- Icosahedron
In

geometry , an**icosahedron**(Greek: "eikosaedron", from "eikosi" twenty + "hedron" seat; IPA|/ˌaɪ.kəʊ.sə.ˈhi.dɹən/; plural: -drons, -dra IPA|/-dɹə/) isanypolyhedron having 20 faces, but usually a**regular icosahedron**is implied, which has equilateral triangles as faces.The regular icosahedron is one of the five

Platonic solid s. It is a convex regularpolyhedron composed oftwenty triangular faces, with five meeting at each of the twelve vertices. It has 30 edges and 12 vertices. Itsdual polyhedron is thedodecahedron .**Dimensions**If the edge length of a regular icosahedron is $a$, the

radius of a circumscribedsphere (one that touches the icosahedron at all vertices) is: $r\_u\; =\; frac\{a\}\{2\}\; sqrt\{varphi\; sqrt\{5\; =\; frac\{a\}\{4\}\; sqrt\{10\; +2sqrt\{5\; approx\; 0.9510565163\; cdot\; a$

and the radius of an inscribed sphere (

tangent to each of the icosahedron's faces) is: $r\_i\; =\; frac\{varphi^2\; a\}\{2\; sqrt\{3\; =\; frac\{1\}\{12\}\; sqrt\{3\}\; left(3+\; sqrt\{5\}\; ight)\; a\; approx\; 0.7557613141cdot\; a$

while the

midradius , which touches the middle of each edge, is: $r\_m\; =\; frac\{a\; varphi\}\{2\}\; =\; frac\{1\}\{4\}\; left(1+sqrt\{5\}\; ight)\; a\; approx\; 0.80901699cdot\; a$

where $varphi$ (also called $au$) is the

golden ratio .**Area and volume**The surface area "A" and the

volume "V" of a regular icosahedron of edge length "a" are::$A\; =\; 5sqrt\{3\}a^2\; approx\; 8.66025404a^2$:$V\; =\; frac\{5\}\{12\}\; (3+sqrt5)a^3\; approx\; 2.18169499a^3$.**Cartesian coordinates**

The followingCartesian coordinates define the vertices of an icosahedron with edge-length 2, centered at the origin:: (0, ±1, ±φ): (±1, ±φ, 0): (±φ, 0, ±1)where φ = (1+√5)/2 is thegolden ratio (also written τ). Note that these vertices form five sets of three mutually centered, mutuallyorthogonal golden rectangle s.The 12 edges of a regular

octahedron can be partitioned 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 partitioning each edge into the golden mean along the direction of its vector. The five octahedra defining any given icosahedron form a regularpolyhedral compound , as do the two icosahedra that can be defined in this way from any given octahedron.**Stellations**According to specific rules defined in the book

The fifty nine icosahedra , 59stellation s were identified for the regular icosahedron. The first form is the icosahedron itself. One is a regularKepler-Poinsot solid . Three are regular compound polyhedra. [*Citation | last1=Coxeter | first1=Harold Scott MacDonald | author1-link=Harold Scott MacDonald Coxeter | last2=Du Val | first2=P. | last3=Flather | first3=H. T. | last4=Petrie | first4=J. F. | title=The fifty-nine icosahedra | publisher=Tarquin | edition=3rd | isbn=978-1-899618-32-3 | id=MathSciNet | id = 676126 | year=1999 (1st Edn University of Toronto (1938))*]**Geometric relations**There are distortions of the icosahedron that, while no longer regular, are nevertheless vertex-uniform. These are invariant under the same

rotation s as thetetrahedron , and are somewhat analogous to thesnub cube andsnub dodecahedron , including some forms which are chiral and some with T_{h}-symmetry, i.e. have different planes of symmetry from the tetrahedron. The icosahedron has a large number ofstellation s, including one of theKepler-Poinsot polyhedra and some of the regular compounds, which could be discussed here.The icosahedron is unique among the

