In
geometry, the small 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-intersecting. Each polyhedron can contain either star polygon faces, star polygon vertex figures, ...
. Its edges are doubled, making it degenerate. The star has 32 faces (20
triangles and 12
pentagon
In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
A pentagon may be simpl ...
s), 60 (doubled) edges and 12 vertices and 4 sharing faces. The faces in it are considered as two overlapping edges as topological polyhedron.
A small complex icosidodecahedron can be
constructed from a number of different
vertex figures.
A very similar figure emerges as a geometrical truncation of the
great stellated dodecahedron, where the pentagram faces become doubly-wound pentagons ( --> ), making the internal pentagonal planes, and the three meeting at each vertex become triangles, making the external triangular planes.
As a compound
The small complex icosidodecahedron can be seen as a
compound of the
icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons".
There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
and the
great dodecahedron where all vertices are precise and edges coincide. The small complex icosidodecahedron resembles an icosahedron, because the great dodecahedron is completely contained inside the icosahedron.
Its two-dimensional analogue would be the compound of a regular
pentagon
In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
A pentagon may be simpl ...
, , representing the icosahedron as the ''n''-dimensional
pentagonal polytope, and regular
pentagram, , as the ''n''-dimensional star. These shapes would share vertices, similarly to how its 3D equivalent shares edges.
See also
*
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 ...
*
Small complex rhombicosidodecahedron
*
Complex rhombidodecadodecahedron
*
Great complex rhombicosidodecahedron
References
* (Table 6, degenerate cases)
*
* {{KlitzingPolytopes, polyhedra-neu.htm, 3D uniform polyhedra, x3/2o5o5*a - cid
Polyhedra
Polyhedral compounds