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Latest revision as of 23:01, 28 November 2014

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Identity element: Difference between revisions

Latest revision as of 23:01, 28 November 2014

Contents

what he was Michael Kors Wallet

Michaela Walsh Longchamp Bag Australia

Feb. 28 Cheap Polo Ralph Lauren Online

However Fred Perry Sydney Australia

and therefore fantasy Jeremy Scott Adidas Alien

Davda Air Jordans Australia

Overall in nutritional terms Celine Luggage Sizes

or flue Nike Shox Shoes

and mental disorders Michael Kors Bags Australia Stockists

Connecticut Buy Christian Louboutin Melbourne

2010 Louis Vuitton Neverfull Gm

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-{{Reg polyhedra db|Reg polyhedron stat table|I}}
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-In [[geometry]], an '''icosahedron''' ({{IPAc-en|ˌ|aɪ|k|ɵ|s|ə|ˈ|h|iː|d|r|ə|n}} or {{IPAc-en|aɪ|ˌ|k|ɒ|s|ə|ˈ|h|iː|d|r|ə|n}}) is a [[polyhedron]] with 20 triangular faces, 30 edges and 12 vertices. A '''regular icosahedron''' with identical equilateral faces is often meant because of its geometrical significance as one of the five [[Platonic solid]]s.
-It has five triangular faces meeting at each vertex. It can be represented by its [[vertex figure]] as 3.3.3.3.3 or 3<sup>5</sup>, and also by [[Schläfli symbol]] {3,5}. It is the [[dual polyhedron|dual]] of the [[dodecahedron]], which is represented by {5,3}, having three pentagonal faces around each vertex.
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-A regular icosahedron is a [[Johnson solid#Names|gyroelongated]] [[pentagonal bipyramid]] and a biaugmented [[pentagonal antiprism]] in any of six orientations.
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-==Dimensions==
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-If the edge length of a regular icosahedron is ''a'', the [[radius]] of a circumscribed [[sphere]] (one that touches the icosahedron at all vertices) is
-:<math>r_u = \frac{a}{2} \sqrt{\varphi \sqrt{5}} = \frac{a}{4} \sqrt{10 +2\sqrt{5}} = a\sin\frac{2\pi}{5} \approx 0.9510565163 \cdot a</math> {{OEIS2C|A019881}}
-and the radius of an inscribed sphere ([[tangent]] to each of the icosahedron's faces) is
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-:<math>r_i = \frac{\varphi^2 a}{2 \sqrt{3}} = \frac{\sqrt{3}}{12} \left(3+ \sqrt{5} \right) a \approx 0.7557613141\cdot a</math> {{OEIS2C|A179294}}
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-:<math>r_m = \frac{a \varphi}{2} = \frac{1}{4} \left(1+\sqrt{5}\right) a = a\cos\frac{\pi}{5} \approx 0.80901699\cdot a</math> {{OEIS2C|A019863}}
-where ''φ'' (also called ''τ'') is the [[golden ratio]].
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-:<math>V = \frac{5}{12} (3+\sqrt5)a^3 \approx 2.18169499a^3.</math> {{OEIS2C|A102208}}
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-inscribed sphere, where the volume of the tetrahedron is one third times the base area {{math| &radic;3a<sup>2</sub>/4}} times its height {{math|r<sub>i</sub>}}.
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-==Cartesian coordinates==
+== or flue Nike Shox Shoes ==
-[[Image:Icosahedron-golden-rectangles.svg|200px|thumb|right|Icosahedron vertices form three orthogonal golden rectangles]]
-The following [[Cartesian coordinates]] define the vertices of an icosahedron with edge-length 2, centered at the origin:<ref>{{mathworld |title=Icosahedral group |urlname=IcosahedralGroup}}</ref>
-:(0, ±1, ±''φ'')
-:(±1, ±''φ'', 0)
-:(±''φ'', 0, ±1)
-where {{nowrap|''φ'' {{=}} (1 + √5) / 2}} is the [[golden ratio]] (also written ''τ''). Note that these vertices form five sets of three concentric, mutually [[orthogonal]] [[golden rectangle]]s, whose edges form [[Borromean rings]].
