# Great icosahedron

Great icosahedron :
| Type | Kepler–Poinsot polyhedron |
| Stellation core | icosahedron |
| Elements | F = 20, E = 30 V = 12 (χ = 2) |
| Faces by sides | 20{3} |
| Schläfli symbol | {3,5⁄2} |
| Face configuration | V(5$^3$)/2 |
| Wythoff symbol | 5⁄2 | 2 3 |
| Coxeter diagram | |
| Symmetry group | I$_h$, H$_3$, [5,3], (*532) |
| References | U$_{53}$, C$_{69}$, W$_{41}$ |
| Properties | Regular nonconvex deltahedron |
| (3$^5$)/2 (Vertex figure) | Great stellated dodecahedron (dual polyhedron) |
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.

## Images

| Transparent model | Density | Stellation diagram | Net |
| A 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. | × 12 Net (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 tiling :
* This polyhedron represents a spherical tiling with a density of 7. (One spherical triangle face is shown above, outlined in blue, filled in yellow)

## Formulas

For a great icosahedron with edge length E (the edge of its dodecahedron core),
${\text{Inradius}}={\frac {{\text{E}}(3{\sqrt {3}}-{\sqrt {15}})}{4}}$
${\text{Midradius}}={\frac {{\text{E}}({\sqrt {5}}-1)}{4}}$
${\text{Circumradius}}={\frac {{\text{E}}{\sqrt {2(5-{\sqrt {5}})}}}{4}}$
${\text{Surface Area}}=3{\sqrt {3}}(5+4{\sqrt {5}}){\text{E}}^{2}$
${\text{Volume}}={\tfrac {25+9{\sqrt {5}}}{4}}{\text{E}}^{3}$

## 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.
| 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 (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.
| Name | Great stellated dodecahedron | Truncated great stellated dodecahedron | Great icosidodecahedron | Truncated great icosahedron | Great icosahedron |
| Coxeter-Dynkin diagram | | | | | |
| Picture | | | | | |