Icosahedron

An icosahedron is a three-dimensional regular polyhedron with 20 faces, each of which is triangular. There are five faces to each vertex. It is called a hydrohedron under the elemental system, and its Bowers name is ike.

Symbols

Dynkin based symbols of the icosahedron include:

o5o3x (regular)
o4s3s (snub octahedron)
s3s3s (snub tetratetrahedron)
oxoo5ooxo&#xt (pentagonal symmetry)
xofo3ofox&#xt (triangular symmetry)
xofox ofxfo&#xt (digonal symmetry)
fxo ofx xof&#zx (block symmetry)

Structure and Subfacets

The icosahedron can be decomposed as a blend of two pentagonal pyramids and a pentagonal antiprism. It is also constructible as a snubbed tetrahedron.

Hypervolumes

vertex count = $12$
edge length = $30l$
surface area = $5{\sqrt {3}}{l}^{2}$
surcell volume = ${\frac {5\left(3+{\sqrt {5}}\right)}{12}}{l}^{3}$

Subfacets

12 points (0D)
30 line segments (1D)
20 triangles (2D)
1 icosahedron (3D)

Radii

Vertex radius: ${\frac {\sqrt {\frac {5+{\sqrt {5}}}{2}}}{2}}l$
Edge radius: ${\frac {\varphi }{2}}l$
Face radius: ${\frac {\sqrt {\frac {7+3{\sqrt {5}}}{6}}}{2}}l$

Angles

dihedral angle: $\arccos({\frac {\sqrt {5}}{3}})$

Vertex coordinates

The vertex coordinates of a regular icosahedron of side 2 are all even permutations of (±φ,±1,0).

Related shapes

Dual: dodecahedron
Vertex figure: Pentagon, edge length 1
Conjugate: Great icosahedron
Caps: Pentagonal pyramid (vertex first)
Diminishings: Pentagonal antiprisms (two opposite vertices diminished), Gyroelongated pentagonal pyramid (single diminishing), metabidiminished icosahedron, tridiminished icosahedron
Regiment members: 2 (other member: Great dodecahedron)


$\{3,2\}$	$\{3,3\}$	$\{3,4\}$	$\{3,5\}$	$\{3,6\}$	$\{3,7\}$	$\{3,8\}$	...	$\{3,\infty \}$	$\{3,{\frac {\pi i}{\lambda }}\}$
Trigonal dihedron	Tetrahedron	Octahedron	Icosahedron	Triangular tiling	Order-7 triangular tiling	Order-8 triangular tiling	...	Infinite-order triangular tiling	Imaginary-order triangular tiling


$\{2,5\}$	$\{3,5\}$	$\{4,5\}$	$\{5,5\}$	$\{6,5\}$	$\{7,5\}$	$\{8,5\}$	...	$\{\infty ,5\}$	$\{{\frac {\pi i}{\lambda }},5\}$
Pentagonal hosohedron	Icosahedron	Order-5 square tiling	Order-5 pentagonal tiling	Order-5 hexagonal tiling	Order-5 heptagonal tiling	Order-5 octagonal tiling	...	Order-5 apeirogonal tiling	Order-5 pseudogonal tiling


Regular $t_{0}\{5,3\}$	Rectified $t_{1}\{5,3\}$	Birectified $t_{2}\{5,3\}$	Truncated $t_{0,1}\{5,3\}$	Bitruncated $t_{1,2}\{5,3\}$	Cantellated $t_{0,2}\{5,3\}$	Cantitruncated $t_{0,1,2}\{5,3\}$
Dodecahedron	Icosidodecahedron	Icosahedron	Truncated dodecahedron	Truncated icosahedron	Rhombicosidodecahedron	Great rhombicosidodecahedron

