icosahedral

• 正二十面体の

関

icosahedron

　

WordNet

1. of or relating to an icosahedron
2. any polyhedron having twenty plane faces

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出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2013/02/19 10:41:56」(JST)

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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}
Wythoff symbol	5 | 2 3
Coxeter–Dynkin
Symmetry	Ih, H3, [5,3], (*532)
Rotation group	I, [5,3]+, (532)
References	U22, C25, W4
Properties	Regular convex deltahedron
Dihedral angle	138.189685° = arccos(-√5/3)
3.3.3.3.3 (Vertex figure)	Dodecahedron (dual polyhedron)
Net

In geometry, an icosahedron (pron.: /ˌaɪkɵsəˈhiːdrən/ or /aɪˌkɒsəˈhiːdrən/) is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids.

It has five triangular faces meeting at each vertex. It can be represented by its vertex figure as 3.3.3.3.3 or 3⁵, and also by Schläfli symbol {3,5}. It is the dual of the dodecahedron, which is represented by {5,3}, having three pentagonal faces around each vertex.

The name comes from the Greek: εικοσάεδρον, from είκοσι (eíkosi) "twenty" and ἕδρα (hédra) "seat". The plural can be either "icosahedrons" or "icosahedra" (-/drə/).

Dimensions

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

 A019881

and the radius of an inscribed sphere (tangent to each of the icosahedron's faces) is

 A179294

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

 A019863

where φ (also called τ) is the golden ratio.

Area and volume

The surface area A and the volume V of a regular icosahedron of edge length a are:

 A010527

 A102208

The latter is F=20 times the volume of a general tetrahedron with apex at the center of the inscribed sphere, where the volume of the tetrahedron is one third times the base area √3a²/4 times its height r_i.

The volume filling factor of the circumscribed sphere is

Cartesian coordinates

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:

(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 concentric, mutually orthogonal golden rectangles, whose edges form Borromean rings.

If the original icosahedron has edge length 1, its dual dodecahedron has edge length , one divided by the golden ratio.

Model of an icosahedron made with metallic spheres and magnetic connectors

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 five octahedra defining any given icosahedron form a regular polyhedral compound, while the two icosahedra that can be defined in this way from any given octahedron form a uniform polyhedron compound.

Spherical Coordinates

Seen as a pentagonal gyroelongated bipyramid, with D_5d, dihedral symmetry, the icosahedron vertices can be positioned in spherical coordinates, with two vertices are placed on the poles of a sphere, the remaining vertices are located at latitude ±arctan(1/2). The longitudes can be found using the antiprismatic symmetry of the remaining ten vertices, thus fixing a point then rotating 36° about the polar axis and reflecting about the equator until we return to the original point.

Orthogonal projections

The icosahedron has three special orthogonal projections, centered on a face, an edge and a vertex:

Orthogonal projections

Centered by	Face	Edge	Vertex
Coxeter plane	A2	A3	H3
Graph
Projective symmetry	[6]	[2]	[10]
Graph	Face normal	Edge normal	Vertex normal

Other facts

• An icosahedron has 43,380 distinct nets.
• To color the icosahedron, such that no two adjacent faces have the same color, requires at least 3 colors.
• 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%).

Construction by a system of equiangular lines

Icosahedron H3 Coxeter plane	6-orthoplex D6 Coxeter plane
This construction can be geometrically seen as the 12 vertices of the 6-orthoplex projected to 3 dimensions. This represents a geometric folding of the D6 to H3 Coxeter groups: Seen by these 2D Coxeter plane orthogonal projections, the two overlapping central vertices define the third axis in this mapping.

The following construction of the icosahedron avoids tedious computations in the number field necessary in more elementary approaches.

The existence of the icosahedron amounts to the existence of six equiangular lines in . 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:

$A=\left(\begin{array}{crrrrr} 0&1&1&1&1&1\\ 1&0&1&-1&-1&1\\ 1&1&0&1&-1&-1\\ 1&-1&1&0&1&-1\\ 1&-1&-1&1&0&1\\ 1&1&-1&-1&1&0\end{array}\right).$

A straightforward computation yields A² = 5I (where I is the 6×6 identity matrix). This implies that A has eigenvalues and , both with multiplicity 3 since A is symmetric and of trace zero.

The matrix induces thus a Euclidean structure on the quotient space which is isomorphic to since the kernel of has dimension 3. The image under the projection of the six coordinate axes in forms thus a system of six equiangular lines in intersecting pairwise at a common acute angle of . Orthogonal projection of ±v₁, ..., ±v₆ onto the -eigenspace of A yields thus the twelve vertices of the icosahedron.

A second straightforward construction of the icosahedron uses representation theory of the alternating group A₅ acting by direct isometries on the icosahedron.

