Great dodecahedron

Great dodecahedron
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

In geometry, the great dodecahedron is one of four Kepler–Poinsot polyhedrons. 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.

Construction

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]

Formulas

Given a great dodecahedron with edge length $a$ . The circumradius of a great dodecahedron $C$ is: $C={\frac {a}{4}}\left({\sqrt {10+2+{\sqrt {5}}}}\right).$ Its surface area $A$ is: $A=15a^{2}\left({\sqrt {5-2{\sqrt {5}}}}\right).$ Its volume $V$ is:^[5] $V={\frac {5}{4}}({\sqrt {5}}-1)a^{3}.$

Appearance

Great dodecahedron in Perspectiva Corporum Regularium

Alexander's Star in solved state

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. It was invented by an American mathematician, Adam Alexander.^{[citation needed]}

Related polyhedra

Great dodecahedron shown solid, surrounding stellated dodecahedron only as wireframe

Animated truncation sequence from {5/2, 5} to {5, 5/2}

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.