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Kepler Poinsot polyhedron with 20 faces

In geometry, the great icosahedron is one of four Kepler–Poinsot polyhedra (nonconvex regular polyhedra), with Schläfli symbol 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.

The edge length of a great icosahedron is times that of the original icosahedron.

Transparent modelDensityStellation diagramNetA 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. × 12Net (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 tilingThis polyhedron represents a spherical tiling with a density of 7. (One spherical triangle face is shown above, outlined in blue, filled in yellow)