Triacistrahedron

3D view of a triacistrahedron ( animation )

The triakis tetrahedron is a convex polyhedron , which is composed of 12 isosceles triangles and belongs to the Catalan solids . It is dual to the truncated tetrahedron and has 8 vertices and 18 edges.

Emergence

Quadruple cut cube

If pyramids with the flank length are placed on all 4 boundary surfaces of a tetrahedron (with edge length ) , a triakis tetrahedron is created, provided the condition is met. ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle {\tfrac {a}{3}}{\sqrt {3}}<b<{\tfrac {a}{2}}{\sqrt {2}}}$

• For the aforementioned minimum value of , the pyramids on top have the height 0, so that only the tetrahedron with the edge length remains. ${\displaystyle b}$ ${\displaystyle a}$
• The special triakis tetrahedron with equal face angles is created when is. ${\displaystyle b={\tfrac {3}{5}}a}$
• Takes the above the maximum value, the triakis tetrahedron degenerates into a cube with the edge length (see graphic on the left); this quadruple cut cube - with an imaginary tetrahedron in the core - is topologically equivalent to the triakis tetrahedron . ${\displaystyle b}$ ${\displaystyle b}$
• If the maximum value is exceeded , the polyhedron is no longer convex and degenerates into a star body . ${\displaystyle b}$

Formulas

General ${\displaystyle {\tfrac {a}{3}}{\sqrt {3}}<b<{\tfrac {a}{2}}{\sqrt {2}}}$ Sizes of a triakis tetrahedron with edge lengths a , b volume ${\displaystyle V={\frac {a^{2}}{12}}\left(a{\sqrt {2}}+4{\sqrt {3b^{2}-a^{2}}}\right)}$ Surface area ${\displaystyle A_{O}=3a{\sqrt {4b^{2}-a^{2}}}}$ Pyramid height ${\displaystyle k={\frac {1}{3}}{\sqrt {9b^{2}-3a^{2}}}}$ Inc sphere radius ${\displaystyle \rho ={\frac {a}{12}}{\sqrt {\frac {48b^{2}-14a^{2}+8a{\sqrt {6b^{2}-2a^{2}}}}{4b^{2}-a^{2}}}}}$ Dihedral angle  (over edge a ) ${\displaystyle \cos \,\alpha _{1}={\frac {5a^{2}-12b^{2}-8a{\sqrt {6b^{2}-2a^{2}}}}{9(4b^{2}-a^{2})}}}$ Dihedral angle  (over edge b ) ${\displaystyle \cos \,\alpha _{2}={\frac {2b^{2}-a^{2}}{4b^{2}-a^{2}}}}$

Special ${\displaystyle b={\tfrac {3}{5}}a}$ Sizes of a triakis tetrahedron with edge length a volume ${\displaystyle V={\frac {3}{20}}\,a^{3}{\sqrt {2}}}$ Surface area ${\displaystyle A_{O}={\frac {3}{5}}\,a^{2}{\sqrt {11}}}$ Pyramid height ${\displaystyle k={\frac {a}{15}}{\sqrt {6}}}$ Inc sphere radius ${\displaystyle \rho ={\frac {3}{4}}\,a\,{\sqrt {\frac {2}{11}}}}$ Edge ball radius ${\displaystyle r={\frac {a}{4}}{\sqrt {2}}}$ Face angle  ≈ 129 ° 31 ′ 16 ″ ${\displaystyle \cos \,\alpha =-{\frac {7}{11}}}$

Web links

Commons : Triakisternraeder  - collection of images, videos and audio files

Wiktionary: Triakisternraeder  - explanations of meanings, word origins, synonyms, translations

• Eric W. Weisstein : Triakisternraeder . In: MathWorld (English).
• Mineralienatlas: Triakistenrahedron Interactive representation of the Triakistenrahedron in the Mineralienatlas