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.

## 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

- 20 points (0D)
- 30 line segments (1D)
- 12 pentagons (2D)
- 1 dodecahedron (3D)

## See Also

... | ||||||||
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Trigonal hosohedron | Tetrahedron | Cube | Dodecahedron
| Hexagonal tiling | Order-3 heptagonal tiling | Order-3 octagonal tiling | ... | Order-3 apeirogonal tiling |

... | ||||||
---|---|---|---|---|---|---|

Pentagonal dihedron | Dodecahedron
| Order-4 pentagonal tiling | Order-5 pentagonal tiling | Order-6 pentagonal tiling | ... | Infinite-order pentagonal tiling |

Regular | Rectified | Birectified | Truncated | Bitruncated | Cantellated | Cantitruncated |
---|---|---|---|---|---|---|

Dodecahedron
| Icosidodecahedron | Icosahedron | Truncated dodecahedron | Truncated icosahedron | Rhombicosidodecahedron | Great rhombicosidodecahedron |

Zeroth | First | Second | Third | Fourth | Fifth | |
---|---|---|---|---|---|---|

Cosmotopes | Point | Line segment | Pentagon | Dodecahedron
| Hecatonicosachoron | Order-3 Hecatonicosachoron honeycomb |

Hydrotopes | Point | Line segment | Pentagon | Icosahedron | Hexacosichoron | Order-5 Pentachoron honeycomb |