geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the great dodecahedron is one of four Kepler–Poinsot polyhedra. It is composed of 12

pentagon In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°. A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...

al faces (six pairs of parallel pentagons), intersecting each other making a

pentagram A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle around ...

mic path, with five pentagons meeting at each vertex.

Construction

One way to construct a great dodecahedron is by faceting the

regular icosahedron The regular icosahedron (or simply ''icosahedron'') is a convex polyhedron that can be constructed from pentagonal antiprism by attaching two pentagonal pyramids with Regular polygon, regular faces to each of its pentagonal faces, or by putting ...

. In other words, it is constructed from the regular icosahedron by removing its polygonal faces without changing or creating new vertices. For each vertex of the icosahedron, the five neighboring vertices become those of a regular pentagon face of the great dodecahedron. The resulting shape has a

as its

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...

, so its

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...

\

. 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.

Formulas

Given a great dodecahedron with edge length

E

\text = \frac\,E

\text = \frac\,E

\text = \frac\,E

\text = 15\sqrt\,E^2

\text = \frac\,E^3

Appearance

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 Johannes Kepler (27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, Natural philosophy, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best know ...

—the

small stellated dodecahedron In geometry, the small stellated dodecahedron is a Kepler–Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol . It is one of four nonconvex List of regular polytopes#Non-convex 2, regular polyhedra. It is composed of 12 pentag ...

and the great stellated dodecahedron. However, the great dodecahedron appeared in the 1568 '' Perspectiva Corporum Regularium'' by Wenzel Jamnitzer, although its drawing is somewhat similar. The great dodecahedron appeared in popular culture and toys. An example is Alexander's Star puzzle, a twisty puzzle that is based on a great dodecahedron.

Related polyhedra

The ''compound of small stellated dodecahedron and great dodecahedron'' is a polyhedron compound where the great dodecahedron is internal to its dual, the

. This can be seen as one of the two three-dimensional equivalents of the compound of two pentagrams ( " decagram"); this series continues into the fourth dimension as compounds of star 4-polytopes. A

truncation In mathematics and computer science, truncation is limiting the number of digits right of the decimal point. Truncation and floor function Truncation of positive real numbers can be done using the floor function. Given a number x \in \mathbb ...

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

. It shares the same edge arrangement as the convex regular

icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical tha ...

; the compound with both is the small complex icosidodecahedron.

References

External links

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Uniform polyhedra and duals

{{Star polyhedron navigator Kepler–Poinsot polyhedra Regular polyhedra Polyhedral stellation Toroidal polyhedra