# Skew icosahedron

Skew icosahedron
Rank3
Dimension6
TypeRegular
Notation
Schläfli symbol${\displaystyle \{3,\{5\}\#\{5/2\}\}}$,
${\displaystyle \left\{3,{\frac {5}{1,2}}\right\}}$
Elements
Faces20 triangles
Edges30
Vertices12
Vertex figurePentagonal-pentagrammic coil, edge length 1
Petrie polygonsSkew decagonal-decagrammic coil ${\displaystyle \left\{{\dfrac {10}{1,3,5}}\right\}}$
Holes12 pentagonal-pentagrammic coils
Measures (as derived from unit-edge hexacontatetrapeton)
Dihedral angle${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
Related polytopes
ArmyGee
DualDodecahedron (6-dimensional)
φ 2 {5,5/2}#{5/2,5}
Convex hullIcosahedral-great icosahedral step prism
Abstract & topological properties
Flag count120
Euler characteristic2
Schläfli type{3,5}
OrientableYes
Genus0
SkeletonIcosahedral graph
Properties
ConvexNo
Dimension vector(4,4,4)

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:

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

However, since it is the blend of 2 pure polytopes, it has a degrees of freedom corresponding to the relative scaling of its components, and as a result has more general coordinates.

## 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 {5,5/2}#{5/2,5}.

The skew icosahedron is one of 6 faithful symmetric realizations of the abstract regular polytope {3,5}, and the only one in 6-dimensions.

Faithful symmetric realizations of {3,5}
Dimension Components Name
3 Icosahedron Icosahedron
3 Great icosahedron Great icosahedron
6
Skew icosahedron
8
8
11