Disheptahedron

A disheptahedron (also anti- cuboctahedron ) is a polyhedron that consists of the same surfaces as the cuboctahedron , i.e. those of a hexahedron (cube) and an octahedron . In the alternative name (anti-cubooctahedron) there are the words cube and octahedron . It is also known as the Johnson body J ₂₇ (triangular double dome (twisted cuboctahedron)).

A disheptahedron

description

A disheptahedron is obtained from a cuboctahedron by cutting along the plane that forms a circumferential edge between the hexahedron and the octahedron, and then rotating both halves by 180 ° against each other. As a result, in the disheptahedron, 3 groups of 2 hexahedron and 2 octahedron faces each have a common edge. The previous cutting plane becomes a mirror plane of the body.

With 14 surfaces (8 equilateral triangles and 6 squares ), 12 corners and 24 edges of the same length, Euler's polyhedron substitution is fulfilled exactly as with the cuboctahedron. ${\displaystyle e+f-k=2}$

Formulas

Sizes of a disheptahedron with edge length a
volume	${\displaystyle V={\frac {5}{3}}\,a^{3}{\sqrt {2}}}$
Surface area	${\displaystyle A_{O}=2a^{2}(3+{\sqrt {3}})}$
Umkugelradius	${\displaystyle \,R=a}$
Edge ball radius	${\displaystyle r={\frac {a}{2}}{\sqrt {3}}}$
3D edge angle  = 120 °	${\displaystyle \cos \,\gamma =-{\frac {1}{2}}}$

Occurrence

The disheptahedron is used in structural chemistry and crystallography as a coordination polyhedron (e.g. in the hexagonal closest packing of spheres hcp ). The associated coordination number is (just like with the cuboctahedron) 12; the limiting radius quotient is also 1.

Web links

• Eric W. Weisstein : Disheptahedron . In: MathWorld (English).
• beuth-hochschule.de (PDF) p. 7