Göttingen Collection of Mathematical Models and Instruments

Icosahedron with inscribed icosidodecahedron

Model 481

Icosahedron with inscribed icosidodecahedron
B II 186


Icosahedron (black wire) with inscribed (12+20)-plane polygon with 30 vertices (red fiber), congruent to the exemplar in model no.480.


Icosahedron  The icosahedron is one of the five Platonic solids, see also model 702.

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

12 pentagons 20 + triangles = 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
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
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.


Showcase of this model is Case number U 7