Dodecahedron with inscribed icosidodecahedron

Dodecahedron  The dodecahedron is one of the five Platonic solids, see also model 702.

Inscribed (12+20)-plane polygon with 30 vertices The solid inscribed in the dodecahedron is a icosidodecahedron. The icosidodecahedron arises from the dodecahedron by truncating its vertices. By truncating the vertices to the middle of the edges 20 new triangles and the previously existing 12 hexagons are transformed to 12 smaller hexagons. The icosidodecahedron is circumscribed by

20 triangles + 12 pentagons = 32 faces.

It has 30 vertices and 60 edges. At each vertice two triangles and two pentagons are meeting (3,3,5,5).

The icosidodecahedron is one of 13 Archimedean solids, see also 482.

rhombic triacontahedron The icosidodecahedron is polar (dual) to the rhombic triacontahedron, see also model 926. To create this new solid a sphere is inscribed in the icosidodecahedron such that the sphere touches each of the faces in exactly one point. These points of contact create the vertices of the dual solid. By connecting these 32 vertices 30 rhombi are formed. These rhombi are the faces of the rhombic triacontahedron. The number of edges remains the same during the transformation into the dual solid, while the number of vertices and faces is exchanged.

 

There are 11 Archimedean solids in the collection.

472	Truncated tetrahedron	inscribed in a tetrahedron
473	Truncated octahedron	inscribed in an octahedron
474 485	Cuboctahedron	474 inscribed in an octahedron 485 inscribed in a cube
475	Truncated cube	inscribed in an octahedron
476	Rhombicuboctahedron	inscribed in a cube
478	Truncated icosahedron	inscribed in an icosahedron
479	truncated dodecahedron	inscribed in a dodecahedron
480 481	Icosidodecahedron	480 inscribed in a dodecahedron 481 inscribed in an icosahedron
482	Rhombicosidodecahedron
483	Snub dodecahedron
484	Truncated cube	inscribed in a cube

The truncated icosidodecahedron and the lost truncated cuboctahedron are not included in the collection.