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 grand 600-cell can be seen as its four-dimensional analogue using the same process.

Construction
The edge length of a great icosahedron is $$\frac{7+3\sqrt{5}}{2}$$ times that of the original icosahedron.

Formulas
For a great icosahedron with edge length E,

$$\text{Circumradius} = {\tfrac{E}{4}\Bigl(\sqrt{50+22\sqrt{5}}\Bigr)}$$

$$\text{Surface Area} = 3\sqrt{3}(5+4\sqrt{5})E^2$$

$$\text{Volume} = {\tfrac{25+9\sqrt{5}}{4}}E^3$$

As a snub
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, 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.

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.