❴6,4∣6❵

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{6,4∣6}
Rank3
Dimension3
TypeRegular
SpaceHyperbolic
Notation
Schläfli symbol{6,4∣6}
Elements
Faces4N  hexagons
Edges12N 
Vertices6N 
Vertex figureSkew square, edge length
HolesHexagons
Measures (edge length 1)
Circumradius
Related polytopes
ArmyHexah
RegimentHexah
Dual{4,6∣6}
φ 2 Hexagonal dihedron
Abstract & topological properties
Flag count48N 
Schläfli type{6,4}
OrientableYes
Genus
Properties
Symmetry[6,3,3]
ConvexNo
Dimension vector(2,1,2)
History
Discovered byCyril Garner
First discovered1967

{6,4∣6} is a regular skew polyhedron in 3-dimensional hyperbolic space. It is paracompact with planar faces.

Vertex coordinates[edit | edit source]

Its vertex coordinates are the same as those of the hexagonal tiling honeycomb.

Related polytopes[edit | edit source]

{6,4∣6} is one of several skew polyhedra of the form {6,4∣n}. It has the greatest n  before the symmetry would become hypercompact.

There is a direct correspondence between these and the regular polyhedra with triangular faces.

As a facet[edit | edit source]

Three copies of {6,4∣6} can be made to tessellate hyperbolic space as the Petrial hexagonal tiling honeycomb.

Bibliography[edit | edit source]

  • Garner, Cyril (1967), "Regular skew polyhedra in hyperbolic three-space" (PDF), Canadian Journal of Mathematics, 19, doi:10.4153/CJM-1967-106-9