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Great icosahedron

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Title: Great icosahedron  
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Subject: Kepler–Poinsot polyhedron, Snub polyhedron, List of regular polytopes and compounds, Great stellated dodecahedron, Great icosidodecahedron
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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 diagram
Symmetry group Ih, H3, [5,3], (*532)
References U53, C69, W41
Properties Regular nonconvex deltahedron

(Vertex figure)

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 . It is composed of 20 intersecting triangular faces, having five triangles meeting at each vertex in a pentagrammic sequence.


  • Images 1
  • As a snub 2
  • Related polyhedra 3
  • References 4
  • External links 5


Transparent model Density Stellation diagram Net

A transparent model of the great icosahedron (See also )

It has a density of 7, as shown in this cross-section.

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.
× 12
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

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: . 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): . It can also be constructed with 2 colors of triangles and pyritohedral symmetry as, or , and is called a retrosnub octahedron.

Tetrahedral Pyritohedral

Related polyhedra

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
Truncated great stellated dodecahedron Great


  1. ^
  • (1st Edn University of Toronto (1938))
  • H.S.M. Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, 3.6 6.2 Stellating the Platonic solids, pp.96-104

External links

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