In the geometry of hyperbolic 3-space, the heptagonal tiling honeycomb or 7,3,3 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 heptagonal tiling honeycomb is {7,3,3}, with three heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is a tetrahedron, {3,3}.
Related polytopes and honeycombs
It is a part of a series of regular polytopes and honeycombs with {p,3,3} Schläfli symbol, and tetrahedral vertex figures:
| {p,3,3} honeycombs
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| Space
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S3
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H3
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| Form
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Finite
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Paracompact
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Noncompact
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| Name
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{3,3,3}
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{4,3,3}
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{5,3,3}
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{6,3,3}
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{7,3,3}
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{8,3,3}
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... {∞,3,3}
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| Image
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Coxeter diagrams
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1
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| 6
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| 12
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| 24
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Cells
{p,3}
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{3,3}
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{4,3}
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{5,3}
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{6,3}
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{7,3}
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{8,3}
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{∞,3}
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It is a part of a series of regular honeycombs, {7,3,p}.
| {7,3,3}
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{7,3,4}
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{7,3,5}
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{7,3,6}
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{7,3,7}
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{7,3,8}
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...{7,3,∞}
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It is a part of a series of regular honeycombs, with {7,p,3}.
Octagonal tiling honeycomb
| Octagonal tiling honeycomb
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| Type |
Regular honeycomb
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| Schläfli symbol |
{8,3,3}
t{8,4,3}
2t{4,8,4}
t{4[3,3]}
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| Coxeter diagram |
(all 4s)
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| Cells |
{8,3} 
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| Faces |
Octagon {8}
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| Vertex figure |
tetrahedron {3,3}
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| Dual |
{3,3,8}
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| Coxeter group |
[8,3,3]
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| Properties |
Regular
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In the geometry of hyperbolic 3-space, the octagonal tiling honeycomb or 8,3,3 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 octagonal tiling honeycomb is {8,3,3}, with three octagonal tilings meeting at each edge. The vertex figure of this honeycomb is an tetrahedron, {3,3}.
Apeirogonal tiling honeycomb
| Apeirogonal tiling honeycomb
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| Type |
Regular honeycomb
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| Schläfli symbol |
{∞,3,3}
t{∞,3,3}
2t{∞,∞,∞}
t{∞[3,3]}
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| Coxeter diagram |
(all ∞)
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| Cells |
{∞,3} 
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| Faces |
Apeirogon {∞}
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| Vertex figure |
tetrahedron {3,3}
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| Dual |
{3,3,∞}
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| Coxeter group |
[∞,3,3]
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| Properties |
Regular
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In the geometry of hyperbolic 3-space, the apeirogonal tiling honeycomb or ∞,3,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an 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 apeirogonal tiling honeycomb is {∞,3,3}, with three apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is an tetrahedron, {3,3}.
The "ideal surface" projection below is a plane-at-infinity, in the Poincare half-space model of H3. It shows an Apollonian gasket pattern of circles inside a largest circle.
