Great icosahedron
In geometry, the great icosahedron is one of four Kepler-Poinsot 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 -D simplex faces of the core nD polytope until the figure regains regular faces. The grand 600-cell can be seen as its four-dimensional analogue using the same process.
Images
| Transparent model | Density | Stellation diagram | Net |
A transparent model of the great icosahedron | It has a density of 7, as shown in this cross-section. | It is a stellation of the icosahedron, counted by Wenninger as model and the 16th of 17 stellations of the icosahedron and 7th of 59 stellations by Coxeter. | Net ; 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. |
This polyhedron represents a spherical tiling with a density of 7. |
As a snub
The great icosahedron can be constructed 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 :. It can also be constructed with 2 colors of triangles and pyritohedral symmetry as, or, and is called a retrosnub octahedron.| Tetrahedral | Pyritohedral |
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 doubled up pentagonal faces as truncations of the original pentagram faces, the latter forming two great dodecahedra inscribed within and sharing the edges of the icosahedron.
| Name | Great stellated dodecahedron | Truncated great stellated dodecahedron | Great icosidodecahedron | Truncated great icosahedron | Great icosahedron |
| Coxeter-Dynkin diagram | |||||
| Picture |