Regular icosahedron | |
---|---|

(Click here for rotating model) | |

Type | Platonic solid |

shortcode | 5<z> |

Elements | F = 20, E = 30V = 12 (χ = 2) |

Faces by sides | 20{3} |

Conway notation | I sT |

Schläfli symbols | {3,5} |

s{3,4} sr{3,3} or | |

Face configuration | V5.5.5 |

Wythoff symbol | 5 | 2 3 |

Coxeter diagram | |

Symmetry | I_{h}, H_{3}, [5,3], (*532) |

Rotation group | I, [5,3]^{+}, (532) |

References | U _{22}, C _{25}, W _{4} |

Properties | regular, convex deltahedron |

Dihedral angle | 138.189685° = arccos(−√5⁄3) |

3.3.3.3.3 (Vertex figure) | Regular dodecahedron (dual polyhedron) |

Net |

In geometry, a **regular icosahedron ** ( /ˌaɪkɒsəˈhiːdrən,-kə-,-koʊ-/ or /aɪˌkɒsəˈhiːdrən/ ^{ [1] }) is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces.

- Dimensions
- Area and volume
- Cartesian coordinates
- Spherical coordinates
- Orthogonal projections
- As a configuration
- Spherical tiling
- Other facts
- Construction by a system of equiangular lines
- Symmetry
- Stellations
- Facetings
- Geometric relations
- Inscribed in other Platonic solids
- Relations to the 600-cell and other 4-polytopes
- Relations to other uniform polytopes
- Relation to the 6-cube and rhombic triacontahedron
- Symmetries
- Uses and natural forms
- Biology
- Chemistry
- Toys and games
- Others
- Icosahedral graph
- Diminished regular icosahedra
- Related polyhedra and polytopes
- See also
- Notes
- Citations
- References
- External links

It has five equilateral triangular faces meeting at each vertex. It is represented by its Schläfli symbol {3,5}, or sometimes by its vertex figure as 3.3.3.3.3 or 3^{5}. It is the dual of the regular dodecahedron, which is represented by {5,3}, having three pentagonal faces around each vertex. In most contexts, the unqualified use of the word "icosahedron" refers specifically to this figure.

A regular icosahedron is a strictly convex deltahedron and a gyroelongated pentagonal bipyramid and a biaugmented pentagonal antiprism in any of six orientations.

The name comes from Greek εἴκοσι* (eíkosi)* 'twenty',andἕδρα* (hédra)* 'seat'. The plural can be either "icosahedrons" or "icosahedra" ( /-drə/ ).

If the edge length of a regular icosahedron is , the radius of a circumscribed sphere (one that touches the icosahedron at all vertices) is

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

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

where is the golden ratio.

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

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 times its height .

The volume filling factor of the circumscribed sphere is:

compared to 66.49% for a dodecahedron. A sphere inscribed in an icosahedron will enclose 89.635% of its volume, compared to only 75.47% for a dodecahedron.

The midsphere of an icosahedron will have a volume 1.01664 times the volume of the icosahedron, which is by far the closest similarity in volume of any platonic solid with its midsphere. This arguably makes the icosahedron the "roundest" of the platonic solids.

The vertices of an icosahedron centered at the origin with an edge length of 2 and a circumradius of are^{ [2] }

where is the golden ratio. Taking all permutations of these coordinates (not just cyclic permutations) results in the Compound of two icosahedra.

The vertices of the icosahedron 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 .

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.

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.

This scheme takes advantage of the fact that the regular icosahedron is a pentagonal gyroelongated bipyramid, with D_{5d} dihedral symmetry—that is, it is formed of two congruent pentagonal pyramids joined by a pentagonal antiprism.