Symmetry

Main article: Icosahedral symmetry

The rotational symmetry group of the regular icosahedron is isomorphic to the alternating group on five letters. This non-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, (Klein 1888). See 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 C₂ of size two, which is generated by the reflection through the center of the icosahedron.

Stellations

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 polyhedron. Three are regular compound polyhedra.

21 of 59 stellations

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.

Geometric relations

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 T_h-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 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.

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.

Uniform colorings and subsymmetries

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.

There are 3 uniform colorings 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 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.

Name	Regular icosahedron	alternated truncated octahedron	snub tetrahedron	Pentagonal gyroelongated bipyramid
Coxeter-Dynkin
Schläfli symbol	{3,5}	h0,1{3,4}	s{3,3}
Wythoff symbol	5 | 3 2		| 3 3 2
Symmetry	Ih [5,3] (*532)	Th [3+,4] (3*2)	T [3,3]+ (332)	D5d [2+,10] (2*5)
Symmetry order	60	24	12	10
Uniform coloring	(11111)	(11212)	(11213)	(11122)&(22222)

Related polyhedra and polytopes

The icosahedron can be transformed by a truncation sequence into its dual, the dodecahedron:

Family of uniform icosahedral polyhedra

Symmetry: [5,3], (*532)							[5,3]+, (532)
{5,3}	t0,1{5,3}	t1{5,3}	t0,1{3,5}	{3,5}	t0,2{5,3}	t0,1,2{5,3}	s{5,3}
Duals to uniform polyhedra
V5.5.5	V3.10.10	V3.5.3.5	V5.6.6	V3.3.3.3.3	V3.4.5.4	V4.6.10	V3.3.3.3.5

As a snub tetrahedron, and alternation of a truncated octahedron it also exists in the tetrahedral and octahedral symmetry families:

Family of uniform tetrahedral polyhedra

Symmetry: [3,3], (*332)							[3,3]+, (332)
{3,3}	t0,1{3,3}	t1{3,3}	t1,2{3,3}	t2{3,3}	t0,2{3,3}	t0,1,2{3,3}	s{3,3}
Duals to uniform polyhedra
V3.3.3	V3.6.6	V3.3.3.3	V3.6.6	V3.3.3	V3.4.3.4	V4.6.6	V3.3.3.3.3

Uniform octahedral polyhedra

Symmetry: [4,3], (*432)							[4,3]+, (432)	[1+,4,3], (*332)	[4,3+], (3*2)
{4,3}	t0,1{4,3}	t1{4,3}	t1,2{4,3}	{3,4}	t0,2{4,3}	t0,1,2{4,3}	s{4,3}	h{4,3}	h1,2{4,3}
Duals to uniform polyhedra
V4.4.4	V3.8.8	V3.4.3.4	V4.6.6	V3.3.3.3	V3.4.4.4	V4.6.8	V3.3.3.3.4	V3.3.3	V3.3.3.3.3

This polyhedron is topologically related as a part of sequence of regular polyhedra with Schläfli symbols {3,n}, continuing into the hyperbolic plane.

{3,3}

{3,4}

{3,5}

{3,6}

{3,7}

{3,8}

{3,9}

The regular icosahedron, seen as a snub tetrahedron, is a member of a sequence of snubbed polyhedra and tilings with vertex figure (3.3.3.3.n) and Coxeter–Dynkin diagram . These figures and their duals have (n32) rotational 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 digons.

Dimensional family of snub polyhedra and tilings: 3.3.3.3.n

Symmetry n32 [n,3]+	Spherical				Euclidean	Hyperbolic
	232 [2,3]+ D3	332 [3,3]+ T	432 [4,3]+ O	532 [5,3]+ I	632 [6,3]+ P6	732 [7,3]+	832 [8,3]+	∞32 [∞,3]+
Snub figure	3.3.3.3.2	3.3.3.3.3	3.3.3.3.4	3.3.3.3.5	3.3.3.3.6	3.3.3.3.7	3.3.3.3.8	3.3.3.3.∞
Coxeter Schläfli	s{2,3}	s{3,3}	s{4,3}	s{5,3}	s{6,3}	s{7,3}	s{8,3}	s{∞,3}
Snub dual figure	V3.3.3.3.2	V3.3.3.3.3	V3.3.3.3.4	V3.3.3.3.5	V3.3.3.3.6	V3.3.3.3.7	V3.3.3.3.8	V3.3.3.3.∞
Coxeter

The icosahedron shares its vertex arrangement with three Kepler–Poinsot solids. The great dodecahedron also has the same edge arrangement.

Picture	Great dodecahedron	Small stellated dodecahedron	Great icosahedron
Coxeter-Dynkin

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 one of four regular tessellations in the hyperbolic 3-space.

It is shown here as an edge framework in a Poincaré disk model, with one icosahedron visible in the center.

