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)

In geometry, this polyhedron can be seen as either a polyhedral stellation or a compound.

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.
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 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

Allikas: http://en.wikipedia.org/wiki/Compound_of_dodecahedron_and_icosahedron
No tags for this post.