Verse and Dimensions Wikia
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Verse and Dimensions Wikia

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

Subfacets

Radii

  • Vertex radius:
  • Edge radius:
  • Face radius:

Angles

  • dihedral angle:

Vertex coordinates

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

Related shapes

See also

Regular polyhedra
Convex regular polyhedra: tet · cube · oct · doe · ike

Self-intersecting regular polyhedra: gad · sissid · gike · gissid

...
Trigonal dihedron Tetrahedron Octahedron Icosahedron Triangular tiling Order-7 triangular tiling Order-8 triangular tiling ... Infinite-order triangular tiling Imaginary-order triangular tiling
...
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
Rectified
Birectified
Truncated
Bitruncated
Cantellated
Cantitruncated
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

Null polytope


Point


Line segment


Triangle

Tetrahedron

Pentachoron

Hexateron

Heptapeton

Octaexon

Enneazetton

Decayotton

Hendecaxennon

Dodecadakon

Tridecahendon

Tetradecadokon

Pentadecatradakon

Hexadecatedakon

Heptadecapedakon

... Omegasimplex
Cross

Square

Octahedron

Hexadecachoron

Pentacross

Hexacross

Heptacross

Octacross

Enneacross

Dekacross

Hendekacross

Dodekacross

Tridekacross

Tetradekacross

Pentadekacross

Hexadekacross

... Omegacross
Hydrotopes

Pentagon

Icosahedron

Hexacosichoron

Order-5 pentachoric tetracomb

Order-5 hexateric pentacomb

...
Hypercube

Square

Cube

Tesseract

Penteract

Hexeract

Hepteract

Octeract

Enneract

Dekeract

Hendekeract

Dodekeract

Tridekeract

Tetradekeract

Pentadekeract

Hexadekeract

... Omegeract
Cosmotopes

Pentagon

Dodecahedron

Hecatonicosachoron

Order-3 hecatonicosachoric tetracomb

Order-3-3 hecatonicosachoric pentacomb

...
Hyperball

Disk

Ball

Gongol

Pentorb

Hexorb

Heptorb

Octorb

Enneorb

Dekorb

Hendekorb

Dodekorb

Tridekorb

Tetradekorb

Pentadekorb

Hexadekorb

... Omegaball

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