Platonic solids in possessing adihedral 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 regularpolychoron 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 convexpolytope 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 thesnub 24-cell ), just as hexagons can be used as faces in semi-regular polyhedra (for example thetruncated icosahedron ). Finally, non-convex polytopes do not carry the same strict requirements as convex polytopes, and icosahedra are indeed the cells of the icosahedral120-cell , one of the ten non-convex regular polychora.An icosahedron can also be called a gyroelongated pentagonal bipyramid. It can be decomposed into a

gyroelongated pentagonal pyramid and apentagonal pyramid or into apentagonal antiprism and two equalpentagonal pyramid s.The icosahedron can also be called a snub tetrahedron, as snubification of a regular tetrahedron gives a regular icosahedron. Alternatively, using the nomenclature for snub polyhedra that refers to a snub cube as a snub cuboctahedron (cuboctahedron = rectified cube) and a snub dodecahedron as a snub icosidodecahedron (icosidodecahedron = rectified dodecahedron), one may call the icosahedron the snub octahedron (octahedron = rectified tetrahedron).

A rectified icosahedron forms an

icosidodecahedron .**Icosahedron vs dodecahedron**When an icosahedron is inscribed in a

sphere , it occupies less of the sphere's volume (60.54%)than adodecahedron inscribed in the same sphere (66.49%).**Natural forms and uses**Many

virus es, e.g.herpes virus, have the shape of an icosahedron. Viral structures are built of repeated identicalprotein subunits 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 viralgenome .In 1904,

Ernst Haeckel described a number of species ofRadiolaria , including "Circogonia icosahedra", whose skeleton is shaped like a regular icosahedron. A copy of Haeckel's illustration for this radiolarian appears in the article onregular polyhedra .In some

roleplaying game s, the twenty-sided die (for short, d20) is 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". Seed20 System .An icosahedron is the three-dimensional game board for

Icosagame , formerly known as the Ico Crystal Game.An icosahedron is used in the board game

Scattergories to choose a letter of the alphabet. Six little-used letters, such as X, Q, and Z, are omitted.Inside a

Magic 8-Ball , various answers to yes-no questions are printed on a regular icosahedron.The icosahedron displayed in a functional form is seen in the

Sol de la Flor light shade. The rosette formed by the overlapping pieces show a resemblance to theFrangipani flower.If each edge of an icosahedron is replaced by a one ohm

resistor , the resistance between opposite vertices is 0.5 ohms, and that between adjacent vertices 11/30 ohms. [*cite journal | last = Klein | first = Douglas J. | year = 2002 | title = Resistance-Distance Sum Rules | journal = Croatica Chemica Acta | volume = 75 | issue = 2 | pages = 633–649 | url = http://public.carnet.hr/ccacaa/CCA-PDF/cca2002/v75-n2/CCA_75_2002_633_649_KLEIN.pdf | format = PDF | accessdate = 2006-09-15*]The

symmetry group of the icosahedron isisomorphic to thealternating group on five letters. This nonabeliansimple group is the only nontrivialnormal subgroup of thesymmetric group on five letters. Since theGalois group of the generalquintic equation is isomorphic to the symmetric group on five letters, and the fact that the icosahedral group is simple and nonabelian means that quintic equations need not have a solution in radicals. The proof of theAbel-Ruffini theorem uses this simple fact, andFelix Klein wrote a book that made use of the theory of icosahedral symmetries to derive an analytical solution to the general quintic equation.**ee also***

Truncated icosahedron

*Regular polyhedron

*Geodesic grid s use an iteratively bisected icosahedron to generate grids on a sphere**References****External links***

* [*http://www.software3d.com/Icosahedron.php Paper models of the icosahedron*]

* [*http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra*]

* [*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://polyhedra.org/poly/show/4/icosahedron Interactive Icosahedron model*] - works right in your web browser

* [*http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra*] The Encyclopedia of Polyhedra

* [*http://www.tulane.edu/~dmsander/WWW/335/335Structure.html Tulane.edu*] A discussion of viral structure and the icosahedron

* [*http://www.korthalsaltes.com/ Paper Models of Polyhedra*] Many links

* [*http://www.flickr.com/photos/pascalin/sets/72157594234292561/ Origami Polyhedra*] - Models made with Modular Origami

* [*http://www.lifeisastoryproblem.org/explore/net_icosahedron.pdf Printable Geometric Net of a Regular Icosahedron*] [*http://www.lifeisastoryproblem.org Life is a Story Problem.org*]

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