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

Centered by | Face | Edge | Vertex |
---|---|---|---|

Coxeter plane | A_{2} | A_{3} | H_{3} |

Graph | |||

Projective symmetry | [6] | [2] | [10] |

Graph | Face normal | Edge normal | Vertex normal |

This configuration matrix represents the icosahedron. The rows and columns correspond to vertices, edges, and faces. The diagonal numbers say how many of each element occur in the whole icosahedron. The nondiagonal numbers say how many of the column's element occur in or at the row's element.^{ [3] }^{ [4] }

Here is the configuration expanded with *k*-face elements and *k*-figures. The diagonal element counts are the ratio of the full Coxeter group H_{3}, order 120, divided by the order of the subgroup with mirror removal.

H_{3} | k-face | f_{k} | f_{0} | f_{1} | f_{2} | k-fig | Notes | |
---|---|---|---|---|---|---|---|---|

A_{2} | ( ) | f_{0} | 12 | 5 | 5 | {5} | H_{3}/H_{2} = 120/10 = 12 | |

A_{1}A_{1} | { } | f_{1} | 2 | 30 | 2 | { } | H_{3}/A_{1}A_{1} = 120/4 = 30 | |

H_{2} | {3} | f_{2} | 3 | 3 | 20 | ( ) | H_{3}/A_{2} = 120/6 = 20 |

The icosahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.

Orthographic projection | Stereographic projection |
---|

- An icosahedron has 43,380 distinct nets.
^{ [5] } - To color the icosahedron, such that no two adjacent faces have the same color, requires at least 3 colors.
^{ [lower-alpha 1] } - 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, Pappus, and Fibonacci, among others.
^{ [6] }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.^{ [7] }Both volumes have formulas involving the golden ratio, but taken to different powers.^{ [8] }As it turns out, the icosahedron occupies less of the sphere's volume (60.54%) than the dodecahedron (66.49%).^{ [9] } - Icosahedral angle - the angle between the closest vertices of the icosahedron, relative to the center of the body of the icosahedron (3D), is equal to the diagonal angle of a double and / or half square (≈ 63.434949°)

Icosahedron H _{3} Coxeter plane | 6-orthoplex D _{6} 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 D_{6} to H_{3} 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 straightforward computation yields (where is the 6 × 6 identity matrix). This implies that has eigenvalues and , both with multiplicity 3 since 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 a system of six equiangular lines in intersecting pairwise at a common acute angle of . Orthogonal projection of the positive and negative basis vectors of onto the -eigenspace of yields thus the twelve vertices of the icosahedron.

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

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 1884 ). 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 of size two, which is generated by the reflection through the center of the icosahedron.

The icosahedron has a large number of stellations. According to specific rules defined in the book * The Fifty-Nine Icosahedra *,^{ [10] } 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.

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. | |||||||

The small stellated dodecahedron, great dodecahedron, and great icosahedron are three facetings of the regular icosahedron. They share the same vertex arrangement. They all have 30 edges. The regular icosahedron and great dodecahedron share the same edge arrangement but differ in faces (triangles vs pentagons), as do the small stellated dodecahedron and great icosahedron (pentagrams vs triangles).

Convex | Regular stars | ||
---|---|---|---|

icosahedron | great dodecahedron | small stellated dodecahedron | great icosahedron |

The regular icosahedron is the dual polyhedron of the regular dodecahedron. An icosahedron can be inscribed in a dodecahedron by placing its vertices at the face centers of the dodecahedron, and vice versa.

An icosahedron can be inscribed in an octahedron by placing its 12 vertices on the 12 edges of the octahedron such that they divide each edge into its two golden sections. Because the golden sections are unequal, there are five different ways to do this consistently, so five disjoint icosahedra can be inscribed in each octahedron.^{ [11] }

An icosahedron of edge length can be inscribed in a unit-edge-length cube by placing six of its edges (3 orthogonal opposite pairs) on the square faces of the cube, centered on the face centers and parallel or perpendicular to the square's edges.^{ [12] } Because there are five times as many icosahedron edges as cube faces, there are five ways to do this consistently, so five disjoint icosahedra can be inscribed in each cube. The edge lengths of the cube and the inscribed icosahedron are in the golden ratio.^{ [lower-alpha 2] }

The icosahedron is the dimensional analogue of the 600-cell, a regular 4-dimensional polytope. The 600-cell has icosahedral cross sections of two sizes, and each of its 120 vertices is an icosahedral pyramid; the icosahedron is the vertex figure of the 600-cell.

