Compound of dodecahedron and icosahedron
Polyhedral compound From Wikipedia, the free encyclopedia
In geometry, this polyhedron can be seen as either a polyhedral stellation or a compound.
| First stellation of icosidodecahedron | |
|---|---|
| |
| Type | Dual compound |
| Coxeter diagram |    ∪  |
| Stellation core | icosidodecahedron |
| Convex hull | Rhombic triacontahedron |
| Index | W47 |
| Polyhedra | 1 icosahedron 1 dodecahedron |
| Faces | 20 triangles 12 pentagons |
| Edges | 60 |
| Vertices | 32 |
| Symmetry group | icosahedral (Ih) |
As a compound
It can be seen as the compound of an icosahedron and dodecahedron. It is one of four compounds constructed from a Platonic solid or Kepler-Poinsot solid, and its dual.
It has icosahedral symmetry (Ih) and the same vertex arrangement as a rhombic triacontahedron.
This can be seen as the three-dimensional equivalent of the compound of two pentagons ({10/2} "decagram"); this series continues into the fourth dimension as the compound of 120-cell and 600-cell and into higher dimensions as compounds of hyperbolic tilings.
A dodecahedron and its dual icosahedron |
The intersection of both solids is the icosidodecahedron, and their convex hull is the rhombic triacontahedron. |
Seen from 2-fold, 3-fold and 5-fold symmetry axes
The decagon on the right is the Petrie polygon of both solids.
The decagon on the right is the Petrie polygon of both solids.
If the edge crossings were vertices, the mapping on a sphere would be the same as that of a deltoidal hexecontahedron.
As a stellation
This polyhedron is the first stellation of the icosidodecahedron, and given as Wenninger model index 47.
The stellation facets for construction are:
As a Faceting
The compound of a Dodecahedron and an Icosahedron shares the same vertices as a list of other polyhedra, including the Rhombic triacontahedron and the Small triambic icosahedron.
In popular culture
In the film Tron (1982), the character Bit took this shape when not speaking.
In the cartoon series Steven Universe (2013-2019), Steven's shield bubble, briefly used in the episode Change Your Mind, had this shape.
See also
References
- Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9.
External links
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