-If the original icosahedron has edge length 1, its dual [[dodecahedron]] has edge length <math>\tfrac{\sqrt{5}-1}{2}</math>, one divided by the [[golden ratio]].
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-[[File:Icosahedron model.JPG|right|200px|thumb|Model of an icosahedron made with metallic spheres and magnetic connectors]]
+== and mental disorders Michael Kors Bags Australia Stockists ==
-The 12 edges of a regular [[octahedron]] can be subdivided 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 subdividing each edge into the golden mean along the direction of its vector. The [[Compound of five octahedra|five octahedra]] defining any given icosahedron form a regular [[polyhedral compound]], while the [[compound of two icosahedra|two icosahedra]] that can be defined in this way from any given octahedron form a [[uniform polyhedron compound]].
-=== Spherical coordinates ===
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-The locations of the vertices of a regular icosahedron can be described using [[spherical coordinates]], for instance as [[latitude and longitude]]. If two vertices are taken to be at the north and south poles (latitude ±90°), then the other ten vertices are at latitude ±[[arctan]](1/2) ≈ ±26.57°. These ten vertices are at evenly spaced longitudes (36° apart), alternating between north and south latitudes.
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-This scheme takes advantage of the fact that the regular icosahedron is a pentagonal [[gyroelongated bipyramid]], with D<sub>5d</sub> [[Dihedral symmetry in three dimensions|dihedral symmetry]]—that is, it is formed of two congruent pentagonal pyramids joined by a pentagonal [[antiprism]].
+== Connecticut  Buy Christian Louboutin Melbourne ==
-==Orthogonal projections==
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-The ''icosahedron'' has three special [[orthogonal projection]]s, centered on a face, an edge and a vertex:
-{|class=wikitable width=360
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-|+ Orthogonal projections
-|-
+   <li>[http://www.3igm.com/forum.php?mod=viewthread&tid=165549 http://www.3igm.com/forum.php?mod=viewthread&tid=165549]</li>
-!Centered by
-!Face
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-!Edge
-!Vertex
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-|- align=center
-![[Coxeter plane]]
+ </ul>
-!A<sub>2</sub>
-!A<sub>3</sub>
-!H<sub>3</sub>
-|- align=center
-!Graph
-|[[File:Icosahedron t0 A2.png|100px]]
-|[[File:Icosahedron graph A3 1.png|100px]]
-|[[File:Icosahedron t0 H3.png|100px]]
-|- align=center
-!Projective<BR>symmetry
-|[6]
-|[2]
-|[10]
-|- align=center
-!Graph
-|[[File:Icosahedron fnormal.png|100px]]<BR>Face normal
-|[[File:Icosahedron graph A3 2.png|100px]]<BR>Edge normal
-|[[File:Icosahedron vnormal.png|100px]]<BR>Vertex normal
-|}
-==Other facts==
+== 2010 Louis Vuitton Neverfull Gm ==
-*An icosahedron has 43,380 distinct [[net (polyhedron)|nets]].<ref>{{mathworld |urlname=Icosahedron |title=Icosahedron}}</ref>
-*To color the icosahedron, such that no two adjacent faces have the same color, requires at least 3 colors.<ref>This is true for all convex polyhedra with triangular faces except for the tetrahedron, by applying [[Brooks' theorem]] to the [[dual graph]] of the polyhedron.</ref>
