Skew icosahedron

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

Skew icosahedron
Rank	3
Dimension	6
Type	Regular
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 $\left\{{\dfrac {10}{1,3,5}}\right\}$
Holes	12 pentagonal-pentagrammic coils
Measures (as derived from unit-edge hexacontatetrapeton)
Dihedral angle	$\arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }$
Related polytopes
Army	Gee
Dual	Dodecahedron (6-dimensional)
φ 2	{5,5/2}#{5/2,5}
Convex hull	Icosahedral-great icosahedral step prism
Abstract & topological properties
Flag count	120
Euler characteristic	2
Schläfli type	{3,5}
Orientable	Yes
Genus	0
Skeleton	Icosahedral graph
Properties
Convex	No
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:

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

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

Symmetric realizations of {3,5}. Click on a node to be taken to the page for that realization.

• : Faithful realization
• : Non-faithful realization
• : Blend with the icosahedron.
• : Blend with the great icosahedron.
• : Blend with the hemiicosahedron.

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	Icosahedron Great icosahedron	Skew icosahedron
8	Icosahedron Hemiicosahedron
8	Great icosahedron Hemiicosahedron
11	Icosahedron Great icosahedron Hemiicosahedron

External links

• Hartley, Michael. "{3,5}*120".

Bibliography

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