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
Symmetry group	Ih, [5,3], (*532)
References	U53, C69, W41
Properties	Regular nonconvex deltahedron
(35)/2 (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

Transparent model	Density	Stellation diagram	Spherical tiling
A transparent model of the great icosahedron (See also Animation)	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.	This polyhedron represents a spherical tiling with a density of 7. (One spherical triangle face is shown above, outlined in blue, filled in yellow)

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) 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
Picture

References

• Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9. 
• Coxeter, Harold Scott MacDonald; Du Val, P.; Flather, H. T.; Petrie, J. F. (1999). The fifty-nine icosahedra (3rd ed.). Tarquin. ISBN 978-1-899618-32-3. MR676126  (1st Edn University of Toronto (1938))

External links

• Eric W. Weisstein, Great icosahedron (Uniform polyhedron) at MathWorld.
  • Weisstein, Eric W., "Fifteen stellations of the icosahedron" from MathWorld.
• Uniform polyhedra and duals

Notable stellations of the icosahedron
Regular	Uniform duals		Regular compounds			Regular star	Others
Icosahedron	Small triambic icosahedron	Great triambic icosahedron	Compound of five octahedra	Compound of five tetrahedra	Compound of ten tetrahedra	Great icosahedron	Excavated dodecahedron	Final stellation
The stellation process on the icosahedron creates a number of related polyhedra and compounds with icosahedral symmetry.