Regular icosahedron

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Regular Icosahedron
Icosahedron.jpg
(Click here for rotating model)
Type Platonic solid
Elements F = 20, E = 30
V = 12 (χ = 2)
Faces by sides 20{3}
Conway notation I
sT
Schläfli symbols {3,5}
s{3,4}
sr{3,3} or s\begin{Bmatrix} 3 \\ 3 \end{Bmatrix}
Wythoff symbol 5 | 2 3
Coxeter diagram CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png
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)
Icosahedron vertfig.svg
3.3.3.3.3
(Vertex figure)
Dodecahedron.png
Dodecahedron
(dual polyhedron)
Icosahedron flat.svg
Net

In geometry, a regular icosahedron (/ˌkɵsəˈhdrən/ or /ˌkɒsəˈhdrən/) is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids.

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 35. It is the dual of the dodecahedron, which is represented by {5,3}, having three pentagonal faces around each vertex.

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

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

Dimensions[edit]

Net folding into icosahedron

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

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 OEISA019881

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

r_i = \frac{\varphi^2 a}{2 \sqrt{3}} = \frac{\sqrt{3}}{12} \left(3+ \sqrt{5} \right) a \approx 0.7557613141\cdot a OEISA179294

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

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 OEISA019863

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

Area and volume[edit]

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

A = 5\sqrt{3}a^2 \approx 8.66025404a^2, OEISA010527
V = \frac{5}{12} (3+\sqrt5)a^3 \approx 2.18169499a^3. OEISA102208

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 ri.

The volume filling factor of the circumscribed sphere is

f=V/(4 \pi r_u^3/3) = \frac{20(3+\surd 5)}{(2\surd 5+10)^{3/2}\pi}\approx 0.6054613829.

Cartesian coordinates[edit]

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:[1]

(0, ±1, ±φ)
(±1, ±φ, 0)
φ, 0, ±1)

where φ=\tfrac{1+\sqrt{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 \tfrac{\sqrt{5}-1}{2}, 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[edit]

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 D5d dihedral symmetry—that is, it is formed of two congruent pentagonal pyramids joined by a pentagonal antiprism.

Orthogonal projections[edit]

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 Icosahedron t0 A2.png Icosahedron graph A3 1.png Icosahedron t0 H3.png
Projective
symmetry
[6] [2] [10]
Graph Icosahedron fnormal.png
Face normal
Icosahedron graph A3 2.png
Edge normal
Icosahedron vnormal.png
Vertex normal

Spherical tiling[edit]

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.

Uniform tiling 532-t2.png Icosahedron stereographic projection.png
Face-centered
Orthographic projection Stereographic projection

Other facts[edit]

  • An icosahedron has 43,380 distinct nets.[2]
  • To color the icosahedron, such that no two adjacent faces have the same color, requires at least 3 colors.[3]
  • 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.[4] 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.[5] Both volumes have formulas involving the golden ratio, but taken to different powers.[6] As it turns out, the icosahedron occupies less of the sphere's volume (60.54%) than the dodecahedron (66.49%).[7]

Construction by a system of equiangular lines[edit]

Icosahedron t0 H3.png
Icosahedron
H3 Coxeter plane
6-cube t5 B5.svg
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: Geometric folding Coxeter graph D6 H3.png

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 \mathbb{Q}[\sqrt{5}] necessary in more elementary approaches.

The existence of the icosahedron amounts to the existence of six equiangular lines in \mathbb R^3. 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 \scriptstyle -\sqrt{5} and \scriptstyle \sqrt{5}, both with multiplicity 3 since A is symmetric and of trace zero.