Uses and natural forms

Twenty-sided die

Gold nanoparticle viewed in electron microscope.

Structure of γ-boron.

Biology

Many viruses, e.g. herpes virus, have icosahedral shells. Viral structures are built of repeated identical protein subunits known as capsomeres, 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. The icosahedral shell encapsulating enzymes and labile intermediates are built of different types of proteins with BMC domains.

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-carboranes are chemical compounds with shape very close to isosahedron. Icosahedral twinning also occurs in crystals, especially nanoparticles.

Many borides and allotropes of boron contain boron B₁₂ icosahedron as a basic structure unit.

Toys and Games

In several roleplaying games, such as Dungeons & Dragons, the twenty-sided die (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 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 questions are inscribed on a regular icosahedron.

Others

R. Buckminster Fuller and Japanese cartographer Shoji Sadao designed a world map in the form of an unfolded icosahedron, called the Fuller projection, whose maximum distortion is only 2%.

The "Sol de la Flor" light shade consists of twenty panels, which meet at the corners of an icosahedron in rosettes resembling the overlapping petals of a frangipani 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.

The company logo of the 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

A planar representation of the icosahedral graph.

The skeleton of the icosahedron—the vertices and edges—form a graph. The high degree of symmetry of the polygon is replicated in the properties of this graph, which is distance-transitive, distance-regular, and symmetric. The automorphism group has order 120. The vertices can be colored with 4 colors, the edges with 5 colors, and the diameter is 3.

The icosahedral graph is Hamiltonian: there is a cycle containing all the vertices. It is also a planar graph.

See also

• Truncated icosahedron
• Regular polyhedron
• Geodesic grids use an iteratively bisected icosahedron to generate grids on a sphere
• Jessen's icosahedron
• Infinite skew polyhedron
• Pseudoicosahedron

References

• Klein, Felix (1888), Lectures on the ikosahedron and the solution of equations of the fifth degree, http://digital.library.cornell.edu/cgi/t/text/text-idx?c=math;cc=math;view=toc;subview=short;idno=03070001, Dover edition ISBN 978-0-486-49528-6.

External links

• Weisstein, Eric W., "Icosahedron" from MathWorld.
• Richard Klitzing, 3D convex uniform polyhedra, x3o5o - ike
• Hartley, Michael. "Dr Mike's Math Games for Kids". http://www.dr-mikes-math-games-for-kids.com/polyhedral-nets.html?net=PKYcLAHcKk9lKCGWdoeJHvnDe7jSJyKXIghPYlKcV5PxkeWqGqwmeoyXUilmvzkXsQnxEoduPHUYFqOf20B8EwLz8CufLOUuc4N5&name=Icosahedron#applet.
• Webb, Robert. "Icosahedron". http://www.software3d.com/Icosahedron.php.
• The Uniform Polyhedra
• K.J.M. MacLean, A Geometric Analysis of the Five Platonic Solids and Other Semi-Regular Polyhedra
• Interactive Icosahedron model - works right in your web browser
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra
• Tulane.edu A discussion of viral structure and the icosahedron
• Paper Models of Polyhedra Many links
• Origami Polyhedra - Models made with Modular Origami
• Video of icosahedral mirror sculpture
• [1] Principle of virus architecture
• Stella: Polyhedron Navigator: Software used to create some of the images on this page.

Notable stellations of the icosahedron
Regular	Uniform duals			Regular compounds			Regular star	Others
Icosahedron	Small triambic icosahedron	Medial triambic icosahedron	Great triambic icosahedron	Compound of five octahedra	Compound of five tetrahedra	Compound of ten tetrahedra	Great icosahedron	Excavated dodecahedron	Final stellation
The stellation process on the icosahedron creates a number of related polyhedra and compounds with icosahedral symmetry.

UpToDate Contents

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Japanese Journal

• ノーベル化学賞 : 準結晶の発見

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• まてりあ : 日本金属学会会報 51(2), 62-63, 2012-02-01
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• NAID 10030121962

Related Links

• Icosahedron -- from Wolfram MathWorld

The icosahedron has the icosahedral group of symmetries. The connectivity of the vertices is given by the icosahedral graph. The dual polyhedron of an icosahedron with unit edge lengths is the dodecahedron with edge lengths ...

• Icosahedral honeycombとは - goo Wikipedia (ウィキペディア)

[ Icosahedral honeycomb ] の検索結果 Icosahedral honeycomb From Wikipedia, the free encyclopedia Jump to: navigation, search Icosahedral honeycomb Poincaré disk model Type regular hyperbolic honeycomb Schläfli symbol ...

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★リンクテーブル★

リンク元	「icosahedron」「正二十面体」

「icosahedron」

　　[★]

• 正二十面体

関

icosahedral

　

「正二十面体」

　　[★]

英

icosahedron、icosahedral