexible vertices, known asjitterbug transformation.^[51]

A regular icosahedron of edge length${\displaystyle 1/\phi \approx 0.618}$ can be inscribed in a unit-edge-length cube by placing six of its edges—three orthogonal opposite pairs—on the square faces of the cube, centered on the face centers and parallel or perpendicular to the square's edges.^[52] 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.^{[citation needed]}

The regular icosahedron has a large number ofstellations, constructed by extending the faces of a regular icosahedron.Coxeter et al. (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. Thefinal 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.^[53]

The six stellations of the regular icosahedron according toCoxeter et al. (1938): regular icosahedron (zeroth), the first stellation, regularcompound of five octahedra,excavated dodecahedron,great icosahedron, andfinal stellation. See thelist for more.

Thegreat dodecahedron can be constructed from the regular icosahedron in other ways. Aside from the stellation, the great dodecahedron can be constructed byfaceting the regular icosahedron, that is, removing the triangle faces of the regular icosahedron without removing the vertices or creating a new one; then forming twelve regular pentagons on sets of five vertices inside of a regular icosahedron. These new faces intersect each other, making apentagram as thevertex figure.^[54]

Top left to bottom right:great dodecahedron,truncated icosahedron,gyroelongated pentagonal pyramid, andedge-contracted icosahedron

Thetriakis icosahedron is aCatalan solid constructed by attaching the base of triangular pyramids onto each face of a regular icosahedron, theKleetope of an icosahedron.^[55] Thetruncated icosahedron is anArchimedean solid constructed by truncating the vertices of a regular icosahedron; the resulting polyhedron may be considered as afootball because of having a pattern of numerous hexagonal and pentagonal faces,^[56] or a structural form ofcarbon known asbuckminsterfullerene that has 60 carbon atoms bonded together.^[57]

AJohnson solid is a polyhedron whose faces are all regular but which is notuniform. In other words, they do not include theArchimedean solids, theCatalan solids, theprisms, or theantiprisms. Some Johnson solids can be derived by removing part of a regular icosahedron, a process known asdiminishment. They aregyroelongated pentagonal pyramid,metabidiminished icosahedron, andtridiminished icosahedron, which remove one, two, and three pentagonal pyramids from the icosahedron, respectively.^[58]

Theedge-contracted icosahedron has a surface like a regular icosahedron but withsome faces lie in the same plane.^[59]

Spherical icosahedron and a toy namedImpossiball

The spherical icosahedron represents a regular icosahedron projected to a sphere, a part ofspherical polyhedron. It can be modeled by thearc ofgreat circles, creating bounds as the edges ofspherical triangle.^[60] Identified byR. Buckminster Fuller, there are31 great circles in a spherical icosahedron.^[61] Its dual is thespherical dodecahedron.^[60] The appearance of this shape may be found inImpossiball, a similar toy toRubik's Cube.^[62]

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 asJessen's icosahedron because of the similar visual, of having the same combinatorial structure and symmetry as Jessen's icosahedron;^[b] the difference is that the non-convex one does not form a tensegrity structure and does not have right-angled dihedrals.^[63]

The regular icosahedron isanalogous to the600-cell, aregular 4-dimensional polytope.^[64] This polytope has six hundred regular tetrahedra as itscells.^[65] The regular icosahedron is acell for the honeycomb in ahyperbolic three-dimensional space.^[66]

• Binary icosahedral group, a double cover of the group of rotations of the icosahedron
  • Icosian, a set of quaternions with the same symmetry
• Dehn invariant
• Disdyakis triacontahedron, a 120-sided convex polyhedron whose faces are the fundamental domains in icosahedral symmetry. Six such regions can be painted on each face of an icosahedron
• Dogic, an icosahedral version ofRubik's Cube
• Geodesic grid
• Geodesic polyhedron
• Goldberg polyhedron, a convex polyhedron made from hexagons and pentagons with icosahedral symmetry
• Hemi-icosahedron
• Polyhedral map projection, map projections taking the globe to a polyhedron
• Rhombic triacontahedron, a 30-sided convex polyhedron with icosahedral symmetry whose faces lie in the 2-fold symmetry planes
• Tensegrity icosahedron
• Tetrahedral-icosahedral honeycomb, another hyperbolic honeycomb involving some icosahedral cells
• Zometool, a construction toy based on the icosahedral symmetry system

Wikimedia Commons has media related toIcosahedron.

Wikisource has the text of the1911Encyclopædia Britannica article "Icosahedron".

Look upicosahedron in Wiktionary, the free dictionary.

• Klitzing, Richard."3D convex uniform polyhedra x3o5o – ike".
• Hartley, Michael."Dr Mike's Math Games for Kids".
• K.J.M. MacLean, A Geometric Analysis of the Five Platonic Solids and Other Semi-Regular Polyhedra
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra
• Tulane.edu A discussion of viral structure and the icosahedron
• Origami Polyhedra – Models made with Modular Origami
• Video of icosahedral mirror sculpture
• [1] Principle of virus architecture

Notablestellations of the icosahedron
Regular	Regular star	Uniform duals
(Convex) icosahedron	Great icosahedron	Small triambic icosahedron	Medial triambic icosahedron	Great triambic icosahedron
Regular compounds			Others
Compound of five octahedra	Compound of five tetrahedra	Compound of ten tetrahedra	Excavated dodecahedron	Final stellation

• Composite polyhedron
• Deltahedra
• Platonic solids
• Regular polyhedra

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