Skew icosahedron

For the icosahedron blended with a digon, see Blended icosahedron.

Skew icosahedron
Rank	3
Type	Regular
Space	6-dimensional Euclidean space
Notation
Schläfli symbol	$\{3,\{5\}\#\{5/2\}\}$, $\left\{3,\frac{5}{1,2} \right\}$
Elements
Faces	20 triangles
Edges	30
Vertices	12
Vertex figure	Pentagonal-pentagrammic coil, edge length 1
Petrie polygons	Skew 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
Army	Gee
Abstract & topological properties
Flag count	120
Euler characteristic	2
Schläfli type	{3,5}
Orientable	Yes
Genus	0
Properties
Convex	No

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.

External links

• Hartley, Michael. "{3,5}*120".

Bibliography

• Coxeter (1950), Self-Dual Configurations and Regular Graphs