The matrix \scriptstyle A+\sqrt{5}I induces thus a Euclidean structure on the quotient space \mathbb R^6/\ker(A+\sqrt{5}I) which is isomorphic to \mathbb R^3 since the kernel \ker(A+\sqrt{5}I) of \scriptstyle {A+\sqrt{5}I} has dimension 3. The image under the projection \pi:\mathbb R^6 \longrightarrow \mathbb R^6/\ker(A+\sqrt{5}I) of the six coordinate axes \mathbb R v_1,\dots,\mathbb R v_6 in \mathbb R^6 forms thus a system of six equiangular lines in \mathbb R^3 intersecting pairwise at a common acute angle of \scriptstyle{\arccos}\tfrac{1}{\sqrt{5}}. Orthogonal projection of ±v1, ..., ±v6 onto the \scriptstyle \sqrt{5}-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[edit]

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[edit]

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.[8]

21 of 59 stellations
Zeroth stellation of icosahedron facets.png
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.
Zeroth stellation of icosahedron.png First stellation of icosahedron.png Second stellation of icosahedron.png Third stellation of icosahedron.png Fourth stellation of icosahedron.png Fifth stellation of icosahedron.png Sixth stellation of icosahedron.png Seventh stellation of icosahedron.png
Eighth stellation of icosahedron.png Ninth stellation of icosahedron.png Tenth stellation of icosahedron.png Eleventh stellation of icosahedron.png Twelfth stellation of icosahedron.png Thirteenth stellation of icosahedron.png Fourteenth stellation of icosahedron.png Fifteenth stellation of icosahedron.png
Sixteenth stellation of icosahedron.png Seventeenth stellation of icosahedron.png First compound stellation of icosahedron.png Second compound stellation of icosahedron.png Third compound stellation of icosahedron.png

Geometric relations[edit]

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.

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[edit]

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
Snub
octahedron
Snub
tetratetrahedron
Pentagonal
Gyroelongated
bipyramid
Snub triangular
antiprism[9]
Coxeter-Dynkin CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png CDel node h.pngCDel split1.pngCDel nodes hh.png
Schläfli symbol {3,5} s{3,4} sr{3,3} ss{2,6}
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)
D3d
[2+,6]
(2*3)
Symmetry order 60 24 12 20 12
Uniform coloring Uniform polyhedron-53-t2.png
(11111)
Uniform polyhedron-43-h01.svg
(11212)
Uniform polyhedron-33-s012.png
(11213)
Pentagonal gyroelongated bipyramid.png
(11122)&(22222)
Snub triangular antiprism.png
(11213)&(11212)

Uses and natural forms[edit]

Gold nanoparticle viewed in electron microscope.
Structure of γ-boron.

Biology[edit]

Many viruses, e.g. herpes virus, have icosahedral shells.[10] 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.[11] 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[edit]

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[edit]

Twenty-sided die

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[edit]

R. Buckminster Fuller and Japanese cartographer Shoji Sadao[12] 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.[13]

The company logo of the TDK Corporation contains a geometric figure which is based on the stellation diagram of the icosahedron.

An icosahedron was used for a logo for the Australian TV company; Grundy Television.

As a graph[edit]

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.[14]

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

Related polyhedra and polytopes[edit]

There are 4 related Johnson solids, including pentagonal faces with a subset of the 12 vertices:

J2 J63 J62 J11
6 of 12 9 of 12 10 of 12 11 of 12
Pentagonal pyramid.png Tridiminished icosahedron.png Metabidiminished icosahedron.png Gyroelongated pentagonal pyramid.png

