# Icosahedron

Icosahedron
Rank3
TypeRegular
Notation
Bowers style acronymIke
Coxeter diagramo5o3x ()
Schläfli symbol{3,5}
Conway notationI
Stewart notationI5
Elements
Faces20 triangles
Edges30
Vertices12
Vertex figurePentagon, edge length 1
Petrie polygons6 skew decagons
Holes12 pentagons
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {5+{\sqrt {5}}}{8}}}\approx 0.95106}$
Edge radius${\displaystyle {\frac {1+{\sqrt {5}}}{4}}\approx 0.80902}$
Inradius${\displaystyle {\frac {3{\sqrt {3}}+{\sqrt {15}}}{12}}\approx 0.75576}$
Volume${\displaystyle 5{\frac {3+{\sqrt {5}}}{12}}\approx 2.18169}$
Dihedral angle${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }}$
Central density1
Number of external pieces20
Level of complexity1
Related polytopes
ArmyIke
RegimentIke
DualDodecahedron
Petrie dualPetrial icosahedron
φ 2 Great dodecahedron
κ ?Petrial small stellated dodecahedron
ConjugateGreat icosahedron
Abstract & topological properties
Flag count120
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
SkeletonIcosahedral graph
Properties
SymmetryH3, order 120
Flag orbits1
ConvexYes
Net count43380
NatureTame

The icosahedron, or ike, is one of the five Platonic solids. It has 20 triangles as faces, joining 5 to a vertex.

An alternate, lower symmetry construction as a snub tetrahedron, furthermore relates the icosahedron to the snub polytopes, most notably to the snub disicositetrachoron, of which it is a cell.

It is the only Platonic solid that does not appear as a cell in one of the convex regular polychora, because its dihedral angle is more than 120° and thus 3 icosahedra cannot fit around an edge in 4D. It does, however, appear as the vertex figure of the hexacosichoron, and as the cell of the non-convex faceted hexacosichoron.

## Vertex coordinates

The vertices of an icosahedron of edge length 1, centered at the origin, are all cyclic permutations of:

• ${\displaystyle \left(0,\,\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}}\right)}$.

## Representations

A regular icosahedron can be represented by the following Coxeter diagrams:

## Related polytopes

The icosahedron is the colonel of a two-member regiment that also includes the great dodecahedron.

### Alternate realizations

The great icosahedron is the icosahedron's conjugate. Thus they are both faithful symmetric realizations of the same abstract regular polytope, {3,5}. There are in total 6 faithful symmetric realizations of the underlying abstract polytope. The icosahedron and the great icosahedron are the only pure faithfully symmetric realizations, the others are the results of blending those two along with the hemiicosahedron.

Faithful symmetric realizations of {3,5}
Dimension Components Name
3 Icosahedron Icosahedron
3 Great icosahedron Great icosahedron
6
Skew icosahedron
8
8
11

There are also realizations that are faithful but not symmetric. The particular case of 3-dimensional realizations with regular faces are called isomorphs. They have been investigated by Jim McNeill[1] and others. For example, one of the pyramids of the icosahedron can be inverted, producing an irregular polyhedron that is concave but with no intersections.

### Stellations

The icosahedron has many stellations, not all of which are true polyhedra. They include the small triambic icosahedron, medial triambic icosahedron, great triambic icosahedron, great icosahedron, chiricosahedron, icosicosahedron, small icosicosahedron, ditrigonal icosahedron and the final stellation of the icosahedron.

### Johnson solids

The icosahedron is related to many Johnson solids. Most obviously, it can be constructed by joining two pentagonal pyramids to a pentagonal antiprism. This means the icosahedron could also be called a gyroelongated pentagonal bipyramid. Joining a single pentagonal pyramid, or diminishing one vertex from the icosahedron, yields the gyroelongated pentagonal pyramid, and replacing the antiprism by a pentagonal prism yields the elongated pentagonal pyramid and the elongated pentagonal bipyramid. Cutting off two pyramids from two non-parallel, non-adjacent vertices yields the metabidiminished icosahedron, and cutting off a further non-adjacent pyramid yields the tridiminished icosahedron.

A much less obvious connection is with the hebesphenomegacorona, which may be derived from the icosahedron by expanding a single edge into a square, thus turning the two adjacent faces into squares as well. Similarly, if we take two opposite edges of the icosahedron and "stretch" them into squares via a partial Stott expansion, we obtain the bilunabirotunda.

### Compounds

Two uniform polyhedron compounds are composed of icosahedra, both using it in pyritohedral symmetry:

### Variations

An icosahedron has several variations as a snub polyhedron:

The icosahedron can be considered to be a snub triangular antiprism, by analogy with the snub disphenoid and snub square antiprism. This can be seen since the icosahedron can be constructed from the octahedron, that is a triangular antiprism, but cutting it into two halves and inserting a set of 12 triangles between the halves.

### In vertex figures

Icosahedra in vertex figures
Name Picture Schläfli symbol Edge length
Great stellated hecatonicosachoron {5/2,3,5} ${\displaystyle {\frac {{\sqrt {5}}-1}{2}}}$
Hexacosichoron {3,3,5} ${\displaystyle 1}$
Order-5 cubic honeycomb {4,3,5}
Order-5 dodecahedral honeycomb {5,3,5}
Order-5 hexagonal tiling honeycomb {6,3,5}