Grande rombicubottaedro stellato

Tipo

Poliedro stellato uniforme

Forma facce

8 triangoli
6+12 quadrati

Nº facce

Nº spigoli

Nº vertici

Caratteristica di Eulero

Incidenza dei vertici

4.4.4.³/₂

Notazione di Wythoff

³/₂ 4 | 2
3 ⁴/₃ | 2

Notazione di Schläfli

rr{4,³/₂}

Diagramma di Coxeter-Dynkin

Gruppo di simmetria

O_h, [4,3], *432

Duale

Grande icositetraedro trapezoidale

Proprietà

Politopi correlati

In geometria, il grande rombicubottaedro stellato è un poliedro stellato uniforme avente 20 facce - 8 triangolari e 18 quadrate - 48 spigoli e 24 vertici.^[1]

Costruzioni di Wythoff

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

Coordinate cartesiane

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

\left(\,\pm 1,\,\pm 1,\,\pm ({\sqrt {2}}-1)\,\right).

Poliedri correlati

Il grande rombicubottaedro stellato, spesso indicato con il simbolo U₁₇, ha la stessa disposizione di vertici del cubo troncato, il suo inviluppo convesso; inoltre, esso condivide anche la posizione degli spigoli con il grande cubicubottaedro, avendo in comune con esso la disposizione delle facce triangolari e di 6 facce quadrate, e con il grande rombiesaedro, con cui condivide la disposizione delle 12 facce quadrate.

Cubo troncato	Grande rombicubottaedro stellato	Grande cubicubottaedro	Grande rombiesaedro

Grande icositetraedro trapezoidale

Grande icositetraedro trapezoidale

Tipo	Poliedro stellato
Forma facce	Dardi
Nº facce	24
Nº spigoli	48
Nº vertici	26
Caratteristica di Eulero	2
Gruppo di simmetria	O_h, [4,3], *432
Duale	Grande rombicubottaedro stellato
Manuale

Il grande icositetraedro trapezoidale è un poliedro stellato isoedro, nonché il duale del grande rombicubottaedro stellato, avente per facce 24 dardi, detti anche aquiloni non convessi.^[2]

Dato un grande rombicubottaedro stellato di spigolo pari a 1, immaginando il grande icositetraedro trapezoidale come composto da 24 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 tre angoli uguali di ampiezza pari a $\arccos({\frac {1}{2}}+{\frac {1}{4}}{\sqrt {2}})\approx 31,399\,714\,809\,92^{\circ }$ e uno di ampiezza pari a $360^{\circ }-\arccos(-{\frac {1}{4}}+{\frac {1}{8}}{\sqrt {2}})\approx 265,800\,855\,570\,24^{\circ }$ , con il rapporto tra lati lunghi e lati corti pari a $2+{\frac {1}{2}}{\sqrt {2}}\approx 2,707\,106\,781\,19$ .

Note

^ Roman Maeder, 17: nonconvex great rhombicuboctahedron, su Mathconsult. URL consultato il 24 marzo 2024.
^ Magnus J. Wenninger, Dual Models, Cambridge University Press, 2004, pp. 59. URL consultato il 20 marzo 2024.

Collegamenti esterni

(EN) Eric W. Weisstein, Grande rombicubottaedro stellato, su MathWorld, Wolfram Research.
(EN) Eric W. Weisstein, Grande icositetraedro trapezoidale, in MathWorld, Wolfram Research. URL consultato il 20 marzo 2024.

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