Göttingen Collection of Mathematical Models and Instruments

Tetrahedron with inscribed truncated tetrahedron

Model 472

B II 177


Tetrahedron (wire) with inscribed (4+4)-plane polygon with 4·3 vertices (brown) that is polar to the triakis tetrahedron.


Tetrahedron: The tetrahedron is one of the five Platonic solids, see also model 702.

Inscribed (4+4)-plane polygon with 4 · 3 vertices: The solid inscribed in the tetrahedron is a truncated tetrahedron. The truncated tetrahedron arises from the tetrahedron by truncating its vertices. By truncating the vertices four new triangles are created and the previously existing triangles are transformed to four hexagons. The truncated tetrahedron is bounded by

4 triangles + 4 hexagons = 8 faces.

It has 4 · 3 = 12 vertices and 18 edges. At each vertice one triangle and two hexagons are meeting (3,6,6).

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

Triakis tetrahedron: The truncated tetrahedron is polar (dual) to the triakis tetrahedron. To create this new solid inscribe a sphere in the truncated tetrahedron 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 eight vertices 12 triangles are formed. These triangles are the faces of the triakis tetrahedron. The dual solid has the same number of edges, 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 20


Heesch, H.. Comm.Math.Helv, 6, p. 147, n=3, Case 1c.

Hess, E.(1883). Kugelteilung, Teubner, mit Figuren, p. 55.