Triakisoktaeder

Das Triakisoktaeder ist ein konvexes Polyeder, das sich aus 24 gleichschenkligen Dreiecken zusammensetzt und zu den Catalanischen Körpern zählt. Es ist dual zum Hexaederstumpf und hat 14 Ecken sowie 36 Kanten.

Entstehung

Werden auf die acht Begrenzungsflächen eines Oktaeders (Kantenlänge $a$ ) Pyramiden mit der Flankenlänge $b$ aufgesetzt, entsteht ein Triakisoktaeder, sofern die Bedingung ${\tfrac {a}{3}}{\sqrt {3}}<b<{\tfrac {a}{4}}{\sqrt {6}}$ erfüllt ist.

Für den zuvor genannten minimalen Wert von $b$ haben die aufgesetzten Pyramiden die Höhe 0, sodass lediglich das Oktaeder mit der Kantenlänge $a$ übrig bleibt.
Das spezielle Triakisoktaeder mit gleichen Flächenwinkeln entsteht, wenn $b=a\,(2-{\sqrt {2}})$ ist.
Nimmt $b$ den o. g. maximalen Wert an, entartet das Triakisoktaeder zu einem Rhombendodekaeder mit der Kantenlänge $b$ .
Überschreitet $b$ den maximalen Wert, so ist das Polyeder nicht mehr konvex und entartet schließlich für $b=a$ zum Sterntetraeder.

Formeln

Allgemein

${\tfrac {a}{3}}{\sqrt {3}}<b<{\tfrac {a}{4}}{\sqrt {6}}$

Größen eines Triakisoktaeders mit Kantenlängen a, b
Volumen	$V={\frac {a^{2}}{3}}\left(a{\sqrt {2}}+2{\sqrt {3b^{2}-a^{2}}}\right)$
Oberflächeninhalt	$A_{O}=6a{\sqrt {4b^{2}-a^{2}}}$
Pyramidenhöhe	$k={\frac {1}{3}}{\sqrt {9b^{2}-3a^{2}}}$
Inkugelradius	$\rho \,=a\,{\sqrt {\frac {a}{2a+4b}}}$
Flächenwinkel (über Kante a)	$\cos \,\alpha _{1}={\frac {12b^{2}-5a^{2}-8a{\sqrt {6b^{2}-2a^{2}}}}{9(4b^{2}-a^{2})}}$
Flächenwinkel (über Kante b)	$\cos \,\alpha _{2}={\frac {2b^{2}-a^{2}}{4b^{2}-a^{2}}}$

Speziell

$b=a\,(2-{\sqrt {2}})$

Größen eines Triakisoktaeders mit Kantenlänge a
Volumen	$V=a^{3}(2-{\sqrt {2}})=a^{2}b$
Oberflächeninhalt	$A_{O}=6a^{2}{\sqrt {23-16{\sqrt {2}}}}$
Inkugelradius	$\rho =a{\sqrt {\frac {5+2{\sqrt {2}}}{34}}}$
Kantenkugelradius	$r={\frac {a}{2}}$
Flächenwinkel ≈ 147° 21′	$\cos \,\alpha =-{\frac {1}{17}}\,(3+8{\sqrt {2}})$
Sphärizität ≈ 0,92444	$\Psi ={\frac {\sqrt[{3}]{9\,\pi \left(3-2{\sqrt {2}}\right)}}{3{\sqrt {23-16{\sqrt {2}}}}}}$

Weblinks

Commons: Triakisoktaeder – Sammlung von Bildern, Videos und Audiodateien

Eric W. Weisstein: Triakisoktaeder. In: MathWorld (englisch).
Mineralienatlas:Triakisoktaeder Interaktive Darstellung des Triakisoktaeders im Mineralienatlas

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