Abgeschrägtes Dodekaeder

3D-Ansicht eines abgeschrägten Dodekaeders (Animation)

Das abgeschrägte Dodekaeder (Dodecaedron simum) ist ein Polyeder (Vielflächner), das zu den archimedischen Körpern zählt. Es setzt sich aus 92 Flächen, nämlich 12 regelmäßigen Fünfecken und 80 gleichseitigen Dreiecken, zusammen und hat 60 Ecken sowie 150 Kanten. Dabei bilden jeweils vier Dreiecke und ein Fünfeck eine Raumecke.

Die folgenden Bilder zeigen zwei zueinander spiegelbildliche abgeschrägte Dodekaeder.

Spiegelvariante 1
Spiegelvariante 2

Der zum abgeschrägten Dodekaeder duale Körper ist das Pentagonhexakontaeder.

Konstruktion

Transformation eines Rhombenikosidodekaeders in ein Abgeschrägtes Dodekaeder

Wie der Name schon andeutet, entsteht dieses Polyeder durch fortwährendes Abschrägen eines Dodekaeders, sodass am Ende zwölf (kleinere) regelmäßige Fünfecke übrigbleiben, die koinzident mit den ursprünglichen Begrenzungsflächen des Dodekaeders sind.
Verdreht man bei einem Rhombenikosidodekaeder alle zwölf Fünfecke – die koinzident mit den Begrenzungsflächen eines umbeschriebenen Dodekaeders sind – jeweils um den gleichen bestimmten Winkel und fügt eine Diagonale in die nun verzerrten Quadrate ein, entsteht auch ein abgeschrägtes Dodekaeder.

Formeln

Nachfolgend bezeichne der Term $t$ den Kosinus des kleineren Zentriwinkels $\zeta$ im Sehnenfünfeck (siehe Grafik oben rechts) mit den Seitenlängen $a$ und $d=\varphi \,a$ (mit $d$ sei die Diagonale im Pentagon, mit $\varphi$ die Goldene Zahl bezeichnet).

$t$ ist die einzige reelle Lösung der kubischen Gleichung $8t^{3}+8t^{2}-\varphi ^{2}=0$ .

t=\cos(\zeta )={\frac {1}{12}}\left({\sqrt[{3}]{44+12\varphi \,(9+{\sqrt {81\varphi -15}})}}+{\sqrt[{3}]{44+12\varphi \,(9-{\sqrt {81\varphi -15}})}}-4\right)\approx 0{,}47157563

Größen eines abgeschrägten Dodekaeders mit Kantenlänge a
Volumen	$V={\frac {a^{3}}{6{\sqrt {1-2t}}}}\left(3{\sqrt {10\,(9t-2+(4t-1){\sqrt {5}})}}+20{\sqrt {2+2t}}\right)$
Oberflächeninhalt	$A_{O}=a^{2}\left(20{\sqrt {3}}+3{\sqrt {25+10{\sqrt {5}}}}\right)$
Umkugelradius	$R={\frac {a}{2}}{\sqrt {\frac {2-2t}{1-2t}}}$
Kantenkugelradius	$r={\frac {a}{2{\sqrt {1-2t}}}}$
1. Flächenwinkel (Trigon–Trigon) ≈ 164° 10′ 31″	$\cos \,\alpha _{1}=-{\frac {1}{3}}\left(1+4t\right)$
2. Flächenwinkel (Pentagon–Trigon) ≈ 152° 55′ 48″	$\cos \,\alpha _{2}={\frac {1}{\sqrt {15}}}\left((1-2t){\sqrt {5+2{\sqrt {5}}}}-2{\sqrt {(1+t)(5t+(2t-1){\sqrt {5}})}}\right)$
Flächen-Kanten-Winkel (Pentagon–Trigon) ≈ 143° 20′ 58″	$\cos \,\beta ={\frac {-4t}{\sqrt {10-2{\sqrt {5}}}}}$
3D-Kantenwinkel (Trigon–Trigon) ≈ 118° 8′ 12″	$\cos \,\gamma =-t$
Eckenraumwinkel ≈ 1,4355 π	$\Omega =\,3\alpha _{1}+2\alpha _{2}-3\pi$
Sphärizität ≈ 0,98201	$\Psi ={\frac {\sqrt[{3}]{36\,\pi \,V^{2}}}{A_{O}}}$

Weblinks

Commons: Abgeschrägtes Dodekaeder – Sammlung von Bildern, Videos und Audiodateien

Wiktionary: abgeschrägtes Dodekaeder – Bedeutungserklärungen, Wortherkunft, Synonyme, Übersetzungen

Eric W. Weisstein: Abgeschrägtes Dodekaeder. In: MathWorld (englisch).

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