Rhombenikosidodekaeder

Körpernetz eines Rhombenikosidodekaeders

Das (kleine) Rhombenikosidodekaeder ist ein Polyeder, das zu den archimedischen Körpern zählt. Es besteht aus 20 gleichseitigen Dreiecken, 30 Quadraten, 12 regelmäßigen Fünfecken und 120 Kanten.

Der Name des Rhombenikosidodekaeders beruht auf der Tatsache, dass die 30 Quadrate deckungsgleich zu den 30 Rhomben eines umbeschriebenen Rhombentriakontaeders sind.

Jeweils zehn Kanten des Rhombenikosidodekaeders bilden ein regelmäßiges Zehneck. Insgesamt gibt es zwölf solcher unabhängiger, gleichseitiger Zehnecke in einem Rhombenikosidodekaeder.

Der zum Rhombenikosidodekaeder duale Körper ist das Deltoidalhexakontaeder.

Formeln

Größen eines Rhombenikosidodekaeders mit Kantenlänge a
Volumen	$V={\frac {a^{3}}{3}}\left(60+29{\sqrt {5}}\right)$
Oberflächeninhalt	$A_{O}=a^{2}\left(30+5{\sqrt {3}}+3{\sqrt {25+10{\sqrt {5}}}}\right)$
Umkugelradius	$R={\frac {a}{2}}{\sqrt {11+4{\sqrt {5}}}}$
Kantenkugelradius	$r={\frac {a}{2}}{\sqrt {10+4{\sqrt {5}}}}$
1. Flächenwinkel (Pentagon–Quadrat) ≈ 148° 16′ 57″	$\cos \,\alpha _{1}=-{\frac {1}{10}}{\sqrt {50+10{\sqrt {5}}}}$
2. Flächenwinkel (Quadrat–Trigon) ≈ 159° 5′ 41″	$\cos \,\alpha _{2}=-{\frac {\sqrt {3}}{6}}(1+{\sqrt {5}})$
Eckenraumwinkel ≈ 1,4153 π	$\Omega =2\pi -\arccos \left({\frac {5-4{\sqrt {5}}}{15}}\right)$
Sphärizität ≈ 0,97924	$\Psi ={\frac {\sqrt[{3}]{20\,\pi \left(1561+696{\sqrt {5}}\right)}}{30+5{\sqrt {3}}+3{\sqrt {25+10{\sqrt {5}}}}}}$

Weblinks

Commons: Rhombenikosidodekaeder – Sammlung von Bildern, Videos und Audiodateien

Eric W. Weisstein: Rhombenikosidodekaeder. In: MathWorld (englisch).

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