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Encyclopedia > Icosahedron
Regular Icosahedron

(Click here for rotating model)
Type Platonic solid
Elements F = 20, E = 30, V = 12 (χ = 2)
Faces by sides 20{3}
Schläfli symbol {3,5} and sbegin{Bmatrix} 3  3 end{Bmatrix}
Wythoff symbol 5 | 2 3
Coxeter-Dynkin
Symmetry Ih
References U22, C25, W4
Properties Regular convex deltahedron
Dihedral angle 138.189685°
Icosahedron
3.3.3.3.3
(Vertex figure)

Dodecahedron
(dual polyhedron)
Icosahedron
Net

[Etymology: 16th century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], "icosa'hedral adjective Download high resolution version (819x791, 71 KB)Icosahedron, made by me using POV-Ray, see image:poly. ... Spinning icosahedron, made by me using POV-Ray, see image:poly. ... In geometry, a Platonic solid is a convex regular polyhedron. ... It has been suggested that Vertex/Face/Edge relation in a convex polyhedron be merged into this article or section. ... In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... In geometry, a Wythoff construction, named after mathematician Willem Abraham Wijthoff, is a method for constructing a uniform polyhedron or plane tiling. ... Coxeter groups in the plane with equivalent diagrams. ... Image File history File links CDW_dot. ... Image File history File links CDW_5. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... // List of symmetry groups on the sphere Spherical symmetry groups are also called point groups (in 3D). ... The icosahedral rotation group I with fundamental domain Apart from the two infinite series of prismatic and antiprismatic symmetry, rotational icosahedral symmetry or chiral icosahedral symmetry of chiral objects and full icosahedral symmetry or achiral icosahedral symmetry are the discrete point symmetries (or equivalently, symmetries on the sphere) with the... A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ... A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ... H.S.M. Coxeter. ... This table contains an indexed list of the Uniform and stellated polyhedra from the book Polyhedron Models, by Magnus J. Wenninger. ... In mathematics, there are three related meanings of the term polyhedron: in the traditional meaning it is a 3-dimensional polytope, and in a newer meaning that exists alongside the older one it is a bounded or unbounded generalization of a polytope of any dimension. ... Look up Convex set in Wiktionary, the free dictionary. ... The triaugmented triangular prism, a convex deltahedron A deltahedron (plural deltahedra) is a polyhedron whose faces are all equilateral triangles. ... In Aerospace engineering, the dihedral is the angle that the two wings make with each other. ... Image File history File links Icosahedron_vertfig. ... In geometry, a vertex figure is most easily thought of as the cut surface exposed when a corner of a polytope is cut off in a certain way. ... Image File history File links Size of this preview: 600 × 600 pixel Image in higher resolution (1000 × 1000 pixel, file size: 211 KB, MIME type: image/png)image for Dodecahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to... A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex. ... In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ... Image File history File links Icosahedron_flat. ... Categories: Polyhedra | Stub ...


An icosahedron /ˌaɪ.kəʊ.sə.ˈhi.dɹən/ noun (plural: -drons, -dra /-dɹə/ ) is any polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. A polyhedron (plural polyhedra or polyhedrons) is a geometric object with flat faces and straight edges. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ...

Contents

In geometry, the regular icosahedron is one of the five Platonic solids. It is a convex regular polyhedron composed of twenty triangular faces, with five meeting at each of the twelve vertices. It has 30 edges and 12 vertices. Its dual polyhedron is the dodecahedron. Calabi-Yau manifold Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... In geometry, a Platonic solid is a convex regular polyhedron. ... Look up Convex set in Wiktionary, the free dictionary. ... A polyhedron (plural polyhedra or polyhedrons) is a geometric object with flat faces and straight edges. ... 20 (twenty) is the natural number following 19 and preceding 21. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ... A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex. ...


Dimensions

If the

r_u = frac{a}{2} sqrt{tau 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 In mathematics, the word tangent has two distinct but etymologically-related meanings: one in geometry and one in trigonometry. ...

r_i = frac{tau^2 a}{2 sqrt{3}} = frac{a}{12} sqrt{3} left(3+ sqrt{5} right) approx 0.7557613141cdot a

while the midradius, which touches the middle of each edge, is

r_m = frac{a tau}{2} = frac{1}{4} left(1+sqrt{5}right) a approx 0.80901699cdot a

where τ (also called φ) is the golden ratio. // Articles with similar titles include Golden mean (philosophy), the felicitous middle between two extremes, and Golden numbers, an indicator of years in astronomy and calendar studies. ...


Area and volume

The surface area A and the volume V of a regular icosahedron of edge length a are: The volume of a solid object is the three-dimensional concept of how much space it occupies, often quantified numerically. ...

