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Order-3-4 heptagonal honeycomb

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In the geometry of hyperbolic 3-space, the order-3-4 heptagonal honeycomb or 7,3,4 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-3-4 heptagonal honeycomb
TypeRegular honeycomb
Schläfli symbol{7,3,4}
Coxeter diagram
=
Cells{7,3} Thumb
Facesheptagon {7}
Vertex figureoctahedron {3,4}
Dual{4,3,7}
Coxeter group[7,3,4]
PropertiesRegular
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Geometry

The Schläfli symbol of the order-3-4 heptagonal honeycomb is {7,3,4}, with four heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is an octahedron, {3,4}.

Thumb
Poincaré disk model
(vertex centered)
Thumb
One hyperideal cell limits to a circle on the ideal surface
Thumb
Ideal surface
Summarize
Perspective

It is a part of a series of regular polytopes and honeycombs with {p,3,4} Schläfli symbol, and octahedral vertex figures:

More information {p,3,4} regular honeycombs, Space ...

Order-3-4 octagonal honeycomb

Order-3-4 octagonal honeycomb
TypeRegular honeycomb
Schläfli symbol{8,3,4}
Coxeter diagram
=

Cells{8,3} Thumb
Facesoctagon {8}
Vertex figureoctahedron {3,4}
Dual{4,3,8}
Coxeter group[8,3,4]
[8,31,1]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-3-4 octagonal honeycomb or 8,3,4 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an 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-3-4 octagonal honeycomb is {8,3,4}, with four octagonal tilings meeting at each edge. The vertex figure of this honeycomb is an octahedron, {3,4}.

Thumb
Poincaré disk model
(vertex centered)

Order-3-4 apeirogonal honeycomb

Order-3-4 apeirogonal honeycomb
TypeRegular honeycomb
Schläfli symbol{∞,3,4}
Coxeter diagram
=

Cells{∞,3} Thumb
Facesapeirogon {∞}
Vertex figureoctahedron {3,4}
Dual{4,3,∞}
Coxeter group[∞,3,4]
[∞,31,1]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-3-4 apeirogonal honeycomb or ∞,3,4 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-3-4 apeirogonal honeycomb is {∞,3,4}, with four order-3 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is an octahedron, {3,4}.

Thumb
Poincaré disk model
(vertex centered)
Thumb
Ideal surface
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See also

References

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