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.
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}.
Related polytopes and honeycombs.
It is a part of a series of regular polytopes and honeycombs with {p,3,4} Schläfli symbol, and octahedral vertex figures:
Order-3-4 octagonal honeycomb.
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}.
Order-3-4 apeirogonal honeycomb.
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}.
References.
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