A=5sqrt3a^2
V=begin{matrix}{5over12}end{matrix}(3+sqrt5)a^3
Golden rectangles in an icosahedron

Image File history File links Icosahedron-golden-rectangles. ...

Cartesian coordinates

The following Cartesian coordinates define the vertices of an icosahedron with edge-length 2, centered at the origin: Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...

(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 orthogonal golden rectangles. // Articles with similar titles include Golden mean (philosophy), the felicitous middle between two extremes, and Golden numbers, an indicator of years in astronomy and calendar studies. ... In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ... The large rectangle BA is a golden rectangle; that is, the proportion b:a is 1:. If we remove square B, what is left, A, is another golden rectangle. ...


The 12 edges of an 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 regular polyhedral compound. An octahedron (plural: octahedra) is a polyhedron with eight faces. ... A polyhedral compound is a polyhedron which is itself composed of several other polyhedra sharing a common centre, the three-dimensional analogs of polygonal compounds such as the hexagram. ...


Geometric relations

Icosahedron as a snub tetrahedron

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 cube and 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 polyhedra and some of the regular compounds, which could be discussed here. Image File history File links Download high resolution version (1000x1000, 274 KB) Licensing Robert Webb produced this image for Wikipedia upon my request, and offered the credit statement below: Colors edited from icosahedron to show Chiral tetrahedral symmetry. ... Image File history File links Download high resolution version (1000x1000, 274 KB) Licensing Robert Webb produced this image for Wikipedia upon my request, and offered the credit statement below: Colors edited from icosahedron to show Chiral tetrahedral symmetry. ... In mathematics, an invariant is something that does not change under a set of transformations. ... A sphere rotating around its axis. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... The snub cube, or snub cuboctahedron, is an Archimedean solid. ... The snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid. ... In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone. ... Stellation is a process of constructing new polygons (in two dimensions), new polyhedra in three dimensions, or in general new polytopes in n dimensions. ... A single face is colored yellow and outlined in red to help identify the faces. ...


The icosahedron is unique among the Platonic solids in possessing a 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 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), 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. In geometry, the defect of a vertex of a polyhedron is the amount by which the sum of the angles of the faces at the vertex falls short of a full circle. ... Cubic honeycomb - four cubic cells per edge hypercube - three cubic cells per edge In geometry, a cell is a three-dimensional element that is part of a higher-dimensional object. ... In geometry, a four-dimensional polytope is sometimes called a polychoron (plural: polychora) (from Greek poly meaning many and choros meaning room or space), 4-polytope, or polyhedroid. ... In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in three dimensions. ... A facet of an n-dimensional simplex is its (n-1)-dimensional face. ... In geometry, a peak is an (n-3)-dimensional element of a polytope. ... In geometry, the snub 24-cell is a convex uniform polychoron composed of 120 regular tetrahedra and 24 icosahedra cells. ... The truncated icosahedron is an Archimedean solid. ... In geometry, the 120-cell (or hecatonicosachoron) is the convex regular 4-polytope with Schläfli symbol {5,3,3}. It is sometimes thought of as the 4-dimensional analog of the dodecahedron. ...


An icosahedron can also be called a gyroelongated pentagonal bipyramid. It can be decomposed into a gyroelongated pentagonal pyramid and a pentagonal pyramid or into a pentagonal antiprism and two equal pentagonal pyramids. In geometry, the gyroelongated dipyramids are an infinite set of polyhedra, constructed by elongating an n-agonal bipyramid (by inserting an n-agonal antiprism between its congruent halves. ... In geometry, the Gyroelongated pentagonal pyramid is one of the Johnson solids (J11). ... In geometry, the Pentagonal pyramid is one of the Johnson solids (J2). ... In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. ... In geometry, the Pentagonal pyramid is one of the Johnson solids (J2). ...


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). Two snub cubes from great rhombicuboctahedron See that red and green dots are placed at alternate vertices. ... In Euclidean geometry, rectification is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. ...


Icosahedron vs dodecahedron

Despite appearances, when an icosahedron is inscribed in a sphere, it occupies less of the sphere's volume (60.54%) than a dodecahedron inscribed in the same sphere (66.49%). A sphere is a perfectly symmetrical geometrical object. ... A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex. ...


