Truncated tetrahedron

The tetrahedron truncated ( truncated tetrahedron or Friauf polyhedron ) is a polyhedron ( polyhedron ), which is among the Archimedean bodies and is created by blunting of the corners of a tetrahedron. Instead of the four corners of the tetrahedron are now there for four equilateral triangles, the triangular faces of the tetrahedron are to regular hexagons.

For the truncated tetrahedron, the special feature is that all lines connecting the centers of adjacent surfaces are equal: they all have the length a and form three equal tetrahedra which fully enclose a fourth of equal size.

The tetrahedron for Stump dual body is the Triakistetraeder.

Formulas

Cartesian coordinates

The Cartesian coordinates of the vertices may be denominated in center at the origin:

And of these 24 coordinates are those select 12 that have an odd number of plus signs ( 1 or 3) and thus a straight at minus sign ( 2 or 0), or vice versa.

Friauf polyhedra

The name Friauf polyhedra for the tetrahedron stump goes back to the chemist James B. Friauf, who described this polyhedron as the basis of the structure of MgZn2. The Friauf polyhedron is a typical coordination polyhedra with coordination number 12 in intermetallic compounds such as Laves phases. In MgNi2 example, the magnesium of 12 nickel atoms is enclosed in the form of a Friauf polyhedron. The next four neighboring magnesium atoms surround the central magnesium atom of the Friauf polyhedron in the form of a tetrahedron and are located just above the hexagons, they are also called caps. For this fourfold capped Friauf polyhedron, this results in a coordination number of 12 4 = 16