Dimensionality	Negative One	Zero	One	Two	Three	Four	Five	Six	Seven	Eight	Nine	Ten	Eleven	Twelve	Thirteen	Fourteen	Fifteen	Sixteen	...	Aleph null
Simplex $\{3^{n-1}\}$	Null polytope $)($ $\emptyset$	Point $()$ $\mathbb {B} ^{0}$	Line segment $\{\}$ $\mathbb {B} ^{1}$	Triangle $\{3\}$	Tetrahedron $\{3^{2}\}$	Pentachoron $\{3^{3}\}$	Hexateron $\{3^{4}\}$	Heptapeton $\{3^{5}\}$	Octaexon $\{3^{6}\}$	Enneazetton $\{3^{7}\}$	Decayotton $\{3^{8}\}$	Hendecaxennon $\{3^{9}\}$	Dodecadakon $\{3^{10}\}$	Tridecahendon $\{3^{11}\}$	Tetradecadokon $\{3^{12}\}$	Pentadecatradakon $\{3^{13}\}$	Hexadecatedakon $\{3^{14}\}$	Heptadecapedakon $\{3^{15}\}$	...	Omegasimplex
Cross $\{3^{n-2},4\}$				Square $\{4\}$	Octahedron $\{3,4\}$	Hexadecachoron $\{3^{2},4\}$	Pentacross $\{3^{3},4\}$	Hexacross $\{3^{4},4\}$	Heptacross $\{3^{5},4\}$	Octacross $\{3^{6},4\}$	Enneacross $\{3^{7},4\}$	Dekacross $\{3^{8},4\}$	Hendekacross $\{3^{9},4\}$	Dodekacross $\{3^{10},4\}$	Tridekacross $\{3^{11},4\}$	Tetradekacross $\{3^{12},4\}$	Pentadekacross $\{3^{13},4\}$	Hexadekacross $\{3^{14},4\}$	...	Omegacross
Hydrotopes $\{3^{n-2},5\}$				Pentagon $\{5\}$	Icosahedron $\{3,5\}$	Hexacosichoron $\{3^{2},5\}$	Order-5 pentachoric tetracomb $\{3^{3},5\}$	Order-5 hexateric pentacomb $\{3^{4},5\}$	...
Hypercube $\{4,3^{n-2}\}$				Square $\{4\}$	Cube $\{4,3\}$	Tesseract $\{4,3^{2}\}$	Penteract $\{4,3^{3}\}$	Hexeract $\{4,3^{4}\}$	Hepteract $\{4,3^{5}\}$	Octeract $\{4,3^{6}\}$	Enneract $\{4,3^{7}\}$	Dekeract $\{4,3^{8}\}$	Hendekeract $\{4,3^{9}\}$	Dodekeract $\{4,3^{10}\}$	Tridekeract $\{4,3^{11}\}$	Tetradekeract $\{4,3^{12}\}$	Pentadekeract $\{4,3^{13}\}$	Hexadekeract $\{4,3^{14}\}$	...	Omegeract
Cosmotopes $\{5,3^{n-2}\}$				Pentagon $\{5\}$	Dodecahedron $\{5,3\}$	Hecatonicosachoron $\{5,3^{2}\}$	Order-3 hecatonicosachoric tetracomb $\{5,3^{3}\}$	Order-3-3 hecatonicosachoric pentacomb $\{5,3^{4}\}$	...
Hyperball $\mathbb {B} ^{n}$				Disk $\mathbb {B} ^{2}$	Ball $\mathbb {B} ^{3}$	Gongol $\mathbb {B} ^{4}$	Pentorb $\mathbb {B} ^{5}$	Hexorb $\mathbb {B} ^{6}$	Heptorb $\mathbb {B} ^{7}$	Octorb $\mathbb {B} ^{8}$	Enneorb $\mathbb {B} ^{9}$	Dekorb $\mathbb {B} ^{10}$	Hendekorb $\mathbb {B} ^{11}$	Dodekorb $\mathbb {B} ^{12}$	Tridekorb $\mathbb {B} ^{13}$	Tetradekorb $\mathbb {B} ^{14}$	Pentadekorb $\mathbb {B} ^{15}$	Hexadekorb $\mathbb {B} ^{16}$	...	Omegaball $\mathbb {B} ^{\aleph _{0}}$

Icosahedron

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