-*A problem dating back to the ancient Greeks is to determine which of two shapes has 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.<ref>{{citation|title=A Mathematical History of the Golden Number|first=Roger|last=Herz-Fischler|publisher=Courier Dover Publications|year=2013|isbn=9780486152325|pages=138–140|url=http://books.google.com/books?id=aYjXZJwLARQC&pg=PA138}}.</ref> [[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.<ref>{{citation|title=Calculus Gems: Brief Lives and Memorable Mathematics|first=George F.|last=Simmons|publisher=Mathematical Association of America|year=2007|isbn=9780883855614|page=50|url=http://books.google.com/books?id=3KOst4Mon90C&pg=PA50}}.</ref> Both volumes have formulas involving the [[golden ratio]], but taken to different powers.<ref>{{citation|title=Platonic & Archimedean Solids|series=Wooden Books|first=Daud|last=Sutton|publisher=Bloomsbury Publishing USA|year=2002|isbn=9780802713865|page=55|url=http://books.google.com/books?id=vgo7bTxDmIsC&pg=PA55}}.</ref> As it turns out, the icosahedron occupies less of the sphere's volume (60.54%) than the dodecahedron (66.49%).<ref>Numerical values for the volumes of the inscribed Platonic solids may be found in {{citation|title=The Platonic Solids (Solution to problem E2053)|first1=W. E.|last1=Buker|first2=R. B.|last2=Eggleton|journal=[[American Mathematical Monthly]]|volume=76|issue=2|year=1969|page=192|jstor= 2317282}}.</ref>
-==Construction by a system of equiangular lines==
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-|[[File:Icosahedron t0 H3.png|160px]]<BR>Icosahedron<BR>H<sub>3</sub> Coxeter plane
-|[[File:6-cube t5 B5.svg|160px]]<BR>[[6-orthoplex]]<BR>D<sub>6</sub> Coxeter plane
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-|-
-|colspan=2|This construction can be geometrically seen as the 12 vertices of the [[6-orthoplex]] projected to 3 dimensions. This represents a [[Coxeter–Dynkin diagram#Geometric folding|geometric folding]] of the D<sub>6</sub> to H<sub>3</sub> [[Coxeter group]]s: [[File:Geometric folding Coxeter graph D6 H3.png]]
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-Seen by these 2D [[Coxeter plane]] orthogonal projections, the two overlapping central vertices define the third axis in this mapping.
-|}
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-The following construction of the icosahedron avoids tedious computations in the [[Algebraic number field|number field]] <math>\mathbb{Q}[\sqrt{5}]</math> necessary in more elementary approaches.
+ </ul>
-The existence of the icosahedron amounts to the existence of six [[equiangular lines]] in <math>\mathbb R^3</math>. Indeed, intersecting such a system of equiangular lines with a Euclidean sphere centered at their common intersection yields the twelve vertices of a regular icosahedron as can easily be checked. Conversely, supposing the existence of a regular icosahedron, lines defined by its six pairs of opposite vertices form an equiangular system.
-In order to construct such an equiangular system, we start with this 6×6 square [[Matrix (mathematics)|matrix]]:
-:<math>A=\left(\begin{array}{crrrrr}
-&1&1&1&1&1\\
-&0&1&-1&-1&1\\
-&1&0&1&-1&-1\\
-&-1&1&0&1&-1\\
-&-1&-1&1&0&1\\
-&1&-1&-1&1&0\end{array}\right).</math>
-A straightforward computation yields {{nowrap|''A''<sup>2</sup> {{=}} 5''I''}} (where ''I'' is the 6×6 identity matrix). This implies that ''A'' has [[Eigenvalue, eigenvector and eigenspace|eigenvalues]] <math>\scriptstyle -\sqrt{5}</math> and <math>\scriptstyle \sqrt{5}</math>, both with multiplicity 3 since ''A'' is [[Symmetric matrix|symmetric]] and of [[Trace (linear algebra)|trace]] zero.
