Petrial octahedron

Petrial octahedron
Rank3
TypeRegular
SpaceSpherical
Notation
Schläfli symbol${\displaystyle \{6,4\}_{3}}$
${\displaystyle \{3,4\}^{\pi }}$
${\displaystyle \left\{{\frac {6}{1,3}},4:3\right\}}$
Elements
Faces4 skew hexagons
Edges12
Vertices6
Vertex figureSquare (edge length 1)
Petrie polygons8 triangles
Holes6 squares
Related polytopes
Petrie dualOctahedron
SkewingHemioctahedron
φ 2 Square dihedron
κ ?Octahedron
Convex hullOctahedron
Abstract & topological properties
Flag count48
Euler characteristic-2
Schläfli type{6,4}
OrientableNo
Genus4
SkeletonK2,2,2
Properties
SymmetryB3
Flag orbits1
ConvexNo
Dimension vector(1,2,2)

The Petrial octahedron is a regular skew polyhedron consisting of 4 skew hexagons. It is the Petrie dual of the octahedron, therefore shares all of its edges and points with the octahedron. The Petrial octahedron has an Euler characteristic of -2.

Vertex coordinates

The vertex coordinates of a Petrial octahedron with unit side length are shared with a regular octahedron, being all permutations of:

• ${\displaystyle \left(\pm {\frac {\sqrt {2}}{2}},\,0,\,0\right)}$.

Related polyhedra

The rectification of the Petrial octahedron is the cubohemioctahedron.

It is the shuriken of the tetrahedron.