Compound of dodecahedron and icosahedron

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:

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.