The unit-radius 600-cell has tetrahedral cells of edge length , 20 of which meet at each vertex to form an icosahedral pyramid (a 4-pyramid with an icosahedron as its base). Thus the 600-cell contains 120 icosahedra of edge length . The 600-cell also contains unit-edge-length cubes and unit-edge-length octahedra as interior features formed by its unit-length chords. In the unit-radius 120-cell (another regular 4-polytope which is both the dual of the 600-cell and a compound of 5 600-cells) we find all three kinds of inscribed icosahedra (in a dodecahedron, in an octahedron, and in a cube).

A semiregular 4-polytope, the snub 24-cell, has icosahedral cells.

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.

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.

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.

The icosahedron can be projected to 3D from the 6D 6-demicube using the same basis vectors that form the hull of the Rhombic triacontahedron from the 6-cube. Shown here including the inner 20 vertices which are not connected by the 30 outer hull edges of 6D norm length . The inner vertices form a dodecahedron.

The 3D projection basis vectors [u,v,w] used are:

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.

Regular | Uniform | 2-uniform | 3-uniform | ||||
---|---|---|---|---|---|---|---|

Name | Regular icosahedron | Snub octahedron | Snub tetratetrahedron | Gyroelongated pentagonal bipyramid | Triangular gyrobianticupola | Snub triangular antiprism ^{ [13] } | Snub square bipyramid |

Image | |||||||

Face coloring | (11111) | (11212) | (11213) | (11122) (22222) | (12332) (23333) | (11213) (11212) | (11424) (22434) (33414) |

Coxeter diagram | |||||||

Schläfli symbol | {3,5} | s{3,4} | sr{3,3} | () || {n} || r{n} || () | ss{2,6} | sdt{2,4} | |

Conway | I | HtO | sT | k5A5 | sY3 = HtA3 | HtdP4 | |

Symmetry | I_{h}[5,3] (*532) | T_{h}[3 ^{+},4](3*2) | T [3,3] ^{+}(332) | D_{5d}[2 ^{+},10](2*5) | D_{3d}[2 ^{+},6](2*3) | D_{3}[3,2] ^{+}(322) | D_{2h}[2,2] (*222) |

Symmetry order | 120 | 24 | 12 | 20 | 12 | 6 | 8 |

Many viruses, e.g. herpes virus, have icosahedral shells.^{ [14] } 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.^{ [15] } 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.

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

Many borides and allotropes of boron contain boron B_{12} icosahedron as a basic structure unit.

Icosahedral dice with twenty sides have been used since ancient times.^{ [16] }

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".

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).

In the *Nintendo 64* game * Kirby 64: The Crystal Shards *, the boss Miracle Matter is a regular icosahedron.

Inside a Magic 8-Ball, various answers to yes–no questions are inscribed on a regular icosahedron.

The "skwish" baby toy is a tensegrity object in the form of a Jessen's icosahedron, which has the same vertex coordinates as a regular icosahedron, and the same number of faces, but with six edges turned 90° to connect to other vertices.

R. Buckminster Fuller and Japanese cartographer Shoji Sadao ^{ [17] } designed a world map in the form of an unfolded icosahedron, called the Fuller projection, whose maximum distortion is only 2%.

The American electronic music duo ODESZA use a regular icosahedron as their logo.

Regular icosahedron graph | |
---|---|

Vertices | 12 |

Edges | 30 |

Radius | 3 |

Diameter | 3 |

Girth | 3 |

Automorphisms | 120 (A_{5} × Z_{2}) |

Chromatic number | 4 |

Properties | Hamiltonian, regular, symmetric, distance-regular, distance-transitive, 3-vertex-connected, planar graph |

Table of graphs and parameters |

The skeleton of the icosahedron (the vertices and edges) forms a graph. It is one of 5 Platonic graphs, each a skeleton of its Platonic solid.