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)
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.png
Uniform polyhedron-53-t0.png Uniform polyhedron-53-t01.png Uniform polyhedron-53-t1.png Uniform polyhedron-53-t12.png Uniform polyhedron-53-t2.png Uniform polyhedron-53-t02.png Uniform polyhedron-53-t012.png Uniform polyhedron-53-s012.png
{5,3} t{5,3} r{5,3} 2t{5,3}=t{3,5} 2r{5,3}={3,5} rr{5,3} tr{5,3} sr{5,3}
Duals to uniform polyhedra
CDel node f1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 5.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Icosahedron.svg Triakisicosahedron.jpg Rhombictriacontahedron.svg Pentakisdodecahedron.jpg Dodecahedron.svg Deltoidalhexecontahedron.jpg Disdyakistriacontahedron.jpg Pentagonalhexecontahedronccw.jpg
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)
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png
Uniform polyhedron-33-t0.png Uniform polyhedron-33-t01.png Uniform polyhedron-33-t1.png Uniform polyhedron-33-t12.png Uniform polyhedron-33-t2.png Uniform polyhedron-33-t02.png Uniform polyhedron-33-t012.png Uniform polyhedron-33-s012.png
{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
CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 3.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Tetrahedron.svg Triakistetrahedron.jpg Hexahedron.svg Triakistetrahedron.jpg Tetrahedron.svg Rhombicdodecahedron.jpg Tetrakishexahedron.jpg Dodecahedron.svg
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{31,1}
t{3,4}
t{31,1}
{3,4}
{31,1}
rr{4,3}
s2{3,4}
tr{4,3} sr{4,3} h{4,3}
{3,3}
h2{4,3}
t{3,3}
s{3,4}
s{31,1}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png
CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
= CDel nodes 11.pngCDel split2.pngCDel node.png
CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
= CDel nodes 11.pngCDel split2.pngCDel node 1.png
CDel node h0.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
= CDel nodes.pngCDel split2.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png =
CDel nodes 10ru.pngCDel split2.pngCDel node.png or CDel nodes 01rd.pngCDel split2.pngCDel node.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png =
CDel nodes 10ru.pngCDel split2.pngCDel node 1.png or CDel nodes 01rd.pngCDel split2.pngCDel node 1.png
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h0.png =
CDel node h.pngCDel split1.pngCDel nodes hh.png
Uniform polyhedron-43-t0.svg Uniform polyhedron-43-t01.svg Uniform polyhedron-43-t1.svg
Uniform polyhedron-33-t02.png
Uniform polyhedron-43-t12.svg
Uniform polyhedron-33-t012.png
Uniform polyhedron-43-t2.svg
Uniform polyhedron-33-t1.png
Uniform polyhedron-43-t02.png
Rhombicuboctahedron uniform edge coloring.png
Uniform polyhedron-43-t012.png Uniform polyhedron-43-s012.png Uniform polyhedron-33-t0.pngUniform polyhedron-33-t2.png Uniform polyhedron-33-t01.pngUniform polyhedron-33-t12.png Uniform polyhedron-43-h01.svg
Uniform polyhedron-33-s012.png
Duals to uniform polyhedra
V43 V3.82 V(3.4)2 V4.62 V34 V3.43 V4.6.8 V34.4 V33 V3.62 V35
CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 3.pngCDel node fh.pngCDel 4.pngCDel node.png
CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 3.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Octahedron.svg Triakisoctahedron.jpg Rhombicdodecahedron.jpg Tetrakishexahedron.jpg Hexahedron.svg Deltoidalicositetrahedron.jpg Disdyakisdodecahedron.jpg Pentagonalicositetrahedronccw.jpg Tetrahedron.svg Triakistetrahedron.jpg Dodecahedron.svg

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

Finite Euclidean Compact hyperbolic Paracompact
Trigonal dihedron.png
{3,2}
Uniform polyhedron-33-t2.png
{3,3}
Uniform polyhedron-43-t2.png
{3,4}
Uniform polyhedron-53-t2.png
{3,5}
Uniform polyhedron-63-t2.png
{3,6}
Uniform tiling 73-t2.png
{3,7}
Uniform tiling 83-t2.png
{3,8}
Uniform tiling 39-t0.png
{3,9}
... H2 tiling 23i-4.png
(3,∞}

