Great Icosahedron
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Great Icosahedron
Great icosahedron
Great icosahedron.png
Type Kepler-Poinsot polyhedron
Stellation core icosahedron
Elements F = 20, E = 30
V = 12 (? = 2)
Faces by sides 20{3}
Schläfli symbol {3,​}
Face configuration V(53)/2
Wythoff symbol ​ | 2 3
Coxeter diagram CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.png
Symmetry group Ih, H3, [5,3], (*532)
References U53, C69, W41
Properties Regular nonconvex deltahedron
Great icosahedron vertfig.svg
(35)/2
(Vertex figure)
Great stellated dodecahedron.png
Great stellated dodecahedron
(dual polyhedron)
3D model of a great icosahedron

In geometry, the great icosahedron is one of four Kepler-Poinsot polyhedra (nonconvex regular polyhedra), with Schläfli symbol {3,​} and Coxeter-Dynkin diagram of CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.png. It is composed of 20 intersecting triangular faces, having five triangles meeting at each vertex in a pentagrammic sequence.

The great icosahedron can be constructed analogously to the pentagram, its two-dimensional analogue, via the extension of the (n-1)-D simplex faces of the core nD polytope (equilateral triangles for the great icosahedron, and line segments for the pentagram) until the figure regains regular faces. The grand 600-cell can be seen as its four-dimensional analogue using the same process.

Images

Transparent model Density Stellation diagram Net
GreatIcosahedron.jpg
A transparent model of the great icosahedron (See also Animation)
Great icosahedron cutplane.png
It has a density of 7, as shown in this cross-section.
Great icosahedron stellation facets.svg
It is a stellation of the icosahedron, counted by Wenninger as model [W41] and the 16th of 17 stellations of the icosahedron and 7th of 59 stellations by Coxeter.

Net (surface geometry); twelve isosceles pentagrammic pyramids, arranged like the faces of a dodecahedron. Each pyramid folds up like a fan: the dotted lines fold the opposite direction from the solid lines.
Spherical tiling
Great icosahedron tiling.png
This polyhedron represents a spherical tiling with a density of 7. (One spherical triangle face is shown above, outlined in blue, filled in yellow)

As a snub

The great icosahedron can be constructed a uniform snub, with different colored faces and only tetrahedral symmetry: CDel node h.pngCDel 3x.pngCDel rat.pngCDel d2.pngCDel node h.pngCDel 3x.pngCDel rat.pngCDel d2.pngCDel node h.png. This construction can be called a retrosnub tetrahedron or retrosnub tetratetrahedron,[1] similar to the snub tetrahedron symmetry of the icosahedron, as a partial faceting of the truncated octahedron (or omnitruncated tetrahedron): CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png. It can also be constructed with 2 colors of triangles and pyritohedral symmetry as, CDel node h.pngCDel 3x.pngCDel rat.pngCDel d2.pngCDel node h.pngCDel 4.pngCDel node.png or CDel node h.pngCDel 3x.pngCDel rat.pngCDel d2.pngCDel node h.pngCDel 4.pngCDel rat.pngCDel 3x.pngCDel node.png, and is called a retrosnub octahedron.

Tetrahedral Pyritohedral
Retrosnub tetrahedron.png Pyritohedral great icosahedron.png
CDel node h.pngCDel 3x.pngCDel rat.pngCDel d2.pngCDel node h.pngCDel 3x.pngCDel rat.pngCDel d2.pngCDel node h.png CDel node h.pngCDel 3x.pngCDel rat.pngCDel d2.pngCDel node h.pngCDel 4.pngCDel node.png

Related polyhedra

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

It shares the same vertex arrangement as the regular convex icosahedron. It also shares the same edge arrangement as the small stellated dodecahedron.

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

The truncated great stellated dodecahedron is a degenerate polyhedron, with 20 triangular faces from the truncated vertices, and 12 (hidden) doubled up pentagonal faces ({10/2}) as truncations of the original pentagram faces, the latter forming two great dodecahedra inscribed within and sharing the edges of the icosahedron.

References

  1. ^ Klitzing, Richard. "uniform polyhedra Great icosahedron".

External links


  This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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