❴4,6∣5❵

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{4,6∣5}
Rank3
Dimension3
TypeRegular
SpaceHyperbolic
Notation
Coxeter diagramx4o6o|o5x
Schläfli symbol{4,6∣5}
Elements
Faces12N  squares
Edges24N 
Vertices4N 
Vertex figureSkew hexagon, edge length
HolesPentagons
Related polytopes
ArmySpidoh
RegimentSpidoh
Dual{6,4∣5}
φ 2 Dodecahedron
Convex hullSmall prismatododecahedral honeycomb
Abstract & topological properties
Flag count96N 
Schläfli type{4,6}
OrientableYes
Genus
Properties
ConvexNo
Dimension vector(2,1,2)
History
Discovered byCyril Garner
First discovered1967

The {4,6∣5} is a compact regular skew polyhedron in 3-dimensional hyperbolic space. Its faces are precisely the square faces of the small prismatododecahedral honeycomb.

Vertex coordinates[edit | edit source]

Its vertex coordinates are the same as those of the small prismatododecahedral honeycomb.

Related polytopes[edit | edit source]

{4,6∣5} is one of several skew polyhedra of the form {4,6∣n}.

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

Each of the latter appears as a pseudocell of the corresponding former. This is a part of a general correspondence where {n,p} appears as a pseudocell of {4,2pn}.

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