Great icosahedron

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Great icosahedron
Great icosahedron
Type Kepler-Poinsot polyhedron
Stellation core icosahedron
Elements F = 20, E = 30
V = 12 (χ = 2)
Faces by sides 20{3}
Schläfli symbol {3,5/2}
Wythoff symbol 5/2 | 2 3
Coxeter-Dynkin 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
(35)/2
(Vertex figure)
Great stellated dodecahedron.png
Great stellated dodecahedron
(dual polyhedron)

In geometry, the great icosahedron is one of four Kepler-Poinsot polyhedra (nonconvex regular polyhedra), with Schläfli symbol {3,5/2} 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.

Images[edit]

Transparent model Density Stellation diagram Spherical tiling
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.
Sixteenth stellation of icosahedron facets.png
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.
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[edit]

Retrosnub tetrahedron.pngSnub-polyhedron-great-icosahedron.png
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, 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 as a partial snub, CDel node h.pngCDel 3x.pngCDel rat.pngCDel d2.pngCDel node h.pngCDel 4.pngCDel node.png or CDel node h.pngCDel 3x.pngCDel node h.pngCDel 4.pngCDel rat.pngCDel 3x.pngCDel node.png. Tetrahedral symmetry icosahedra in general are called pseudo-icosahedra.[citation needed]

Related polyhedra[edit]

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.

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[edit]

External links[edit]

Notable stellations of the icosahedron
Regular Uniform duals Regular compounds Regular star Others
Icosahedron Small triambic icosahedron Medial triambic icosahedron Great triambic icosahedron Compound of five octahedra Compound of five tetrahedra Compound of ten tetrahedra Great icosahedron Excavated dodecahedron Final stellation
Zeroth stellation of icosahedron.png First stellation of icosahedron.png Ninth stellation of icosahedron.png First compound stellation of icosahedron.png Second compound stellation of icosahedron.png Third compound stellation of icosahedron.png Sixteenth stellation of icosahedron.png Third stellation of icosahedron.png Seventeenth stellation of icosahedron.png
Zeroth stellation of icosahedron facets.png First stellation of icosahedron facets.png Ninth stellation of icosahedron facets.png First compound stellation of icosahedron facets.png Second compound stellation of icosahedron facets.png Third compound stellation of icosahedron facets.png Sixteenth stellation of icosahedron facets.png Third stellation of icosahedron facets.png Seventeenth stellation of icosahedron facets.png
The stellation process on the icosahedron creates a number of related polyhedra and compounds with icosahedral symmetry.