Icosahedron

Icosahedron
(3D model) (OFF file)
Rank	3
Type	Regular
Space	Spherical
Notation
Bowers style acronym	Ike
Coxeter diagram	o5o3x ()
Schläfli symbol	{3,5}
Elements
Faces	20 triangles
Edges	30
Vertices	12
Vertex figure	Pentagon, edge length 1
Measures (edge length 1)
Circumradius	$\sqrt{\frac{5+\sqrt5}{8}} \approx 0.95106$
Edge radius	$\frac{1+\sqrt5}{4} \approx 0.80902$
Inradius	$\frac{3\sqrt3+\sqrt{15}}{12} \approx 0.75576$
Volume	$5\frac{3+\sqrt5}{12} \approx 2.18169$
Dihedral angle	$\arccos\left(-\frac{\sqrt5}{3}\right) \approx 138.18969^\circ$
Central density	1
Number of pieces	20
Level of complexity	1
Related polytopes
Army	Ike
Regiment	Ike
Dual	Dodecahedron
Petrie dual	Petrial icosahedron
Conjugate	Great icosahedron
Abstract properties
Flag count	120
Net count	43380
Euler characteristic	2
Topological properties
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	H3, order 120
Convex	Yes
Nature	Tame

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:

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

Representations

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

• o5o3x (regular)
• o4s3s (B₃/2 symmetry, alternated truncated octahedron)
• s3s3s (A₃+, snub tetrahedron)
• oxoo5ooxo&#xt (H₂ axial, seen as gyroelongated pentagonal bipyramid)
• xofo3ofox&#xt (A₂ axial, face-first)
• xofox ofxfo&#xt (A₁×A₁ axial, edge-first)
• fxo ofx xof&#zx (A₁×A₁×A₁ subsymmetry)

In vertex figures

Icosahedra in vertex figures

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

Variations

An icosahedron has several variations as a snub polyhedron:

• Pyritohedral icosahedron - formed by alternating a truncated octahedron, has pyritohedral symmetry
• Snub tetrahedron - formed by alternating a great rhombitetratetrahedron, has chiral tetrahedral symmetry

Related polyhedra

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

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.

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.

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

• Snub disoctahedron (2)
• Snub icosicosahedron (5)

The icosahedron has a multitude of 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.

The great icosahedron is the icosahedron's conjugate, meaning it is isomorphic. The icosahedron and great icosahedron are only two of the many polyhedra made of equilateral triangles with that abstract structure. For example, one of the pyramids of the icosahedron can be inverted, producing an irregular polyhedron that is concave but with no intersections. More can be found here.

o5o3o truncations

Name	OBSA	Schläfli symbol	CD diagram	Picture
Dodecahedron	doe	{5,3}	x5o3o
Truncated dodecahedron	tid	t{5,3}	x5x3o
Icosidodecahedron	id	r{5,3}	o5x3o
Truncated icosahedron	ti	t{3,5}	o5x3x
Icosahedron	ike	{3,5}	o5o3x
Small rhombicosidodecahedron	srid	rr{5,3}	x5o3x
Great rhombicosidodecahedron	grid	tr{5,3}	x5x3x
Snub dodecahedron	snid	sr{5,3}	s5s3s

o3o3o truncations

Name	OBSA	Schläfli symbol	CD diagram	Picture
Tetrahedron	tet	{3,3}	x3o3o
Truncated tetrahedron	tut	t{3,3}	x3x3o
Tetratetrahedron = Octahedron	oct	r{3,3}	o3x3o
Truncated tetrahedron	tut	t{3,3}	o3x3x
Tetrahedron	tet	{3,3}	o3o3x
Small rhombitetratetrahedron = Cuboctahedron	co	rr{3,3}	x3o3x
Great rhombitetratetrahedron = Truncated octahedron	toe	tr{3,3}	x3x3x
Snub tetrahedron = Icosahedron	ike	sr{3,3}	s3s3s

External links

• Bowers, Jonathan. "Polyhedron Category 1: Regulars" (#5).

• Bowers, Jonathan. "Batch 2: Ike and Sissid Facetings" (#1 under ike).

• Klitzing, Richard. "Ike".

• Quickfur. "The Icosahedron".

• Nan Ma. "Icosahedron {3, 5}".

• McCooey, David. "Icosahedron"

• Hi.gher.Space Wiki Contributors. "Icosahedron".

• Wikipedia Contributors. "Regular icosahedron".