-The matrix <math>\scriptstyle A+\sqrt{5}I</math> induces thus a [[Euclidean space|Euclidean structure]] on the [[Quotient space (linear algebra)|quotient space]] <math>\mathbb R^6/\ker(A+\sqrt{5}I)</math> which is [[Isomorphism|isomorphic]] to <math>\mathbb R^3</math> since the [[Kernel (linear operator)|kernel]] <math>\ker(A+\sqrt{5}I)</math> of <math>\scriptstyle {A+\sqrt{5}I}</math> has [[dimension]] 3. The image under the [[Projection (linear algebra)|projection]] <math>\pi:\mathbb R^6 \longrightarrow \mathbb R^6/\ker(A+\sqrt{5}I)</math> of the six coordinate axes <math>\mathbb R v_1,\dots,\mathbb R v_6</math> in <math>\mathbb R^6</math> forms thus a system of six equiangular lines in <math>\mathbb R^3</math> intersecting pairwise at a common acute angle of <math>\scriptstyle{\arccos}\tfrac{1}{\sqrt{5}}</math>. Orthogonal projection of ±''v''<sub>1</sub>, ..., ±''v''<sub>6</sub> onto the [[Eigenvalue, eigenvector and eigenspace|<math>\scriptstyle \sqrt{5}</math>-eigenspace]] of ''A'' yields thus the twelve vertices of the icosahedron.
-A second straightforward construction of the icosahedron uses [[Group representation|representation theory]] of the [[alternating group]] ''A''<sub>5</sub> acting by direct [[Isometry|isometries]] on the icosahedron.
-==Symmetry==
-{{main|Icosahedral symmetry}}
-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. 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, 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, {{Harv|Klein|1888}}. See [[Icosahedral symmetry#Related geometries|icosahedral symmetry: related geometries]] for further history, and related symmetries on seven and eleven letters.
-The full symmetry group of the icosahedron (including reflections) is known as the [[full icosahedral group]], and is isomorphic to the product of the rotational symmetry group and the group ''C''<sub>2</sub> of size two, which is generated by the reflection through the center of the icosahedron.
-== Pseudoicosahedron ==
-{| class=wikitable align=right width="320"
-!bgcolor=#e7dcc3 colspan=2|Pseudoicosahedron
-|-
-|colspan=2 align=center|[[File:Pseudoicosahedron-2.png|140px]][[File:Pseudoicosahedron-1.png|140px]]<BR>[[File:Pseudoicosahedron-4.png|140px]][[File:Pseudoicosahedron-3.png|140px]]<BR>Four views of the pseudoicosahedron, with eight equilateral triangles (red and yellow), and 12 blue isosceles triangles.
-|-
-|bgcolor=#e7dcc3|[[Coxeter diagram]]s||{{CDD|node|4|node_h|3|node_h}} (pyritohedral) [[File:Uniform polyhedron-43-h01.svg|24px]]<BR>{{CDD|node_h|3|node_h|3|node_h}} (tetrahedral) [[File:Uniform polyhedron-33-s012.svg|24px]]
-|-
-|bgcolor=#e7dcc3|[[Face (geometry)|Faces]]||20 triangles:<BR>8 equilateral<BR>12 isosceles
-|-
-|bgcolor=#e7dcc3|[[Edge (geometry)|Edges]]||30 (6 short + 24 long)
-|-
-|bgcolor=#e7dcc3|[[Vertex (geometry)|Vertices]]||12
-|-
-|bgcolor=#e7dcc3|[[List of spherical symmetry groups#Polyhedral symmetry|Symmetry group]]||[[Pyritohedral symmetry|T<sub>h</sub>]], [4,3<sup>+</sup>], (3*2), order 24
-|-
-|bgcolor=#e7dcc3|[[Point groups in three dimensions#Rotation groups|Rotation group]]||[[Tetrahedral symmetry|T<sub>d</sub>]], [3,3]<sup>+</sup>, (332), order 12
-|-
-|bgcolor=#e7dcc3|[[Dual polyhedron]]||[[Pyritohedron]]
-|-
-|bgcolor=#e7dcc3|Properties||[[convex set|convex]]
-|-
-!colspan=2|[[File:Pseudoicosahedron flat.png|200px]]<br>[[Net (polyhedron)|Net]]
-|}
-[[File:Snub-polyhedron-icosahedron.png|150px|thumb|left|The pseudoicosahedron as an [[alternation (geometry)|alternated]] [[truncated octahedron]].]]
-[[File:P1-P5.gif|thumb|left|The icosahedron can be constructed from the [[tetrahedron]] by a rotation of the triangular faces, inserting pairs of new triangles in place of the original 6 edges.]]
-A '''pseudoicosahedron''' is an icosahedron with [[pyritohedral symmetry]]. The 20 triangular faces are divided into two groups of 8 [[equilateral triangle]]s and 12 [[isosceles triangle]]s. If all the triangles are [[equilateral triangle]]s, the symmetry can also be distinguished by coloring the 8 and 12 triangle sets differently. The pseudoicosahedron is an [[alternation (geometry)|alternated]] [[truncated octahedron]].
