Truncated tetrahedron

Truncated tetrahedron ( animation )

Folding template

The truncated tetrahedron ( truncated tetrahedron or Friauf polyhedron ) is a polyhedron ( polyhedron ) that is one of the Archimedean solids and is created by blunting the corners of a tetrahedron . Instead of the four corners of the tetrahedron, there are now four equilateral triangles , the triangular surfaces of the tetrahedron become regular hexagons . The Friauf polyhedra are named after the physicist and chemist James Byron Friauf (1896–1972).

The special feature of the truncated tetrahedron is that all connecting lines between the centers of adjacent surfaces are of the same length: they all have the length  a and form three tetrahedra of the same size that completely enclose a fourth of the same size.

The body that is dual to the truncated tetrahedron is the triakis tetrahedron .

Formulas

Sizes of a truncated tetrahedron with edge length  a
volume	${\displaystyle V={\frac {23}{12}}\,a^{3}{\sqrt {2}}}$
Surface area	${\displaystyle A_{O}=7\,a^{2}{\sqrt {3}}}$
Umkugelradius	${\displaystyle R={\frac {a}{4}}{\sqrt {22}}}$
Edge ball radius	${\displaystyle r={\frac {3}{4}}\,a\,{\sqrt {2}}}$
1. Diagonal angle  ( hexagon - hexagon )  ≈ 70 ° 31 ′ 44 ″	${\displaystyle \cos \,\alpha _{1}={\frac {1}{3}}}$
2.  Face angle (hexagon - trigon )  ≈ 109 ° 28 ′ 16 ″	${\displaystyle \cos \,\alpha _{2}=-{\frac {1}{3}}}$
Corners solid angle  ≈ 0.6082 π	${\displaystyle \cos \,\Omega =-{\frac {1}{3}}}$

Cartesian coordinates

The Cartesian coordinates of the corner points can be, with center at origin:

(± 3, ± 1, ± 1),

(± 1, ± 3, ± 1),

(± 1, ± 1, ± 3),

of these 24 coordinates, the 12 that have an odd number of plus signs (1 or 3) and thus an even number of minus signs (2 or 0) must be selected, or vice versa.

Friauf polyhedron

The name Friauf polyhedron for the truncated tetrahedron goes back to the chemist James B. Friauf , who described this polyhedron as the basis for the structure of MgZn ₂ . The Friauf polyhedron is a typical coordination polyhedron with the coordination number 12 in intermetallic compounds such as the Laves phases . In MgNi _2, for example, the magnesium is surrounded by 12 nickel atoms 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 exactly above the hexagons, they are also known as caps . For this fourfold capped Friauf polyhedron, the result is a coordination number of 12 + 4 = 16.

Web links

Commons : Truncated Tetrahedron  - Collection of images, videos and audio files

Wiktionary: truncated tetrahedron  - explanations of meanings, word origins, synonyms, translations

• Eric W. Weisstein : Truncated Tetrahedron . In: MathWorld (English).

Individual evidence

1. ↑ laves.mpie.de ( Memento of the original from June 19, 2010 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. 
2. ↑ resolver.caltech.edu