The high degree of symmetry of the polygon is replicated in the properties of this graph, which is distance-transitive 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.^{ [18] }

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

There are 4 related Johnson solids, including pentagonal faces with a subset of the 12 vertices. The similar dissected regular icosahedron has 2 adjacent vertices diminished, leaving two trapezoidal faces, and a bifastigium has 2 opposite sets of vertices removed and 4 trapezoidal faces. The pentagonal antiprism is formed by removing two opposite vertices.

Form | J2 | Bifastigium | J63 | J62 | Dissected icosahedron | s{2,10} | J11 |
---|---|---|---|---|---|---|---|

Vertices | 6 of 12 | 8 of 12 | 9 of 12 | 10 of 12 | 11 of 12 | ||

Symmetry | C_{5v}, [5], (*55)order 10 | D_{2h}, [2,2], *222order 8 | C_{3v}, [3], (*33)order 6 | C_{2v}, [2], (*22)order 4 | D_{5d}, [2^{+},10], (2*5)order 20 | C_{5v}, [5], (*55)order 10 | |

Image |

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} | t{5,3} | r{5,3} | t{3,5} | {3,5} | rr{5,3} | tr{5,3} | sr{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} | t{3,3} | r{3,3} | t{3,3} | {3,3} | rr{3,3} | tr{3,3} | sr{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] = [3,3](*332) | [3^{+},4](3*2) | |||||||

{4,3} | t{4,3} | r{4,3} r{3 ^{1,1}} | t{3,4} t{3 ^{1,1}} | {3,4} {3 ^{1,1}} | rr{4,3} s _{2}{3,4} | tr{4,3} | sr{4,3} | h{4,3} {3,3} | h_{2}{4,3} t{3,3} | s{3,4} s{3 ^{1,1}} |

= | = | = | = or | = or | = | |||||

| | | | | ||||||

Duals to uniform polyhedra | ||||||||||

V4^{3} | V3.8^{2} | V(3.4)^{2} | V4.6^{2} | V3^{4} | V3.4^{3} | V4.6.8 | V3^{4}.4 | V3^{3} | V3.6^{2} | V3^{5} |

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

*n32 symmetry mutation of regular tilings: {3,n} | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Spherical | Euclid. | Compact hyper. | Paraco. | Noncompact hyperbolic | |||||||

3.3 | 3^{3} | 3^{4} | 3^{5} | 3^{6} | 3^{7} | 3^{8} | 3^{∞} | 3^{12i} | 3^{9i} | 3^{6i} | 3^{3i} |

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 (*n*32) rotational symmetry, being in the Euclidean plane for , and hyperbolic plane for any higher . The series can be considered to begin with , with one set of faces degenerated into digons.

n32 symmetry mutations of snub tilings: 3.3.3.3.n | ||||||||
---|---|---|---|---|---|---|---|---|

Symmetryn32 | Spherical | Euclidean | Compact hyperbolic | Paracomp. | ||||

232 | 332 | 432 | 532 | 632 | 732 | 832 | ∞32 | |

Snub figures | ||||||||

Config. | 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.∞ |

Gyro figures | ||||||||

Config. | 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.∞ |

Spherical | Hyperbolic tilings | |||||||
---|---|---|---|---|---|---|---|---|

{2,5} | {3,5} | {4,5} | {5,5} | {6,5} | {7,5} | {8,5} | ... | {∞,5} |

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. |

- Great icosahedron
- Geodesic grids use an iteratively bisected icosahedron to generate grids on a sphere
- Icosahedral twins
- Infinite skew polyhedron
- Jessen's icosahedron
- Regular polyhedron
- Truncated icosahedron

- ↑ 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.
- ↑ Reciprocally, the edge length of a cube inscribed in a dodecahedron is in the golden ratio to the dodecahedron's edge length. The cube's edges lie in pentagonal face planes of the dodecahedron as regular pentagon diagonals, which are always in the golden ratio to the regular pentagon's edge. So when a cube is inscribed in a dodecahedron and an icosahedron is inscribed in the cube, the dodecahedron and icosahedron (which do not share any vertices) have the same edge length.