-=== Pyritohedral symmetry ===
-[[File:Truncated octahedron internal rectangles.png|150px|thumb|left|The 3 rectangles with 12 vertices of the 24 vertex [[truncated octahedron]], corresponding to the vertices of the pseudoicosahedron. The rectangle edge length ratios of 2:1.]]
-If the 8 equilateral triangles are geometrically identical, the pseudoicosahedron has [[pyritohedral symmetry]] (3*2), [4,3<sup>+</sup>], with order 24. A lower [[tetrahedral symmetry]] (332), [3,3]<sup>+</sup>, exists as well, seen as the 8 triangles marked (colored) in alternating pairs of four, with order 12. These symmetries offer [[Coxeter-Dynkin diagram]]s: {{CDD|node|4|node_h|3|node_h}} and {{CDD|node_h|3|node_h|3|node_h}} respectfully, each representing the lower symmetry to the regular icosahedron {{CDD|node_1|3|node|5|node}}, (*532), [5,3] [[icosahedral symmetry]] of order 120.
-=== Cartesian coordinates ===
-The coordinates of the 12 vertices can be defined by the vectors defined by all the possible cyclic permutations and sign-flips of coordinates of the form (2, 1, 0). These coordinates represent the [[truncated octahedron]] with [[Alternation (geometry)|alternated]] vertices deleted.
-This construction is called a ''snub tetrahedron'' in its regular icosahedron form, generated by the same operations carried out starting with the vector (&phi;, 1, 0), where &phi; is the [[golden ratio]].<ref>{{cite web|url=http://math.ucr.edu/home/baez/golden.html|title=Fool's Gold|author=John Baez|date=September 11, 2011}}</ref>
-The regular [[star polyhedron]], [[great icosahedron]], {{CDD|node_h|3x|rat|d2|node_h|3x|rat|d2|node_h}} or {{CDD|node_h|3x|rat|d2|node_h|4|node}} can be considered a special case of parametric ''pseudoicosahedron'' with a [[vertex figure]] that overlaps into a regular [[pentagram]]. The [[tetrahedral symmetry]] of this form is called a [[retrosnub tetrahedron]]:
-:[[File:Retrosnub tetrahedron.png|160px]]
-=== Crystal pyrite ===
-[[Iron pyrites]] have been observed to have formed crystals in the form of pseudoicosahedra.<ref>{{cite web|url=http://math.ucr.edu/home/baez/icosahedron/|title=Who Discovered the Icosahedron?|author=John Baez|date=November 7, 2009}}</ref>
-==Stellations==
-According to specific rules defined in the book ''[[The Fifty-Nine Icosahedra]]'', 59 [[stellation]]s were identified for the regular icosahedron. The first form is the icosahedron itself. One is a regular [[Kepler–Poinsot polyhedron]]. Three are [[Polyhedral compound#Regular compounds|regular compound polyhedra]].<ref>{{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 |mr=676126  |year=1999}} (1st Edn University of Toronto (1938))</ref>
-{| class=wikitable
-|+ 21 of 59 stellations
-|-
-|rowspan=3 width=200|[[Image:Zeroth stellation of icosahedron facets.png|200px]]<br>The faces of the icosahedron extended outwards as planes intersect, defining regions in space as shown by this [[stellation diagram]] of the intersections in a single plane.
-|[[Image:Zeroth stellation of icosahedron.png|50px]]
-|[[Image:First stellation of icosahedron.png|50px]]
-|[[Image:Second stellation of icosahedron.png|50px]]
-|[[Image:Third stellation of icosahedron.png|50px]]
-|[[Image:Fourth stellation of icosahedron.png|50px]]
-|[[Image:Fifth stellation of icosahedron.png|50px]]
-|[[Image:Sixth stellation of icosahedron.png|50px]]
-|[[Image:Seventh stellation of icosahedron.png|50px]]
-|-
-|[[Image:Eighth stellation of icosahedron.png|50px]]
-|[[Image:Ninth stellation of icosahedron.png|50px]]
