Great icosahedron

The great icosahedron, or gike, is one of the four Kepler–Poinsot solids. It has 20 triangles as faces, joining 5 to a vertex in a pentagrammic fashion.

It has the same edges as the small stellated dodecahedron, and the same vertices as the convex icosahedron. It is also one of the stellations of the icosahedron, and the only Kepler-Poinsot solid to be a stellation of the icosahedron as opposed to the dodecahedron.

Great icosahedra appear as cells in only one of the regular star polychora, namely the great faceted hexacosichoron.

Vertex coordinates
Its vertices are the same as those of its regiment colonel, the small stellated dodecahedron.

Variations
The great icosahedron can also be considered to be a kind of retrosnub tetrahedron, by analogy with the snub cube and snub dodecahedron. It is the result of alternating the vertices of a degenerate uniform polyhedron with 8 degenerate hexagrams and 6 doubled-up squares and then adjusting edge lengths to be equal. It can be represented as s3/2s3/2s or s3/2s4o, with chiral tetrahedral and pyritohedral symmetry respectively, the conjugate of the icosahedron being viewed as a snub tetrahedron.

In vertex figures
The great icosahedron appears as a vertex figure of two Schläfli–Hess polychora.

Related polyhedra
Two uniform polyhedron compounds are composed of great icosahedra:


 * Small retrosnub disoctahedron (2)
 * Small retrosnub icosicosahedron (5)

The great icosahedron can be constructed by joining pentagrammic pyramids to the bases of a pentagrammic retroprism, conjugate to the icosahedron's view as a pentagonal antiprism augmented with pentagonal pyramids.