- ↑ Jones, Daniel (2003) [1917], Peter Roach; James Hartmann; Jane Setter (eds.),
*English Pronouncing Dictionary*, Cambridge: Cambridge University Press, ISBN 3-12-539683-2 - ↑ Weisstein, Eric W. "Icosahedral group".
*MathWorld*. - ↑ Coxeter, Regular Polytopes, sec 1.8 Configurations
- ↑ Coxeter, Complex Regular Polytopes, p.117
- ↑ Weisstein, Eric W. "Regular Icosahedron".
*MathWorld*. - ↑ Herz-Fischler, Roger (2013),
*A Mathematical History of the Golden Number*, Courier Dover Publications, pp. 138–140, ISBN 9780486152325 . - ↑ Simmons, George F. (2007),
*Calculus Gems: Brief Lives and Memorable Mathematics*, Mathematical Association of America, p. 50, ISBN 9780883855614 . - ↑ Sutton, Daud (2002),
*Platonic & Archimedean Solids*, Wooden Books, Bloomsbury Publishing USA, p. 55, ISBN 9780802713865 . - ↑ Numerical values for the volumes of the inscribed Platonic solids may be found in Buker, W. E.; Eggleton, R. B. (1969), "The Platonic Solids (Solution to problem E2053)",
*American Mathematical Monthly*,**76**(2): 192, doi:10.2307/2317282, JSTOR 2317282 . - ↑ Coxeter et al. 1938.
- ↑ Coxeter et al. 1938, p. 4; "Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the "golden section". Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a compound of five octahedra, which comes under our definition of stellated icosahedron. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated triacontahedron.) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a stella octangula, thus forming a compound of ten tetrahedra. Further, we can choose one tetrahedron from each stella octangula, so as to derive a compound of five tetrahedra, which still has all the rotation symmetry of the icosahedron (i.e. the icosahedral group), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be
*chiral*." - ↑ Borovik 2006, pp. 8–9, §5. How to draw an icosahedron on a blackboard.
- ↑ Snub Anti-Prisms
- ↑ C. Michael Hogan. 2010.
*Virus*. Encyclopedia of Earth. National Council for Science and the Environment. eds. S. Draggan and C. Cleveland - ↑ Bobik, T.A. (2007), "Bacterial Microcompartments",
*Microbe*, Am. Soc. Microbiol.,**2**: 25–31, archived from the original on 2013-07-29 - ↑ Cromwell, Peter R. "Polyhedra" (1997) Page 327.
- ↑ "Fuller and Sadao: Partners in Design". September 19, 2006. Archived from the original on August 16, 2010. Retrieved 2010-01-26.
- ↑ Weisstein, Eric W. "Icosahedral Graph".
*MathWorld*.

In geometry, a **dodecahedron** or **duodecahedron** is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.

In mathematics, two quantities are in the **golden ratio** if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities and with

In geometry, an **icosidodecahedron** is a polyhedron with twenty (*icosi*) triangular faces and twelve (*dodeca*) pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such it is one of the Archimedean solids and more particularly, a quasiregular polyhedron.

In geometry, a **Kepler–Poinsot polyhedron** is any of four regular star polyhedra.

In geometry, an **octahedron** is a polyhedron with eight faces. The term is most commonly used to refer to the **regular** octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.

In geometry, a **Platonic solid** is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent regular polygons, and the same number of faces meet at each vertex. There are only five such polyhedra:

In geometry, the **rhombicuboctahedron**, or **small rhombicuboctahedron**, is an Archimedean solid with eight triangular, six square, and twelve rectangular faces. There are 24 identical vertices, with one triangle, one square, and two rectangles meeting at each one. The polyhedron has octahedral symmetry, like the cube and octahedron. Its dual is called the deltoidal icositetrahedron or trapezoidal icositetrahedron, although its faces are not really true trapezoids.