-|[[Image:Tenth stellation of icosahedron.png|50px]]
-|[[Image:Eleventh stellation of icosahedron.png|50px]]
-|[[Image:Twelfth stellation of icosahedron.png|50px]]
-|[[Image:Thirteenth stellation of icosahedron.png|50px]]
-|[[Image:Fourteenth stellation of icosahedron.png|50px]]
-|[[Image:Fifteenth stellation of icosahedron.png|50px]]
-|-
-|[[Image:Sixteenth stellation of icosahedron.png|50px]]
-|[[Image:Seventeenth stellation of icosahedron.png|50px]]
-|[[Image:First compound stellation of icosahedron.png|50px]]
-|[[Image:Second compound stellation of icosahedron.png|50px]]
-|[[Image:Third compound stellation of icosahedron.png|50px]]
-|}
-==Geometric relations==
-There are distortions of the icosahedron that, while no longer regular, are nevertheless [[vertex-uniform]]. These are [[invariant (mathematics)|invariant]] under the same [[rotation]]s as the tetrahedron, and are somewhat analogous to the [[snub cube]] and [[snub dodecahedron]], including some forms which are [[chirality (mathematics)|chiral]] and some with T<sub>h</sub>-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.
-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 (geometry)|defect]] for folding in three dimensions, icosahedra cannot be used as the [[cell (geometry)|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 [[facet (mathematics)|facets]] must meet at a [[peak (geometry)|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 [[Schläfli–Hess polychoron|non-convex regular polychora]].
-An icosahedron can also be called a [[Gyroelongated dipyramid|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.
-==Uniform colorings and subsymmetries==
-There are 3 [[uniform coloring]]s of the icosahedron. These colorings can be represented as 11213, 11212, 11111, naming the 5 triangular faces around each vertex by their color.
-The icosahedron can be considered a snub tetrahedron, as [[snub (geometry)|snubification]] of a regular tetrahedron gives a regular icosahedron having chiral [[tetrahedral symmetry]]. It can also be constructed as an alternated truncated octahedron, having [[pyritohedral symmetry]]. The pyritohedral symmetry version is sometimes called a [[pseudoicosahedron]], and is dual to the [[pyritohedron]].
-{| class=wikitable
-|- align=center
-!Name
-|Regular icosahedron
-|[[snub octahedron]]
-|[[Snub tetratetrahedron]]
-|Pentagonal<br>[[gyroelongated bipyramid]]
-|- align=center
-![[Coxeter-Dynkin diagram|Coxeter-Dynkin]]
-|{{CDD|node|5|node|3|node_1}}
-|{{CDD|node|4|node_h|3|node_h}}
-|{{CDD|node_h|3|node_h|3|node_h}}
-|
-|- align=center
-![[Schläfli symbol]]
-|{3,5}
-|s{3,4}
-|sr{3,3}
-|
-|- align=center
-![[Wythoff symbol]]
-| 5 &#124; 3 2
-|
-| &#124; 3 3 2
-|
-|- align=center
-![[List of spherical symmetry groups|Symmetry]]
-|I<sub>h</sub><br>[5,3]<br>(*532)
-|T<sub>h</sub><br>[3<sup>+</sup>,4]<br>(3*2)
-|T<br>[3,3]<sup>+</sup><br>(332)
-|D<sub>5d</sub><br>[2<sup>+</sup>,10]<br>(2*5)
-|- align=center
-!Symmetry order
-|60
-|24
-|12
-|10
-|- align=center
-![[Uniform coloring]]
-|[[Image:Uniform polyhedron-53-t2.png|100px]]<br>(11111)
-|[[Image:Uniform polyhedron-43-h01.svg|100px]]<br>(11212)
-|[[Image:Uniform polyhedron-33-s012.png|100px]]<br>(11213)
-|valign=top|[[Image:Pentagonal gyroelongated bipyramid.png|100px]]<br>(11122)&(22222)
-|}
-==Related polyhedra and polytopes==
-The icosahedron can be transformed by a [[Truncation (geometry)|truncation]] sequence into its [[Dual polyhedron|dual]], the dodecahedron:
-{{Icosahedral truncations}}
-As a snub tetrahedron, and alternation of a truncated octahedron it also exists in the tetrahedral and octahedral symmetry families:
-{{Tetrahedron family}}
-{{Octahedral truncations}}
