Compound of dodecahedron and icosahedron

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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
IndexW47
Polyhedra1 icosahedron
1 dodecahedron
Faces20 triangles
12 pentagons
Edges60
Vertices32
Symmetry group icosahedral (Ih)

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

Contents

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.

Polyhedron 12.png
Polyhedron 20.png
A dodecahedron and its dual icosahedron
Polyhedron 12-20 blue.png
Polyhedron 12-20 dual blue.png
The intersection of both solids is the icosidodecahedron, and their convex hull is the rhombic triacontahedron.
Polyhedron pair 12-20 from blue.png
Polyhedron pair 12-20 from yellow.png
Polyhedron pair 12-20 from red.png
Seen from 2-fold, 3-fold and 5-fold symmetry axes
The decagon on the right is the Petrie polygon of both solids.
Polyhedron pair 12-20 big.png
Polyhedron small rhombi 12-20 dual max.png
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.png First stellation of icosidodecahedron pentfacets.png First stellation of icosidodecahedron.png

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