❴6,6∣6❵

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{6,6∣6}
Rank3
Dimension3
TypeRegular
SpaceHyperbolic
Notation
Schläfli symbol{6,6∣6}
Elements
FacesN  hexagons
Edges3N 
VerticesN 
Vertex figureSkew hexagon
HolesHexagons
Related polytopes
ArmyShexah
RegimentShexah
Dual{6,6∣6}
Convex hullOrder-4 hexagonal tiling honeycomb
Abstract & topological properties
Flag count12N 
Schläfli type{6,6}
OrientableYes
Genus
Properties
Symmetry
ConvexNo
Dimension vector(2,1,2)
History
Discovered byCyril Garner
First discovered1967

The {6,6∣6} is a paracompact regular skew apeirohedron in 3-dimensional hyperbolic space. Its faces are half of the hexagonal faces of the order-4 hexagonal tiling honeycomb. It is a self-dual polyhedron, and it also shares a symmetry group with another hyperbolic regular skew apeirohedron: {12,12∣3}.

Related polytopes[edit | edit source]

Four copies of {6,6∣6} can be made to tessellate hyperbolic space as the facets of the regular Petrial order-4 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