-This polyhedron is topologically related as a part of sequence of regular polyhedra with [[Schläfli symbol]]s {3,''n''}, continuing into the [[Hyperbolic space|hyperbolic plane]].
-{{Triangular regular tiling}}
-The regular icosahedron, seen as a ''snub tetrahedron'', is a member of a sequence of [[Snub (geometry)|snubbed]] polyhedra and tilings with vertex figure (3.3.3.3.''n'') and [[Coxeter–Dynkin diagram]] {{CDD|node_h|n|node_h|3|node_h}}. These figures and their duals have (n32) rotational [[Orbifold notation|symmetry]], being in the Euclidean plane for n=6, and hyperbolic plane for any higher n. The series can be considered to begin with n=2, with one set of faces degenerated into [[digon]]s.
-{{Snub table}}
-{{Order-5 regular tilings}}
-The icosahedron shares its [[vertex arrangement]] with three [[Kepler–Poinsot solid]]s. The [[great dodecahedron]] also has the same [[edge arrangement]].
-{| class="wikitable"
-|- align=center
-!Picture
-| [[Image:Great dodecahedron.png|100px]]<br>[[Great dodecahedron]]
-| [[Image:Small stellated dodecahedron.png|100px]]<br>[[Small stellated dodecahedron]]
-| [[Image:Great icosahedron.png|100px]]<br>[[Great icosahedron]]
-|- align=center
-![[Coxeter-Dynkin diagram|Coxeter-Dynkin]]
-|{{CDD|node_1|5|node|5|rat|d2|node}}
-|{{CDD|node|5|node|5|rat|d2|node_1}}
-|{{CDD|node_1|3|node|5|rat|d2|node}}
-|}
-The icosahedron can tessellate hyperbolic space in the [[order-3 icosahedral honeycomb]], with 3 icosahedra around each edge, 12 icosahedra around each vertex, with [[Schläfli symbol]] {3,5,3}. It is [[List of regular polytopes#Tessellations of hyperbolic 3-space|one of four regular tessellations]] in the hyperbolic 3-space.
-{| class=wikitable width=240
-|[[File:Hyperb icosahedral hc.png|240px]]<br>It is shown here as an edge framework in a [[Poincaré disk model]], with one icosahedron visible in the center.
-|}
-==Uses and natural forms==
-[[Image:20-sided dice 250.jpg|left|thumb|Twenty-sided [[dice]]]]
-[[File:Twin2.jpg|thumb|upright|[[Gold]] nanoparticle viewed in electron microscope. ]]
-[[File:Gamma-bor.jpg|thumb|upright|Structure of γ-boron.]]
-=== Biology ===
-Many [[virus]]es, e.g. [[herpes]] virus, have icosahedral [[capsid|shells]].<ref>C. Michael Hogan. 2010. [http://www.eoearth.org/article/Virus?topic=49496 ''Virus''. Encyclopedia of Earth. National Council for Science and the Environment]. eds. S. Draggan and C. Cleveland</ref> Viral structures are built of repeated identical [[protein]] subunits known as [[capsomere]]s, 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]].
-Various bacterial organelles with an icosahedral shape were also found.<ref name=Bobik2007>{{Citation |author=Bobik, T.A. |title=Bacterial Microcompartments |year=2007 |journal=Microbe |volume=2 |pages=25–31 |url=http://www.microbemagazine.org/index.php/01-2007-home/2308-bacterial-microcompartments|publisher=Am. Soc. Microbiol.}}</ref> The icosahedral shell encapsulating enzymes and labile intermediates are built of different types of proteins with [[BMC domain]]s.
-In 1904, [[Ernst Haeckel]] described 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]].
-=== Chemistry ===
-The [[closo cluster|closo]]-[[carboranes]] are chemical compounds with shape very close to isosahedron. [[icosahedral twins|Icosahedral]] [[Crystal twinning|twinning]] also occurs in crystals, especially [[nanoparticle]]s.
-Many [[Crystal structure of boron-rich metal borides|borides]] and [[allotropes of boron]] contain boron B<sub>12</sub> icosahedron as a basic structure unit.
-=== Toys and games ===
-[[File:ScatDice.JPG|thumb|[[Scattergories]] die]]
