Great icosahedron

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.

The great icosahedron can be constructed analogously to the pentagram, its two-dimensional analogue, via the extension of the (n–1)-dimensional simplex faces of the core n-polytope (equilateral triangles for the great icosahedron, and line segments for the pentagram) until the figure regains regular faces.

The great icosahedron can be constructed as 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.

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.

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.