In geometry, the **truncated icosahedron** is an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares. In general usage, the degree of truncation is assumed to be uniform unless specified.

In geometry, the **rhombicosidodecahedron** is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces.

In geometry, the **truncated icosidodecahedron** is an Archimedean solid, one of thirteen convex, isogonal, non-prismatic solids constructed by two or more types of regular polygon faces.

In geometry, the **snub dodecahedron**, or **snub icosidodecahedron**, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.

A **regular polyhedron** is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex.

In geometry, the **truncated dodecahedron** is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges.

In geometry, the **rhombic triacontahedron**, sometimes simply called the **triacontahedron** as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Catalan solid, and the dual polyhedron of the icosidodecahedron. It is a zonohedron.

In geometry, a **disdyakis triacontahedron**, **hexakis icosahedron**, **decakis dodecahedron** or **kisrhombic triacontahedron** is a Catalan solid with 120 faces and the dual to the Archimedean truncated icosidodecahedron. As such it is face-uniform but with irregular face polygons. It slightly resembles an inflated rhombic triacontahedron: if one replaces each face of the rhombic triacontahedron with a single vertex and four triangles in a regular fashion, one ends up with a disdyakis triacontahedron. That is, the disdyakis triacontahedron is the Kleetope of the rhombic triacontahedron. It also has the most faces among the Archimedean and Catalan solids, with the snub dodecahedron, with 92 faces, in second place.

In geometry, the **bilunabirotunda** is one of the Johnson solids. A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra. They were named by Norman Johnson, who first listed these polyhedra in 1966.

In geometry, the **complete** or **final stellation of the icosahedron** is the outermost stellation of the icosahedron, and is "complete" and "final" because it includes all of the cells in the icosahedron's stellation diagram. That is, every three intersecting face planes of the icosahedral core intersect either on a vertex of this polyhedron, or inside of it.

A **regular dodecahedron** or **pentagonal dodecahedron** is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges, and 160 diagonals. It is represented by the Schläfli symbol {5,3}.

In geometry, the **medial rhombic triacontahedron** is a nonconvex isohedral polyhedron. It is a stellation of the rhombic triacontahedron, and can also be called **small stellated triacontahedron**. Its dual is the dodecadodecahedron.

In geometry, an **icosahedron** is a polyhedron with 20 faces. The name comes from Ancient Greek εἴκοσι* (eíkosi)* 'twenty' and from Ancient Greek ἕδρα* (hédra)* ' seat'. The plural can be either "icosahedra" or "icosahedrons".

- Klein, Felix (1888),
*Lectures on the ikosahedron and the solution of equations of the fifth degree*, ISBN 978-0-486-49528-6, Dover edition`{{citation}}`

: CS1 maint: postscript (link), translated from Klein, Felix (1884).*Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade*. Teubner. - Coxeter, H.S.M.; du Val, Patrick; Flather, H.T.; Petrie, J.F. (1938).
*The Fifty-Nine Icosahedra*. Vol. 6. University of Toronto Studies (Mathematical Series). - Borovik, Alexandre (2006). "Coxeter Theory: The Cognitive Aspects". In Davis, Chandler; Ellers, Erich (eds.).
*The Coxeter Legacy*. Providence, Rhode Island: American Mathematical Society. pp. 17–43. ISBN 978-0821837221.

Wikimedia Commons has media related to Icosahedron .

Look up ** icosahedron ** in Wiktionary, the free dictionary.

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- Tulane.edu A discussion of viral structure and the icosahedron
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Notable stellations of the icosahedron | |||||||||

Regular | Uniform duals | Regular compounds | Regular star | Others | |||||

(Convex) 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. |

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