In geometry, the small complex rhombicosidodecahedron (also known as the small complex ditrigonal rhombicosidodecahedron) is a
degenerate
Degeneracy, degenerate, or degeneration may refer to:
Arts and entertainment
* ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed
* Degenerate art, a term adopted in the 1920s by the Nazi Party in Germany to descr ...
uniform star polyhedron
In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting. Each polyhedron can contain either star polygon faces, star polygon vertex figure ...
. It has 62 faces (20
triangles, 12
pentagrams and 30
squares), 120 (doubled) edges and 20 vertices. All edges are doubled (making it degenerate), sharing 4 faces, but are considered as two overlapping edges as a
topological polyhedron.
It can be constructed from the vertex figure 3(
5/
2.4.3.4), thus making it also a
cantellated great icosahedron. The "3" in front of this vertex figure indicates that each vertex in this degenerate polyhedron is in fact three
coincident vertices. It may also be given the
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...
rr or t
0,2.
As a compound
It can be seen as a
compound of the
small ditrigonal icosidodecahedron
In geometry, the small ditrigonal icosidodecahedron (or small ditrigonary icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U30. It has 32 faces (20 triangles and 12 pentagrams), 60 edges, and 20 vertices. It has extended Schläfl ...
, U
30, and the
compound of five cubes. It is also a
faceting of the
dodecahedron.
As a cantellation
It can also be seen as a
cantellation of the
great icosahedron (or, equivalently, of the
great stellated dodecahedron).
Related degenerate uniform polyhedra
Two other degenerate uniform polyhedra are also facettings of the dodecahedron. They are the complex rhombidodecadodecahedron (a compound of the
ditrigonal dodecadodecahedron
In geometry, the ditrigonal dodecadodecahedron (or ditrigonary dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U41. It has 24 faces (12 pentagons and 12 pentagrams), 60 edges, and 20 vertices. It has extended Schläfli symbol ...
and the compound of five cubes) with vertex figure (.4.5.4)/3 and the great complex rhombicosidodecahedron (a compound of the
great ditrigonal icosidodecahedron
In geometry, the great ditrigonal icosidodecahedron (or great ditrigonary icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U47. It has 32 faces (20 triangles and 12 pentagons), 60 edges, and 20 vertices. It has 4 Schwarz triangle ...
and the compound of five cubes) with vertex figure (.4..4)/3. All three degenerate uniform polyhedra have each vertex in fact being three coincident vertices and each edge in fact being two coincident edges.
They can all be constructed by
cantellation of regular polyhedra. The complex rhombidodecadodecahedron may be given the Schläfli symbol rr or t
0,2, while the great complex rhombicosidodecahedron may be given the Schläfli symbol rr or t
0,2.
See also
*
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. ...
*
Great complex icosidodecahedron
In geometry, the great complex icosidodecahedron is a degenerate uniform star polyhedron
In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-interse ...
*
Complex rhombidodecadodecahedron
*
Great complex rhombicosidodecahedron
References
*
*
* {{KlitzingPolytopes, polyhedra-neu.htm, 3D uniform polyhedra, gicdatrid
Polyhedra
Polyhedral compounds