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In geometry, the **great dodecahedron** is one of four Kepler–Poinsot polyhedra. It is composed of 12 pentagonal faces (six pairs of parallel pentagons), intersecting each other making a pentagrammic path, with five pentagons meeting at each vertex.

Great dodecahedron | |
---|---|

Type | Kepler–Poinsot polyhedron |

Faces | 12 |

Edges | 30 |

Vertices | 12 |

Symmetry group | icosahedral symmetry |

Dual polyhedron | small stellated dodecahedron |

Properties | regular, non-convex |

Vertex figure | |

One way to construct a great dodecahedron is by faceting the regular icosahedron. In other words, it is constructed from the regular icosahedron by removing its polygonal faces without changing or creating new vertices.^{[1]} Another way is to form a regular pentagon by each of the five vertices inside of a regular icosahedron, and twelve regular pentagons intersecting each other, making a pentagram as its vertex figure.^{[2]}^{[3]}

The great dodecahedron may also be interpreted as the *second stellation of dodecahedron*. The construction started from a regular dodecahedron by attaching 12 pentagonal pyramids onto each of its faces, known as the *first stellation*. The second stellation appears when 30 wedges are attached to it.^{[4]}

Given a great dodecahedron with edge length . The circumradius of a great dodecahedron is:
Its surface area is:
Its volume is:^{[5]}

Historically, the great dodecahedron is one of two solids discovered by Louis Poinsot in 1810, with some people named it after him, *Poinsot solid*. As for the background, Poinsot rediscovered two other solids that were already discovered by Johannes Kepler—the small stellated dodecahedron and the great stellated dodecahedron.^{[3]} However, the great dodecahedron appeared in the 1568 *Perspectiva Corporum Regularium* by Wenzel Jamnitzer, although its drawing is somewhat similar.^{[6]}

The great dodecahedron appeared in popular culture and toys. An example is Alexander's Star puzzle, a Rubik's Cube that is based on a great dodecahedron.^{[7]}

The *compound of small stellated dodecahedron and great dodecahedron* is a polyhedron compound where the great dodecahedron is internal to its dual, the small stellated dodecahedron. This can be seen as one of the two three-dimensional equivalents of the compound of two pentagrams ({10/4} "decagram"); this series continues into the fourth dimension as compounds of star 4-polytopes.

A truncation process applied to the great dodecahedron produces a series of nonconvex uniform polyhedra. Truncating edges down to points produces the dodecadodecahedron as a rectified great dodecahedron. The process completes as a birectification, reducing the original faces down to points, and producing the small stellated dodecahedron.

It shares the same edge arrangement as the convex regular icosahedron; the compound with both is the small complex icosidodecahedron.

**^**Inchbald, Guy (2006). "Facetting Diagrams".*The Mathematical Gazette*.**90**(518): 253–261. JSTOR 40378613.**^**Pugh, Anthony (1976).*Polyhedra: A Visual Approach*. University of California Press. p. 85.- ^
^{a}^{b}Barnes, John (2012).*Gems of Geometry*(2nd ed.). Springer. p. 46. doi:10.1007/978-3-642-30964-9. ISBN 978-3-642-30964-9. **^**Cromwell, Peter (1997).*Polyhedra*. Cambridge University Press. p. 265.**^**French, Doug; Jordan, David (2010). "Dodecahedral slices and polyhedral pieces".*The Mathematical Gazette*.**92**(529): 5–17. JSTOR 27821883.**^**Scriba, Christoph; Schreiber, Peter (2015).*5000 Years of Geometry: Mathematics in History and Culture*. Springer. p. 305. doi:10.1007/978-3-0348-0898-9. ISBN 978-3-0348-0898-9.**^**"Alexander's star".*Games*. No. 32. October 1982. p. 56.

- Weisstein, Eric W., "Great dodecahedron" ("Uniform polyhedron") at
*MathWorld*. - Weisstein, Eric W. "Three dodecahedron stellations".
*MathWorld*. - Uniform polyhedra and duals
- Metal sculpture of Great Dodecahedron