# Skew icosahedron

Skew icosahedron
Rank3
TypeRegular
Space6-dimensional Euclidean space
Notation
Schläfli symbol$\{3,\{5\}\#\{5/2\}\}$ ,
$\left\{3,\frac{5}{1,2} \right\}$ Elements
Faces20 triangles
Edges30
Vertices12
Vertex figurePentagonal-pentagrammic coil, edge length 1
Petrie polygonsSkew decagonal-decagrammic coil $\left\{\dfrac{10}{1,3,5}\right\}$ Measures (as derived from unit-edge hexacontatetrapeton)
Dihedral angle$\arccos\left(-\frac13\right) \approx 109.47122^\circ$ Related polytopes
ArmyGee
Abstract & topological properties
Flag count120
Euler characteristic2
Schläfli type{3,5}
OrientableYes
Genus0
Properties
ConvexNo

The skew icosahedron is a regular skew polyhedron in 6-dimensional Euclidean space. It is abstractly equivalent to the regular icosahedron in 3-dimensional Euclidean space. It can be constructed as the blend of the icosahedron and the great icosahedron $\{3,5 \} \# \{ 3, 5/2 \}$ .

## Vertex coordinates

Vertex coordinates for a skew icosahedron with unit edge length centered at the origin can be given as:

• $\pm\dfrac{\sqrt{2}}{4}\left(1,\,1,\,1,\,1,\,1,\,1\right)$ ,
• $\pm\dfrac{\sqrt{2}}{4}\left(-1,\,-1,\,1,\,1,\,1,\,1\right)$ ,
• $\pm\dfrac{\sqrt{2}}{4}\left(1,\,-1,\,-1,\,1,\,1,\,1\right)$ ,
• $\pm\dfrac{\sqrt{2}}{4}\left(1,\,1,\,-1,\,-1,\,1,\,1\right)$ ,
• $\pm\dfrac{\sqrt{2}}{4}\left(1,\,1,\,1,\,-1,\,-1,\,1\right)$ ,
• $\pm\dfrac{\sqrt{2}}{4}\left(-1,\,1,\,1,\,1,\,-1,\,1\right)$ .

## Related polyhedra

Just as the 3-dimensional icosahedron can be facetted to form the great dodecahedron, the skew icosahedron can be facetted to form a regular skew great dodecahedron.