Truncated hexahedron

Truncated Hexahedron ( animation )

The room is tiled with truncated hexahedra and octahedra

The truncated hexahedron is a polyhedron ( polyhedron ) that is created by blunting the corners of a cube (hexahedron) and is one of the Archimedean solids . Instead of the eight corners of the cube, there are now eight equilateral triangles ; the six squares of the cube become regular octagons .

If you put the cut corner pieces together again in a suitable way, you get an octahedron . From this it follows that the entire space can be completely filled ( parquetted ) using truncated hexahedra and octahedra (each with the same edge length) : eight truncated hexahedra enclose exactly one octahedron.

The body that is dual to the truncated hexahedron is the triakis octahedron .

Formulas

Sizes of a truncated hexahedron with edge length a
volume	${\displaystyle V={\frac {a^{3}}{3}}\left(21+14{\sqrt {2}}\right)}$
Surface area	${\displaystyle A_{O}=2\,a^{2}(6+6{\sqrt {2}}+{\sqrt {3}})}$
Umkugelradius	${\displaystyle R={\frac {a}{2}}{\sqrt {7+4{\sqrt {2}}}}}$
Edge ball radius	${\displaystyle r={\frac {a}{2}}\left(2+{\sqrt {2}}\right)}$
1. Diagonal angle  ( octagon - octagon )  = 90 °	${\displaystyle \cos \,\alpha _{1}=0}$
2.  Diagonal angle (octagon - trine )  ≈ 125 ° 15 ′ 52 ″	${\displaystyle \cos \,\alpha _{2}=-{\frac {1}{3}}{\sqrt {3}}}$
Surface-edge angle  ≈ 144 ° 44 ′ 8 ″	${\displaystyle \cos \,\beta =-{\frac {1}{3}}{\sqrt {6}}}$
Corners solid angle  ≈ 0.8918 π	${\displaystyle \cos \,\Omega =-{\frac {2}{3}}{\sqrt {2}}}$

Web links

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

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

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