Great dodecahedron

Great dodecahedron

Type	Kepler–Poinsot polyhedron
Stellation core	regular dodecahedron
Elements	F = 12, E = 30 V = 12 (χ = -6)
Faces by sides	12{5}
Schläfli symbol	{5,5⁄2}
Face configuration	V(5⁄2)⁵
Wythoff symbol	5⁄2 \| 2 5
Coxeter diagram
Symmetry group	I_h, H₃, [5,3], (*532)
References	U₃₅, C₄₄, W₂₁
Properties	Regular nonconvex
(5⁵)/2 (Vertex figure)	Small stellated dodecahedron (dual polyhedron)

In geometry, the great dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol ${5,5/2}$ and Coxeter–Dynkin diagram of . It is one of four nonconvex regular 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.

The discovery of the great dodecahedron is sometimes credited to Louis Poinsot in 1810, though there is a drawing of something very similar to a great dodecahedron in the 1568 book Perspectiva Corporum Regularium by Wenzel Jamnitzer.

The great dodecahedron can be constructed analogously to the pentagram, its two-dimensional analogue, via the extension of the $(n - 1)$ -pentagonal polytope faces of the core $n$ -polytope (pentagons for the great dodecahedron, and line segments for the pentagram) until the figure again closes.

Images edit

Transparent model	Spherical tiling
(With animation)	This polyhedron represents a spherical tiling with a density of 3. (One spherical pentagon face is shown above in yellow)
Net	Stellation
× 20 Net for surface geometry; twenty isosceles triangular pyramids, arranged like the faces of an icosahedron	It can also be constructed as the second of three stellations of the dodecahedron, and referenced as Wenninger model [W21].

Formulas edit

For a great dodecahedron with edge length E,

{\text{Circumradius}}={\tfrac {E}{4}}{\Bigl (}{\sqrt {10+2+{\sqrt {5}}}}{\Bigr )}

{\text{Surface Area}}=15{\Bigl (}{\sqrt {5-2{\sqrt {5}}}}{\Bigr )}E^{2}

{\text{Volume}}={\tfrac {5}{4}}({\sqrt {5}}-1)E^{3}

Related polyhedra edit

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

If only the visible surface is considered, it has the same topology as a triakis icosahedron with concave pyramids rather than convex ones. The excavated dodecahedron can be seen as the same process applied to a regular dodecahedron, although this result is not regular.

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.