Tetrakis hexahedron

3D view of a tetrakis hexahedron ( animation )

The Tetrakishexaeder (from Greek τετράκις tetrakis "four times" and hexahedron "hexahedron"), even pyramidal cube or Disdyakishexaeder ( Greek δίς dis "twice" and δυάκις dyakis "twice") is a convex polyhedron , which consists of 24 equilateral triangles composed and the Catalan bodies counts. It is dual to the truncated octahedron and has 14 vertices and 36 edges.

Emergence

If square pyramids with the flank length are placed on the 6 boundary surfaces of a cube (edge ​​length ) , a tetrakis hexahedron is created, provided the condition is met. ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle {\tfrac {a}{2}}{\sqrt {2}}<b<{\tfrac {a}{2}}{\sqrt {3}}}$

• For the aforementioned minimum value of , the pyramids on top have the height 0, so that only the cube with the edge length remains. ${\displaystyle b}$ ${\displaystyle a}$
• The special tetrakis hexahedron with equal face angles is created when is. ${\displaystyle b={\tfrac {3}{4}}\,a}$
• Takes the above At the maximum value, the tetrakis hexahedron degenerates into a rhombic dodecahedron with the edge length . ${\displaystyle b}$ ${\displaystyle b}$
• If the maximum value is exceeded , the polyhedron is no longer convex and degenerates into a star body . ${\displaystyle b}$

Formulas

General ${\displaystyle {\tfrac {a}{2}}{\sqrt {2}}<b<{\tfrac {a}{2}}{\sqrt {3}}}$ Sizes of a tetrakis hexahedron with edge length a , b volume ${\displaystyle V=a^{2}\left(a+{\sqrt {4b^{2}-2a^{2}}}\right)}$ Surface area ${\displaystyle A_{O}=6a{\sqrt {4b^{2}-a^{2}}}}$ Pyramid height ${\displaystyle k={\frac {1}{2}}{\sqrt {4b^{2}-2a^{2}}}}$ Inc sphere radius ${\displaystyle \rho ={\frac {a\left(a+{\sqrt {4b^{2}-2a^{2}}}\right)}{2{\sqrt {4b^{2}-a^{2}}}}}}$ Dihedral angle  (over edge a ) ${\displaystyle \cos \,\alpha _{1}={\frac {2a{\sqrt {4b^{2}-2a^{2}}}}{a^{2}-4b^{2}}}}$ Dihedral angle  (over edge b ) ${\displaystyle \cos \,\alpha _{2}={\frac {a^{2}}{a^{2}-4b^{2}}}}$

Special ${\displaystyle b={\tfrac {3}{4}}\,a}$ Sizes of a tetrakis hexahedron with edge length a volume ${\displaystyle V={\frac {3}{2}}\,a^{3}}$ Surface area ${\displaystyle A_{O}=3a^{2}{\sqrt {5}}}$ Pyramid height ${\displaystyle k={\frac {a}{4}}}$ Inc sphere radius ${\displaystyle \rho ={\frac {3}{10}}\,a{\sqrt {5}}}$ Edge ball radius ${\displaystyle r={\frac {a}{2}}{\sqrt {2}}}$ Face angle  ≈ 143 ° 7 ′ 48 ″ ${\displaystyle \cos \,\alpha =-{\frac {4}{5}}}$

application

• In nature the tetrakis hexahedron occurs as a special form {hk0} in crystals of the classes 432, 4 2m and m 3 m, e.g. B. with fluorite .
• The tetrakis hexahedron is also used as a game die (D24).

Web links

Commons : Tetrakis Hexahedron  - collection of images, videos, and audio files

• Eric W. Weisstein : Tetrakis hexahedron . In: MathWorld (English).
• Interactive representation of the tetrakis hexahedron in the mineral atlas
• Fluorite in the Wölsendorfer fluorspar district