Compound of dodecahedron and icosahedron

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Short description: Polyhedral compound
First stellation of icosidodecahedron
Compound of dodecahedron and icosahedron.png
Type Dual compound
Coxeter diagram CDel nodes 10ru.pngCDel split2-53.pngCDel node.pngCDel nodes 01rd.pngCDel split2-53.pngCDel node.png
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:

First stellation of icosidodecahedron facets.png300pxFirst stellation of icosidodecahedron.png

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