geometry, an icosahedron (Greek: "eikosaedron", from "eikosi" twenty + "hedron" seat; IPA|/ˌaɪ.kəʊ.sə.ˈhi.dɹən/; plural: -drons, -dra IPA|/-dɹə/) isany polyhedronhaving 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces.
The regular icosahedron is one of the five
Platonic solids. It is a convex regular polyhedroncomposed of twentytriangular faces, with five meeting at each of the twelve vertices. It has 30 edges and 12 vertices. Its dual polyhedronis the dodecahedron.
If the edge length of a regular icosahedron is , the
radiusof a circumscribed sphere(one that touches the icosahedron at all vertices) is
and the radius of an inscribed sphere (
tangentto each of the icosahedron's faces) is
midradius, which touches the middle of each edge, is
where (also called ) is the
Area and volume
The surface area "A" and the
volume"V" of a regular icosahedron of edge length "a" are:::.
Cartesian coordinatesdefine 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 the golden ratio(also written τ). Note that these vertices form five sets of three mutually centered, mutually orthogonal golden rectangles.
The 12 edges of a regular
octahedroncan 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 regular polyhedral compound, as do the two icosahedra that can be defined in this way from any given octahedron.
According to specific rules defined in the book
The fifty nine icosahedra, 59 stellations were identified for the regular icosahedron. The first form is the icosahedron itself. One is a regular Kepler-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))]
There are distortions of the icosahedron that, while no longer regular, are nevertheless vertex-uniform. These are invariant under the same
rotations as the tetrahedron, and are somewhat analogous to the snub cubeand snub dodecahedron, including some forms which are chiral and some with Th-symmetry, i.e. have different planes of symmetry from the tetrahedron. The icosahedron has a large number of stellations, including one of the Kepler-Poinsot polyhedraand some of the regular compounds, which could be discussed here.
The icosahedron is unique among the
Platonic solidsin possessing a dihedral anglenot 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 polychoronbecause, 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 polytopein "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), just as hexagons can be used as faces in semi-regular polyhedra (for example the 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, 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 pyramidand a pentagonal pyramidor into a pentagonal antiprismand two equal pentagonal pyramids.
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
Icosahedron vs dodecahedron
When an icosahedron is inscribed in a
sphere, it occupies less of the sphere's volume (60.54%)than a dodecahedroninscribed in the same sphere (66.49%).
Natural forms and uses
viruses, e.g. herpesvirus, have the shape of an icosahedron. Viral structures are built of repeated identical proteinsubunits 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 genome.
Ernst Haeckeldescribed a number of species of 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.
roleplaying games, 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". See d20 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
Scattergoriesto choose a letter of the alphabet. Six little-used letters, such as X, Q, and Z, are omitted.
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 Florlight shade. The rosette formed by the overlapping pieces show a resemblance to the Frangipaniflower.
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]
symmetry groupof the icosahedron is isomorphicto the alternating groupon five letters. This nonabelian simple groupis the only nontrivial normal subgroupof the symmetric groupon five letters. Since the Galois groupof the general quintic equationis 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 the Abel-Ruffini theoremuses this simple fact, and Felix Kleinwrote a book that made use of the theory of icosahedral symmetries to derive an analytical solution to the general quintic equation.
Geodesic grids use an iteratively bisected icosahedron to generate grids on a sphere
* [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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