Natural forms and uses

Many viruses, e.g. herpes virus, have the shape of an icosahedron. Viral structures are built of repeated identical protein 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 viral genome. Groups I: dsDNA viruses II: ssDNA viruses III: dsRNA viruses IV: (+)ssRNA viruses V: (-)ssRNA viruses VI: ssRNA-RT viruses VII: dsDNA-RT viruses A virus (from the Latin noun virus, meaning toxin or poison) is a microscopic particle (ranging in size from 20 - 300 nm) that can infect the... Genera Subfamily Alphaherpesvirinae    Simplexvirus    Varicellovirus    Mardivirus    Iltovirus Subfamily Betaherpesvirinae    Cytomegalovirus    Muromegalovirus    Roseolovirus Subfamily Gammaherpesvirinae    Lymphocryptovirus    Rhadinovirus Unassigned    Ictalurivirus The Herpesviridae are a family of DNA viruses that cause diseases in humans and animals. ... A representation of the 3D structure of myoglobin, showing coloured alpha helices. ... In biology the genome of an organism is the whole hereditary information of an organism that is encoded in the DNA (or, for some viruses, RNA). ...


In some 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, but most modern versions are labeled from "1" to "20". A roleplaying game (RPG) is a type of game in which players assume the roles of characters and collaboratively create stories. ... Two standard six-sided pipped dice with rounded corners. ... Two standard six-sided pipped dice with rounded corners. ... Two standard six-sided pipped dice with rounded corners. ...


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. Scattergories is a crowd-pleasing, fast-thinking categories game. ...


The die inside of a Magic 8-Ball that has printed on it 20 answers to yes-no questions is a regular icosahedron. Two standard six-sided pipped dice with rounded corners. ... The Magic 8-Ball, manufactured by Mattel is a toy used for fortune-telling. ...


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 the Frangipani flower. Species 7-8 species including: Plumeria (common name: Frangipani) is a small genus of 7-8 species, native to tropical and subtropical America. ...


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.[1] The ohm (symbol: Ω) is the SI unit of electric resistance. ... Resistor symbols (non-European) Resistor symbols (Europe, IEC) Axial-lead resistors on tape. ...


The symmetry group of the icosahedron is isomorphic to the alternating group on five letters. This nonabelian simple group is the only nontrivial normal subgroup of the symmetric group on five letters. Since the Galois group of the general quintic 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 the Abel-Ruffini theorem uses this simple fact, 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. The symmetry group of an object (e. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... In mathematics an alternating group is the group of even permutations of a finite set. ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... In mathematics, a simple group is a group which is not the trivial group and whose only normal subgroups are the trivial group and the group itself. ... In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are... In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ... In mathematics, a Galois group is a group associated with a certain type of field extension. ... Polynomial of degree 5: f(x) = (x+4)(x+2)(x+1)(x-1)(x-3)/20+2 In mathematics, a quintic equation is a polynomial equation in which the greatest exponent on the independent variable is five. ... The Abel-Ruffini theorem states that there is no general solution in radicals to polynomial equations of degree five or higher. ... Felix Christian Klein (April 25, 1849, Düsseldorf, Germany – June 22, 1925, Göttingen) was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory. ...


See also

Look up icosahedron in Wiktionary, the free dictionary.

Wikipedia does not have an article with this exact name. ... Wiktionary (a portmanteau of wiki and dictionary) is a multilingual, Web-based project to create a free content dictionary, available in over 150 languages. ... Spinning icosahedron, made by me using POV-Ray, see image:poly. ... The truncated icosahedron is an Archimedean solid. ... Icosahedral–Hexagonal Grids in Weather Prediction - numerical approach to weather, ocean and climat prediction which uses geodesic grids generated from an icosahedron and could become an attractive alternative to current climate models. ...

References

  1. ^ Klein, Douglas J. (2002). "Resistance-Distance Sum Rules" (PDF). Croatica Chemica Acta 75 (2): 633–649. Retrieved on 2006-09-15. 

For the Manfred Mann album, see 2006 (album). ... September 15 is the 258th day of the year (259th in leap years) in the Gregorian calendar. ...

External links

Wikimedia Commons has media related to:
Icosahedron

  Results from FactBites:
 
Stellations of the Icosahedron (435 words)
A few of these we have seen elsewhere: the great icosahedron is one of the Kepler-Poinsot polyhedra, and the first stellation in the official list is just the icosahedron itself.
For the five-colored versions, each of the 20 icosahedral planes is assigned a color according to this five-coloring of the icosahedron, which in turn comes from the most famous of its stellations, the compound of five tetrahedra.
Exercise: Note that both the shape of the compound of five tetrahedra and the color pattern of the five-colored icosahedron are chiral.
PlanetMath: icosahedron (37 words)
An icosahedron is a polyhedron with twenty faces.
This is version 3 of icosahedron, born on 2004-06-22, modified 2007-05-02.
Object id is 5945, canonical name is Icosahedron.
  More results at FactBites »

 
 

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