Great icosahedron

The great icosahedron can be constructed analogously to the pentagram, its two-dimensional analogue, via the extension of the (n-1)-D simplex faces of the core nD 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 great icosahedron can be constructed a uniform snub, with different colored faces and only tetrahedral symmetry: CDel node h.pngCDel 3x.pngCDel rat.pngCDel d2.pngCDel node h.pngCDel 3x.pngCDel rat.pngCDel d2.pngCDel node h.png. 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): CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png. It can also be constructed with 2 colors of triangles and pyritohedral symmetry as, CDel node h.pngCDel 3x.pngCDel rat.pngCDel d2.pngCDel node h.pngCDel 4.pngCDel node.png or CDel node h.pngCDel 3x.pngCDel rat.pngCDel d2.pngCDel node h.pngCDel 4.pngCDel rat.pngCDel 3x.pngCDel node.png, 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. 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.