Grande icosicosidodecaedro

Tipo

Poliedro stellato uniforme

Forma facce

20 triangoli
12 pentagoni
20 esagoni

Nº facce

Nº spigoli

120

Nº vertici

Caratteristica di Eulero

-8

Incidenza dei vertici

5.6.³/₂.6

Notazione di Wythoff

³/₂ 5 | 3
3 ⁵/₄ | 3

Diagramma di Coxeter-Dynkin

Gruppo di simmetria

I_h, [5,3], *532

Duale

Grande esacontaedro icosacronico

Proprietà

Politopi correlati

In geometria, il grande icosicosidodecaedro è un poliedro stellato uniforme avente 52 facce - 20 triangolari, 12 pentagonali e 20 esagonali - 120 spigoli e 60 vertici.^[1]

Costruzioni di Wythoff

Utilizzando la costruzione di Wythoff, il grande icosicosidodecaedro si può ottenere utilizzando tre famiglie di triangoli di Schwarz: 3/2 5 | 3 e 3 5/4 | 3, ottenendo sempre lo stesso risultato.

Coordinate cartesiane

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

\left(\,0,\,\pm (\varphi -1),\,\pm (\varphi +2)\,\right)

\left(\,\pm (\varphi -1),\,\pm \varphi ,\,\pm 2\varphi \,\right)

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

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

Poliedri correlati

Il grande icosicosidodecaedro, spesso indicato con il simbolo U₄₈, ha la stessa disposizione di vertici del dodecaedro troncato, una cui versione non regolare è il suo inviluppo convesso, e condivide la posizione degli spigoli con il grande dodecicosidodecaedro ditrigonale, con cui ha in comune la disposizione delle facce triangolari e pentagonali, e con il grande dodecicosaedro, con cui ha in comune la disposizione delle facce esagonali.

Dodecaedro troncato	Grande icosicosidodecaedro	Grande dodecicosidodecaedro ditrigonale	Grande dodecicosaedro

Grande esacontaedro icosacronico

Grande esacontaedro icosacronico

Tipo	Poliedro stellato
Forma facce	Dardi
Nº facce	60
Nº spigoli	120
Nº vertici	52
Caratteristica di Eulero	-8
Gruppo di simmetria	I_h, [5,3], *532
Duale	Grande icosicosidodecaedro
Manuale

Il grande esacontaedro icosacronico è un poliedro stellato isoedro, nonché il duale del grande icosicosidodecaedro, avente per facce 60 dardi, detti anche aquiloni non convessi.^[2] Dato un grande icosicosidodecaedro di spigolo pari a 1, immaginando il grande esacontaedro icosacronico come composto da 60 facce intersecanti a forma di dardo, come riportato nella figura sottostante, di cui solo una parte visibile all'esterno del solido, le facce risultanti hanno una coppia di angoli uguali di ampiezza pari a $\arccos({\frac {3}{4}}+{\frac {1}{20}}{\sqrt {5}})\approx 30,480\,324\,565\,36^{\circ }$ e due angoli di ampiezza $\arccos(-{\frac {1}{12}}+{\frac {19}{60}}{\sqrt {5}})\approx 51,335\,802\,942\,83^{\circ }$ e $360^{\circ }-\arccos(-{\frac {5}{12}}+{\frac {1}{60}}{\sqrt {5}})\approx 247,703\,547\,926\,46^{\circ }$ , con il rapporto tra lati lunghi e lati corti pari a ${\frac {31+5{\sqrt {5}}}{22}}\approx 1,917\,288\,176\,70$ .

Note

^ Roman Maeder, 48: great icosicosidodecahedron, su Mathconsult. URL consultato il 24 marzo 2024.
^ Magnus J. Wenninger, Dual Models, Cambridge University Press, 2004, pp. 65. URL consultato il 20 marzo 2024.

Collegamenti esterni

(EN) Eric W. Weisstein, Grande esacontaedro icosacronico, in MathWorld, Wolfram Research. URL consultato il 20 marzo 2024.

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