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 CDel node h.pngCDel n.pngCDel node h.pngCDel 3.pngCDel node h.png. 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 Compact hyperbolic Paracompact
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
Spherical trigonal antiprism.png
3.3.3.3.2
Spherical snub tetrahedron.png
3.3.3.3.3
Spherical snub cube.png
3.3.3.3.4
Spherical snub dodecahedron.png
3.3.3.3.5
Uniform tiling 63-snub.png
3.3.3.3.6
Uniform tiling 73-snub.png
3.3.3.3.7
Uniform tiling 83-snub.png
3.3.3.3.8
Uniform tiling i32-snub.png
3.3.3.3.∞
Coxeter
Schläfli
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{2,3}
CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{3,3}
CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{4,3}
CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{5,3}
CDel node h.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{6,3}
CDel node h.pngCDel 7.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{7,3}
CDel node h.pngCDel 8.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{8,3}
CDel node h.pngCDel infin.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{∞,3}
Snub
dual
figure
Uniform tiling 432-t0.png
V3.3.3.3.2
Uniform tiling 532-t0.png
V3.3.3.3.3
Spherical pentagonal icositetrahedron.png
V3.3.3.3.4
Spherical pentagonal hexecontahedron.png
V3.3.3.3.5
Tiling Dual Semiregular V3-3-3-3-6 Floret Pentagonal.svg
V3.3.3.3.6
Ord7 3 floret penta til.png
V3.3.3.3.7
V3.3.3.3.8 Order-3-infinite floret pentagonal tiling.png
V3.3.3.3.∞
Coxeter CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 3.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 5.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 6.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 7.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 8.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel infin.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Spherical Hyperbolic tilings
Spherical pentagonal hosohedron.png
{2,5}
CDel node 1.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.png
Uniform tiling 532-t2.png
{3,5}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
H2 tiling 245-1.png
{4,5}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node.png
H2 tiling 255-1.png
{5,5}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node.png
H2 tiling 256-1.png
{6,5}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 5.pngCDel node.png
H2 tiling 257-1.png
{7,5}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 5.pngCDel node.png
H2 tiling 258-1.png
{8,5}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 5.pngCDel node.png
... H2 tiling 25i-1.png
{∞,5}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 5.pngCDel node.png

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

Picture Great dodecahedron.png
Great dodecahedron
Small stellated dodecahedron.png
Small stellated dodecahedron
Great icosahedron.png
Great icosahedron
Coxeter-Dynkin CDel node 1.pngCDel 5.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.png CDel node.pngCDel 5.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node 1.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.png

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.

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

See also[edit]

References[edit]

  1. ^ Weisstein, Eric W., "Icosahedral group", MathWorld.
  2. ^ Weisstein, Eric W., "Icosahedron", MathWorld.
  3. ^ 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.
  4. ^ Herz-Fischler, Roger (2013), A Mathematical History of the Golden Number, Courier Dover Publications, pp. 138–140, ISBN 9780486152325 .
  5. ^ Simmons, George F. (2007), Calculus Gems: Brief Lives and Memorable Mathematics, Mathematical Association of America, p. 50, ISBN 9780883855614 .
  6. ^ Sutton, Daud (2002), Platonic & Archimedean Solids, Wooden Books, Bloomsbury Publishing USA, p. 55, ISBN 9780802713865 .
  7. ^ 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, JSTOR 2317282 .
  8. ^ Coxeter, Harold Scott MacDonald; Du Val, P.; Flather, H.T.; Petrie, J.F. (1999), The Fifty-Nine Icosahedra (3rd ed.), Tarquin, ISBN 978-1-899618-32-3, MR 676126  (1st Edn University of Toronto (1938))
  9. ^ Snub Anti-Prisms
  10. ^ C. Michael Hogan. 2010. Virus. Encyclopedia of Earth. National Council for Science and the Environment. eds. S. Draggan and C. Cleveland
  11. ^ Bobik, T.A. (2007), Bacterial Microcompartments, Microbe (Am. Soc. Microbiol.) 2: 25–31 
  12. ^ "Fuller and Sadao: Partners in Design". September 19, 2006. Retrieved 2010-01-26. 
  13. ^ Klein, Douglas J. (2002), Resistance-Distance Sum Rules (PDF), Croatica Chemica Acta 75 (2): 633–649, retrieved 2006-09-15 
  14. ^ Weisstein, Eric W., "Icosahedral Graph", MathWorld.

External links[edit]

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
Zeroth stellation of icosahedron.png First stellation of icosahedron.png Ninth stellation of icosahedron.png First compound stellation of icosahedron.png Second compound stellation of icosahedron.png Third compound stellation of icosahedron.png Sixteenth stellation of icosahedron.png Third stellation of icosahedron.png Seventeenth stellation of icosahedron.png
Zeroth stellation of icosahedron facets.png First stellation of icosahedron facets.png Ninth stellation of icosahedron facets.png First compound stellation of icosahedron facets.png Second compound stellation of icosahedron facets.png Third compound stellation of icosahedron facets.png Sixteenth stellation of icosahedron facets.png Third stellation of icosahedron facets.png Seventeenth stellation of icosahedron facets.png
The stellation process on the icosahedron creates a number of related polyhedra and compounds with icosahedral symmetry.