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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 $ \{5, 3\} $, 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: $ \frac{\varphi\sqrt{3}}{2}l $
  • Edge radius: $ \frac{\varphi^2}{2}l $
  • Face radius: $ \frac{\sqrt{\frac{25+11\sqrt{5}}{10}}{2}}l $

Where $ \varphi $ is the golden ratio.

Angles

  • Dihedral angle: $ \arccos(\frac{\sqrt{5}}{5}) $

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

$ \{2,3\} $ $ \{3,3\} $ $ \{4,3\} $ $ \{5,3\} $ $ \{6,3\} $ $ \{7,3\} $ $ \{8,3\} $ ... $ \{\aleph_0,3\} $
Trigonal hosohedron Tetrahedron Cube Dodecahedron Hexagonal tiling Order-3 heptagonal tiling Order-3 octagonal tiling ... Order-3 apeirogonal tiling
$ \{5,2\} $ $ \{5,3\} $ $ \{5,4\} $ $ \{5,5\} $ $ \{5,6\} $ ... $ \{5,\aleph_0\} $
Pentagonal dihedron Dodecahedron Order-4 pentagonal tiling Order-5 pentagonal tiling Order-6 pentagonal tiling ... Infinite-order pentagonal 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 First Zeroth First Second Third Fourth Fifth Sixth Seventh Eighth Ninth Tenth Eleventh Twelfth Thirteenth Fourteenth Fifteenth Sixteenth ... Omegath
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

$ \{3^{\aleph_0}\} $

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

$ \{3^{\aleph_0}, 4\} $

Hydrotopes

$ \{3^{n-2}, 5\} $

Pentagon

$ \{5\} $

Icosahedron

$ \{3, 5\} $

Hexacosichoron

$ \{3^2, 5\} $

Order-5 pentachoric tetracomb

$ \{3^3, 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

$ \{4, 3^{\aleph_0}\} $

Cosmotopes

$ \{5, 3^{n-2}\} $

Pentagon

$ \{5\} $

Dodecahedron

$ \{5, 3\} $

Hecatonicosachoron

$ \{5, 3^2\} $

Order-3 hecatonicosachoric tetracomb

$ \{5, 3^3\} $

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} $