Verse and Dimensions Wikia
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Verse and Dimensions Wikia


A dodecahedron is a regular polyhedron, one of the five three-dimensional platonic solids, and has twelve congruent pentagonal faces. It has a schläfli symbol of , meaning that 3 pentagons join at each vertex. It is the dual of the icosahedron.

Under the elemental naming scheme, it is called a cosmogon.

Properties

It is possible to inscribe 5 cubes into a dodecahedron such that all of the edges of the cubes are diagonals of the dodecahedron's faces.

It is also the only platonic solid to have pentagonal faces.

Symbols

Symbols for the dodecahedron include:

  • x5o3o (regular)
  • xfoo5oofx&Ext (pentagon symmetry)
  • ofxfoo3oofxfo&#xt (triangular symmetry)
  • xfoFofx ofFxFxo&#xt (digonal symmetry)
  • oxfF xFfo Fofx&#zx (block symmetry)

Subfacets and Structure

Structure

The dodecahedron has 12 pentagonal faces, joining three to a vertex. When seen face first, the layers start with one face, then the five surrounding it, then five more, and finally the opposite face.

Hypervolumes

Subfacets

Radii

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

Where is the golden ratio.

Angles

  • Dihedral angle:

vertex coordinates

The vertex coordinates of a dodecahedron of side 2 are:

  • (±φ,±φ,±φ)
  • (±φ^2,±1,0) and all even permutations

where φ = (1+√5)/2.

Related shapes

See Also

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

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

...
Trigonal hosohedron Tetrahedron Cube Dodecahedron Hexagonal tiling Order-3 heptagonal tiling Order-3 octagonal tiling ... Order-3 apeirogonal tiling Order-3 pseudogonal tiling
...
Pentagonal dihedron Dodecahedron Order-4 pentagonal tiling Order-5 pentagonal tiling Order-6 pentagonal tiling Order-7 pentagonal tiling Order-8 pentagonal tiling ... Infinite-order pentagonal tiling Imaginary-order pentagonal 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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