❴6,4∣5❵

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{6,4∣5}
Rank3
Dimension3
SpaceHyperbolic
Notation
Coxeter diagramo4o6x|x5o
Schläfli symbol{6,4∣5}
Elements
Faces Hexagons
Edges
Vertices
Vertex figureSkew square, edge length
Holespentagons
Related polytopes
ArmyBitped
RegimentBitped
Dual{4,6∣5}
φ 2 Pentagonal dihedron
Abstract & topological properties
Schläfli type{6,4}
OrientableYes
Genus
Properties
Symmetry[5,3,5]
ConvexNo
Dimension vector(2,1,2)
History
Discovered byCyril Garner
First discovered1967

The {6,4∣5} is a regular skew apeirohedron in 3-dimensional hyperbolic space.

The {6,4∣5} can be constructed from the bitruncated order-5 dodecahedral honeycomb. Its faces are the hexagonal faces of the bitruncated order-5 dodecahedral honeycomb. The pentagonal faces form holes in the {6,4∣5}.

It can also be constructed from order-5 hexagonal tiling by identifying faces as to make pentagonal holes.

Vertex coordinates[edit | edit source]

It's vertex coordinates are the same as the bitruncated order-5 dodecahedral honeycomb.

Related polytopes[edit | edit source]

{6,4∣3} is one of several skew polyhedra of the form {6,4∣n}.

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

External links[edit | edit source]

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