Grande dodecicosidodecaedro

Tipo

Poliedro stellato uniforme

Forma facce

20 triangoli
12 pentagrammi
12 decagrammi

Nº facce

Nº spigoli

120

Nº vertici

Caratteristica di Eulero

-6

Incidenza dei vertici

3.¹⁰/₃.⁵/₂.¹⁰/₇

Notazione di Wythoff

⁵/₂ 3 | ⁵/₃
⁵/₃ ³/₂ | ⁵/₃

Diagramma di Coxeter-Dynkin

Gruppo di simmetria

I_h, [5,3], *532

Duale

Grande esacontaedro dodecacronico

Proprietà

Politopi correlati

In geometria, il grande dodecicosidodecaedro è un poliedro stellato uniforme avente 44 facce - 20 triangolari, 12 forma di pentagramma e 12 a forma di decagramma - 120 spigoli e 60 vertici.^[1]

Costruzioni di Wythoff

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

Poliedri correlati

Il grande dodecicosidodecaedro, spesso indicato con il simbolo U₆₁, ha la stessa disposizione di vertici di altri tre poliedri uniformi, ossia il grande rombicosidodecaedro non convesso, il grande dodecaedro troncato e il grande rombidodecaedro, nonché di altri due composti uniformi, ossia il composto di sei prismi pentagonali e il composto di dodici prismi pentagonali. Con il grande rombicosidodecaedro non convesso e con il grande rombidodecaedro, esso condivide anche la posizione degli spigoli, avendo in comune con il primo anche la disposizione delle sue facce triangolari e pentagrammiche, e con il secondo quella delle sue facce a forma di decagramma.

Grande rombicosidodecaedro non convesso	Grande dodecicosidodecaedro	Grande rombidodecaedro
Grande dodecaedro troncato	Composto di sei prismi pentagonali	Composto di dodici prismi pentagonali

Grande esacontaedro dodecacronico

Grande esacontaedro dodecacronico

Tipo	Poliedro stellato
Forma facce	Aquiloni
Nº facce	60
Nº spigoli	120
Nº vertici	44
Caratteristica di Eulero	-6
Gruppo di simmetria	I_h, [5,3], *532
Duale	Grande dodecicosidodecaedro
Manuale

Il grande esacontaedro dodecacronico è un poliedro stellato isoedro, nonché il duale del grande dodecicosidodecaedro, avente per facce 60 aquiloni.

Immaginando il poliedro come composto da 60 facce a forma di aquiloni intersecanti, come riportato nella figura sottostante, di cui solo una parte visibile all'esterno del solido, i due angoli uguali di tali facce, ossia i due angoli acuti, hanno un'ampiezza di $\arccos({\frac {5}{8}}-{\frac {1}{8}}{\sqrt {5}})\approx 69,788^{\circ }$ , mentre i due angoli ottusi misurano $\arccos(-{\frac {1}{4}}+{\frac {1}{10}}{\sqrt {5}})\approx 91,512^{\circ }$ e $\arccos(-{\frac {1}{8}}-{\frac {9}{40}}{\sqrt {5}})\approx 128,911^{\circ }$ . Il rapporto tra un lato lungo e un lato corto è invece pari a ${\frac {21+3{\sqrt {5}}}{22}}\approx 1,259$ .^[2]

Note

^ Roman Maeder, 61: great dodecicosidodecahedron, su Mathconsult. URL consultato il 24 maggio 2022.
^ Magnus Wenninger, Dual Models, Cambridge University Press, 1983, DOI:10.1017/CBO9780511569371, ISBN 978-0-521-54325-5, MR 730208.

Collegamenti esterni

(EN) Eric W. Weisstein, Grande dodecicosidodecaedro, su MathWorld, Wolfram Research.

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