-In several [[roleplaying game]]s, such as ''[[Dungeons & Dragons]]'', the twenty-sided die ([[Dice#Non-cubical dice|d20]] for short) is commonly 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 [[Dice#Non-cubical dice|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 [[Scattergories]] to choose a letter of the alphabet. Six letters are omitted (Q, U, V, X, Y, and Z).
-Inside a [[Magic 8-Ball]], various answers to [[yes-no question]]s are inscribed on a regular icosahedron.
-=== Others ===
-[[R. Buckminster Fuller]] and Japanese [[cartographer]] Shoji Sadao<ref>{{Cite web |url=http://library.stanford.edu/depts/spc/exhibits/fullersadao.html |title=Fuller and Sadao: Partners in Design |date= September 19, 2006 |accessdate=2010-01-26}}</ref> designed a world map in the form of an unfolded icosahedron, called the [[Fuller projection]], whose maximum [[distortion#Map projections|distortion]] is only 2%.
-The "Sol de la Flor" light shade consists of twenty panels, which meet at the corners of an icosahedron in [[Rosette (design)|rosette]]s resembling the overlapping petals of a [[frangipani]] flower.
-If each edge of an icosahedron is replaced by a one [[ohm (unit)|ohm]] [[resistor]], the resistance between opposite vertices is 0.5 ohms, and that between adjacent vertices 11/30 ohms.<ref>{{Citation |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}}</ref>
-The company logo of the [[TDK|TDK Corporation]] contains a geometric figure which is based on the stellation diagram of the icosahedron.
-A icosahedron was used for a logo for the Australian TV company; [[Grundy Television]].
-==As a graph==
-[[Image:Icosahedron graph.svg|right|120px|thumb|A planar representation of the icosahedral graph.]]
-The skeleton of the icosahedron—the vertices and edges—form a [[Graph (mathematics)|graph]]. The high degree of symmetry of the polygon is replicated in the properties of this graph, which is [[distance-transitive graph|distance-transitive]], [[distance-regular graph|distance-regular]], and [[symmetric graph|symmetric]]. The [[graph automorphism|automorphism group]] has order 120. The vertices can be [[graph coloring|colored]] with 4 colors, the edges with 5 colors, and the [[graph diameter|diameter]] is 3.<ref>{{MathWorld |urlname=IcosahedralGraph |title=Icosahedral Graph}}</ref>
-The icosahedral graph is [[Hamiltonian graph|Hamiltonian]]: there is a cycle containing all the vertices.  It is also a [[planar graph]].
-==See also==
-*[[Geodesic grid]]s use an iteratively bisected icosahedron to generate grids on a sphere
-*[[Infinite skew polyhedron]]
-*[[Jessen's icosahedron]]
-*[[Regular polyhedron]]
-*[[Truncated icosahedron]]
-*[[Icosahedral twins]]
-==References==
-<!--See [[Wikipedia:Footnotes]] for instructions.-->
-{{reflist}}
-{{refbegin}}
-*{{Citation |last=Klein |first=Felix |authorlink=Felix Klein |title=Lectures on the ikosahedron and the solution of equations of the fifth degree |year=1888 |url=http://digital.library.cornell.edu/cgi/t/text/text-idx?c=math;cc=math;view=toc;subview=short;idno=03070001 |postscript=, Dover edition ISBN 978-0-486-49528-6.}}
-{{refend}}
-==External links==
-{{commons category|Icosahedron}}
-{{Wikisource1911Enc|Icosahedron}}
-{{Wiktionary|icosahedron}}
-*{{mathworld |urlname=Icosahedron |title=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://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.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]
-*[http://web.uct.ac.za/depts/mmi/stannard/virarch.html] Principle of virus architecture
-{{Polyhedra}}
-{{Polyhedron navigator}}
-{{Polytopes}}
-{{Icosahedron stellations}}
-[[Category:Deltahedra]]
-[[Category:Planar graphs]]
-[[Category:Platonic solids]]
-[[Category:Pyramids and bipyramids]]

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