Grande dodecicosaedro

Grande dodecicosaedro
TipoPoliedro stellato uniforme
Forma facce20 esagoni
12 decagrammi
Nº facce32
Nº spigoli120
Nº vertici60
Caratteristica di Eulero-28
Incidenza dei vertici6.10/3.6/5.10/7
Notazione di Wythoff3 5/3 (3/2 5/2) |
Diagramma di Coxeter-Dynkin
Gruppo di simmetriaIh, [5,3], *532
DualeGrande dodecicosacrono
ProprietàNon convessità
Politopi correlati
Figura al vertice
Poliedro duale

In geometria, il grande dodecicosaedro è un poliedro stellato uniforme avente 32 facce - 20 esagonali e 12 forma di decagramma - 120 spigoli e 60 vertici.[1]

Costruzioni di Wythoff

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

Coordinate cartesiane

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

dove è la sezione aurea.

Poliedri correlati

Il grande dodecicosaedro, spesso indicato con il simbolo U63, 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 decagrammiche, e con il grande icosicosidodecaedro, con cui ha in comune la disposizione delle facce esagonali.


Dodecaedro troncato

Grande icosicosidodecaedro

Grande dodecicosidodecaedro ditrigonale

Grande dodecicosaedro

Grande dodecicosacrono

Grande dodecicosacrono
TipoPoliedro stellato
Forma facceAntiparallelogrammi
Nº facce60
Nº spigoli120
Nº vertici32
Caratteristica di Eulero-28
Gruppo di simmetriaIh, [5,3], *532
DualeGrande dodecicosaedro

Il grande dodecicosacrono è un poliedro stellato isoedro, nonché il duale del grande dodecicosaedro, avente per facce 60 antiparallelogrammi.[2] Dato un grande dodecicosaedro di spigolo pari a 1, immaginando il grande dodecicosacrono come composto da 60 facce intersecanti a forma di antiparalleogramma, come riportato nella figura sottostante, di cui solo una parte visibile all'esterno del solido, le facce risultanti hanno due coppie di angoli uguali di ampiezza pari a e , con il rapporto tra lati lunghi e lati corti pari a , e le due diagonali che si incontrano con un angolo di .

Note

  1. ^ Roman Maeder, 63: great dodecicosahedron, su Mathconsult. URL consultato il 24 marzo 2024.
  2. ^ Magnus J. Wenninger, Dual Models, Cambridge University Press, 2004, pp. 67. URL consultato il 20 marzo 2024.

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