❴6,4∣3❵

{6,4∣3}
Rank3
Dimension4
TypeRegular
Notation
Schläfli symbol{6,4∣3}
Elements
Faces20 hexagons
Edges60
Vertices30
Vertex figureSkew square, edge length ${\displaystyle {\sqrt {3}}}$
HolesTriangles
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {2}}\approx 1.41421}$
Related polytopes
ArmyDeca
RegimentDeca
Dual{4,6∣3}
Petrie dual{10,4}6,3
Abstract & topological properties
Flag count240
Euler characteristic-10
Schläfli type{6,4}
OrientableYes
Genus6
Properties
SymmetryA4×2, order 240
ConvexNo
Dimension vector(3,2,3)

{6,4∣3} is a regular skew polyhedron in 4-space. Its faces are precisely the hexagonal faces of the decachoron.

Vertex coordinates

Its vertex coordinates are the same as those of the decachoron.

Related polytopes

{6,4∣3} is the smallest of several skew polyhedra of the form {6,4∣n}, and the only finite case.

There is a direct correspondence between these and the regular polyhedra with triangular faces: {6,4∣n} contains truncated {3,n} polyhedra as pseudocells.