Great stellated dodecahedron























































Great stellated dodecahedron

Great stellated dodecahedron.png
Type
Kepler–Poinsot polyhedron

Stellation core

regular dodecahedron
Elements
F = 12, E = 30
V = 20 (χ = 2)
Faces by sides 125
Schläfli symbol {5/2,3}
Face configuration (35)/2
Wythoff symbol 3 | 25/2
Coxeter diagram
CDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node 1.png
Symmetry group
Ih, H3, [5,3], (*532)
References
U52, C68, W22
Properties
Regular nonconvex

Great stellated dodecahedron vertfig.png
(5/2)3
(Vertex figure)

Great icosahedron.png
Great icosahedron
(dual polyhedron)

In geometry, the great stellated dodecahedron is a Kepler-Poinsot polyhedron, with Schläfli symbol {5/2,3}. It is one of four nonconvex regular polyhedra.


It is composed of 12 intersecting pentagrammic faces, with three pentagrams meeting at each vertex.


It shares its vertex arrangement with the regular dodecahedron, as well as being a stellation of a (smaller) dodecahedron. It is the only dodecahedral stellation with this property, apart from the dodecahedron itself. Its dual, the great icosahedron, is related in a similar fashion to the icosahedron.


Shaving the triangular pyramids off results in an icosahedron.


If the pentagrammic faces are broken into triangles, it is topologically related to the triakis icosahedron, with the same face connectivity, but much taller isosceles triangle faces. If the triangles are instead made to invert themselves and excavate the central icosahedron, the result is a great dodecahedron.


The great stellated dodecahedron can be constructed analogously to the pentagram, its two-dimensional analogue, by attempting to stellate the n-dimensional pentagonal polytope which has pentagonal polytope faces and simplex vertex figures until it can no longer be stellated; that is, it is its final stellation.




Contents






  • 1 Images


  • 2 Related polyhedra


  • 3 References


  • 4 External links





Images




















Transparent model

Tiling

GreatStellatedDodecahedron.jpg
Transparent great stellated dodecahedron (Animation)

Great stellated dodecahedron tiling.png
This polyhedron can be made as spherical tiling with a density of 7. (One spherical pentagram face is shown above, outlined in blue, filled in yellow)

Net
Stellation facets

Great stellated dodecahedron net.png × 20
A net of a great stellated dodecahedron (surface geometry); twenty isosceles triangular pyramids, arranged like the faces of an icosahedron

Third stellation of dodecahedron facets.svg
It can be constructed as the third of three stellations of the dodecahedron, and referenced as Wenninger model [W22].

Geometric Net of a Great Stellated Dodecahedron
Complete net of a great stellated dodecahedron.


Related polyhedra




Animated truncation sequence from {5/2, 3} to {3, 5/2}


A truncation process applied to the great stellated dodecahedron produces a series of uniform polyhedra. Truncating edges down to points produces the great icosidodecahedron as a rectified great stellated dodecahedron. The process completes as a birectification, reducing the original faces down to points, and producing the great icosahedron.


The truncated great stellated dodecahedron is a degenerate polyhedron, with 20 triangular faces from the truncated vertices, and 12 (hidden) pentagonal faces as truncations of the original pentagram faces, the latter forming a great dodecahedron inscribed within and sharing the edges of the icosahedron.



























Name
Great
stellated
dodecahedron

Truncated great stellated dodecahedron

Great
icosidodecahedron

Truncated
great
icosahedron

Great
icosahedron

Coxeter-Dynkin
diagram

CDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node 1.png

CDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node 1.png

CDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.png

CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.png

CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.png
Picture

Great stellated dodecahedron.png

Icosahedron.png

Great icosidodecahedron.png

Great truncated icosahedron.png

Great icosahedron.png


References



  • Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}


External links




  • Eric W. Weisstein, Great stellated dodecahedron (Uniform polyhedron) at MathWorld.
    • Weisstein, Eric W. "Three stellations of the dodecahedron". MathWorld.


  • Uniform polyhedra and duals




























Stellations of the dodecahedron

Platonic solid

Kepler–Poinsot solids

Dodecahedron

Small stellated dodecahedron

Great dodecahedron

Great stellated dodecahedron

Zeroth stellation of dodecahedron.png

First stellation of dodecahedron.svg

Second stellation of dodecahedron.png

Third stellation of dodecahedron.png

Zeroth stellation of dodecahedron facets.png

First stellation of dodecahedron facets.png

Second stellation of dodecahedron facets.png

Third stellation of dodecahedron facets.png



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