Grande icosidodecaedro troncato

Tipo

Poliedro stellato uniforme

Forma facce

30 quadrati
20 esagoni
12 decagrammi

Nº facce

Nº spigoli

180

Nº vertici

120

Caratteristica di Eulero

Incidenza dei vertici

4.6.¹⁰/₃

Notazione di Wythoff

2 3 ⁵/₃ |

Notazione di Schläfli

t_0,1,2{⁵/₃,3}

Diagramma di Coxeter-Dynkin

Gruppo di simmetria

I_h, [5,3], *532

Duale

Grande disdiacistricontaedro

Proprietà

Politopi correlati

In geometria, il grande icosidodecaedro troncato è un poliedro stellato uniforme avente 54 facce - 30 quadrate, 20 esagonali e 12 a forma di decagramma - 180 spigoli e 120 vertici.^[1]

Coordinate cartesiane

Le coordinate cartesiane per i vertici del grande icosidodecaedro troncato sono date da tutte le permutazioni pari di:

\left(\,\pm \varphi ,\,\pm \varphi ,\,\pm (3-\varphi ^{-1})\,\right)

\left(\,\pm 2\varphi ,\,\pm \varphi ^{-1},\,\pm \varphi ^{-3}\,\right)

\left(\,\pm \varphi ,\,\pm \varphi ^{-2},\,\pm (1+3\varphi ^{-1})\,\right)

\left(\,\pm {\sqrt {5}},\,\pm 2\,\pm {\sqrt {5}}\varphi ^{-1}\,\right)

\left(\,\pm \varphi ^{-1},\,\pm 3\,\pm 2\varphi ^{-1}\,\right)

dove $\varphi ={\tfrac {1+{\sqrt {5}}}{2}}$ è la sezione aurea.

Inviluppo convesso

L'inviluppo convesso del grande icosidodecaedro troncato, spesso indicato con il simbolo U₆₈, è un icosidodecaedro troncato non uniforme.

Icosidodecaedro troncato	Inviluppo convesso	Grande icosidodecaedro troncato

Poliedri correlati

Grande disdiacistricontaedro

Grande disdiacistricontaedro

Tipo	Poliedro stellato
Forma facce	Triangoli scaleni
Nº facce	120
Nº spigoli	180
Nº vertici	62
Caratteristica di Eulero	2
Gruppo di simmetria	I_h, [5,3], *532
Duale	Grande icosidodecaedro troncato
Manuale

Il grande disdiacistricontaedro è un poliedro stellato isoedro, nonché il duale del grande icosidodecaedro troncato, avente per facce 120 triangoli scaleni.^[2]

Dato un grande icosidodecaedro troncato di spigolo pari a 1, immaginando il grande disdiacistricontaedro come composto da 120 facce intersecanti a forma di triangolo scaleno, come riportato nella figura sottostante, di cui solo una parte visibile all'esterno del solido, le facce risultanti hanno angoli di ampiezza pari a $\arccos \left({\tfrac {1}{6}}+{\tfrac {1}{15}}{\sqrt {5}}\right)\approx 71,594\,636\,220\,88^{\circ }$ , $\arccos \left({\tfrac {3}{4}}+{\tfrac {1}{10}}{\sqrt {5}}\right)\approx 13,192\,999\,040\,74^{\circ }$ e $\arccos \left({\tfrac {3}{8}}-{\tfrac {5}{24}}{\sqrt {5}}\right)\approx 95,212\,364\,738\,38^{\circ }$ .

Note

^ Roman Maeder, 68: great truncated icosidodecahedron, su Mathconsult. URL consultato il 24 marzo 2024.
^ Magnus J. Wenninger, Dual Models, Cambridge University Press, 2004, pp. 96. URL consultato il 20 marzo 2024.

Collegamenti esterni

(EN) Eric W. Weisstein, Grande icosidodecaedro troncato, su MathWorld, Wolfram Research.
(EN) Eric W. Weisstein, Grande disdiacistricontaedro, in MathWorld, Wolfram Research. URL consultato il 20 marzo 2024.

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