Truncated octahedron

3D view of a truncated octahedron ( animation )

Formation of the truncated octahedron from an octahedron

The truncated octahedron is a polyhedron ( polyhedron ) that is one of the Archimedean solids and is created by truncating the six corners of an octahedron . Instead of the corners there are now six squares ; the eight regular triangles become regular hexagons . The truncated octahedron stands out from related polyhedra in that it fills the space without gaps and has a very small surface area per volume.

properties

The truncated octahedron has several symmetries . Its 24 corners are all equivalent: a square and two regular hexagons meet at every corner, and every corner can be mapped to any other corner by rotating the body. In the Cartesian coordinate system with x , y and z axes, the truncated octahedron can be centered at the coordinate origin so that the coordinates of its corners are permutations of (0, ± 1, ± 2). The edge length is then . ${\displaystyle a={\sqrt {2}}}$

If, when truncating an octahedron, the six cut pyramids are put together in pairs in such a way that their square bases meet, then three full-fledged octahedra are created.

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

Room filling

Truncated octahedra fill the room without gaps if they are joined together to parquet the room, as shown in the following graphics. For this reason, its shape was proposed, among other things, as a basic building block for foam , materials and for modular spaceships or space stations .

In addition, there are several room fillings with gaps based on truncated octahedra. These correspond to the crystal structure of zeolite A , zeolite X , zeolite Y , sodalite and faujasite .

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Formulas

Sizes of a truncated octahedron with edge length a
Volume  ≈ 11.31 a 3	${\displaystyle V=8\,a^{3}{\sqrt {2}}}$
Surface area  ≈ 26.78 a 2	${\displaystyle A_{O}=6\,a^{2}\left(1+2{\sqrt {3}}\right)}$
Umkugelradius  ≈ 1.58 a	${\displaystyle R={\frac {a}{2}}{\sqrt {10}}}$
Edge ball radius  = 1.5 a	${\displaystyle r={\frac {3}{2}}\,a}$
1. Diagonal angle  ( hexagon - hexagon )  ≈ 109 ° 28 ′ 16 ″	${\displaystyle \cos \,\alpha _{1}=-{\frac {1}{3}}}$
2.  Face angle (hexagon - square )  ≈ 125 ° 15 ′ 52 ″	${\displaystyle \cos \,\alpha _{2}=-{\frac {1}{3}}{\sqrt {3}}}$
Corners solid angle  = π	${\displaystyle \cos \,\Omega =-1}$

See also

• Rhombic dodecahedron

literature

• Norbert Treitz: Kelvin's assumption .
• Robert A. Freitas, Jr .: Uniform space-filling using only truncated octahedra . Figure 5.5 of Nanomedicine, Volume I: Basic Capabilities , Landes Bioscience, Georgetown TX 1999. Retrieved July 31, 2011.
• George W. Hart: VRML model of truncated octahedron . Virtual Polyhedra: The Encyclopedia of Polyhedra . Retrieved July 31, 2011.

Web links

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

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

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

Individual evidence

1. ↑ Patent DE102006050393 : Load- bearing space lattice structure, lightweight construction element and method for its production. Published April 30, 2008 . And based thereon company http://www.octamold.com/
2. ↑ Patent DE4431290 : Nuclear reactor with loose fill in the reactor core. Published on June 4, 1996 , applicant: Forschungszentrum Jülich . ‌
3. ↑ OL de Weck, WD Nadir, JG Wong, G. Bounova and TM Coffee: Modular Structures for Manned Space Exploration: The Truncated Octahedron as a Building Block . In: 1st Space Exploration Conference: Continuing the Voyage of Discovery . Orlando, Florida 2005 ( spacelogistics.mit.edu (PDF)).