  Göttingen Collection of Mathematical Models and Instruments

# Tetrahedron with inscribed truncated tetrahedron

### Model 472

 Category: B II 177

### Description

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 474485 Cuboctahedron 474 inscribed in an octahedron485 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 480481 Icosidodecahedron 480 inscribed in a dodecahedron481 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

### References

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

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