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Order-3-6 heptagonal honeycomb

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In the geometry of hyperbolic 3-space, the order-3-6 heptagonal honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of a heptagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.

Order-3-6 heptagonal honeycomb
TypeRegular honeycomb
Schläfli symbol{7,3,6}
{7,3[3]}
Coxeter diagram
=
Cells{7,3} Thumb
Faces{7}
Vertex figure{3,6}
Dual{6,3,7}
Coxeter group[7,3,6]
[7,3[3]]
PropertiesRegular
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Geometry

The Schläfli symbol of the order-3-6 heptagonal honeycomb is {7,3,6}, with six heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is a triangular tiling, {3,6}.

It has a quasiregular construction, , which can be seen as alternately colored cells.

Thumb
Poincaré disk model
Thumb
Ideal surface
Summarize
Perspective

It is a part of a series of regular polytopes and honeycombs with {p,3,6} Schläfli symbol, and triangular tiling vertex figures.

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Order-3-6 octagonal honeycomb

Order-3-6 octagonal honeycomb
TypeRegular honeycomb
Schläfli symbol{8,3,6}
{8,3[3]}
Coxeter diagram
=
Cells{8,3} Thumb
FacesOctagon {8}
Vertex figuretriangular tiling {3,6}
Dual{6,3,8}
Coxeter group[8,3,6]
[8,3[3]]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-3-6 octagonal honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-6 octagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.

The Schläfli symbol of the order-3-6 octagonal honeycomb is {8,3,6}, with six octagonal tilings meeting at each edge. The vertex figure of this honeycomb is a triangular tiling, {3,6}.

It has a quasiregular construction, , which can be seen as alternately colored cells.

Thumb
Poincaré disk model

Order-3-6 apeirogonal honeycomb

Order-3-6 apeirogonal honeycomb
TypeRegular honeycomb
Schläfli symbol{∞,3,6}
{∞,3[3]}
Coxeter diagram
=
Cells{∞,3} Thumb
FacesApeirogon {∞}
Vertex figuretriangular tiling {3,6}
Dual{6,3,∞}
Coxeter group[∞,3,6]
[∞,3[3]]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-3-6 apeirogonal honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-3 apeirogonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.

The Schläfli symbol of the order-3-6 apeirogonal honeycomb is {∞,3,6}, with six order-3 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is a triangular tiling, {3,6}.

Thumb
Poincaré disk model
Thumb
Ideal surface

It has a quasiregular construction, , which can be seen as alternately colored cells.

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See also

References

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