![Great dodecahedron](https://upload.wikimedia.org/wikipedia/commons/c/c5/Great_dodecahedron.stl)
In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the great dodecahedron is a
Kepler–Poinsot polyhedron
In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra.
They may be obtained by stellating the regular convex dodecahedron and icosahedron, and differ from these in having regular pentagrammic faces or vertex figures. ...
, with
Schläfli symbol and
Coxeter–Dynkin diagram
In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes). It describe ...
of . It is one of four
nonconvex
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...
regular polyhedra
A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equival ...
. It is composed of 12
pentagonal 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 arou ...
mic path, with five pentagons meeting at each
vertex
Vertex, vertices or vertexes may refer to:
Science and technology Mathematics and computer science
*Vertex (geometry), a point where two or more curves, lines, or edges meet
* Vertex (computer graphics), a data structure that describes the positio ...
.
The discovery of the great dodecahedron is sometimes credited to
Louis Poinsot
Louis Poinsot (3 January 1777 – 5 December 1859) was a French mathematician and physicist. Poinsot was the inventor of geometrical mechanics, showing how a system of forces acting on a rigid body could be resolved into a single force and a c ...
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
Wenzel Jamnitzer (sometimes Jamitzer, or Wenzel ''Gemniczer'') (1507/1508 – 19 December 1585) was a Northern Mannerist goldsmith, artist, and printmaker in etching, who worked in Nuremberg. He was the best known German goldsmith of his e ...
.
The great dodecahedron can be constructed analogously to the pentagram, its two-dimensional analogue, via the extension of the -
pentagonal polytope
In geometry, a pentagonal polytope is a regular polytope in ''n'' dimensions constructed from the H''n'' Coxeter group. The family was named by H. S. M. Coxeter, because the two-dimensional pentagonal polytope is a pentagon. It can be named by i ...
faces of the core -polytope (pentagons for the great dodecahedron, and line segments for the pentagram) until the figure again closes.
Images
Related polyhedra
![Small stellated dodecahedron truncations](https://upload.wikimedia.org/wikipedia/commons/d/d8/Small_stellated_dodecahedron_truncations.gif)
It shares the same
edge arrangement
In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes.
For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equa ...
as the convex regular
icosahedron; the compound with both is the
small complex icosidodecahedron
In geometry, the small complex icosidodecahedron is a degenerate uniform star polyhedron. Its edges are doubled, making it degenerate. The star has 32 faces (20 triangles and 12 pentagons), 60 (doubled) edges and 12 vertices and 4 sharing faces. ...
.
If only the visible surface is considered, it has the same topology as a
triakis icosahedron
In geometry, the triakis icosahedron (or kisicosahedronConway, Symmetries of things, p.284) is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated dodecahedron.
Cartesian coordinates
Let \phi be the golden ratio. The 12 p ...
with concave pyramids rather than convex ones. The
excavated dodecahedron
In geometry, the excavated dodecahedron is a star polyhedron that looks like a dodecahedron with concave pentagonal pyramids in place of its faces. Its exterior surface represents the Ef1g1 stellation of the icosahedron. It appears in Magnus We ...
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
In geometry, the dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U36. It is the rectification of the great dodecahedron (and that of its dual, the small stellated dodecahedron). It was discovered independently by , and .
The ed ...
as a rectified great dodecahedron. The process completes as a birectification, reducing the original faces down to points, and producing 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 regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeti ...
.
Usage
* This shape was the basis for the
Rubik's Cube-like
Alexander's Star
Alexander's Star is a puzzle similar to the Rubik's Cube, in the shape of a great dodecahedron.
History
Alexander's Star was invented by Adam Alexander, an American mathematician, in 1982. It was patented on 26 March 1985, with US patent n ...
puzzle.
* The great dodecahedron provides an easy mnemonic for the
binary Golay code[* Baez, John]
Golay code
" ''Visual Insight'', December 1, 2015.
See also
*
References
External links
*
*
Uniform polyhedra and duals
{{Star polyhedron navigator
Kepler–Poinsot polyhedra
Regular polyhedra
Polyhedral stellation
Toroidal polyhedra