Heptagonal tiling honeycomb
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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.
| Heptagonal tiling honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {7,3,3} |
| Coxeter diagram |     |
| Cells | {7,3}  |
| Faces | Heptagon {7} |
| Vertex figure | tetrahedron {3,3} |
| Dual | {3,3,7} |
| Coxeter group | [7,3,3] |
| Properties | Regular |
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}.
 Poincaré disk model (vertex centered) |
 Rotating |
 Ideal surface |
It is a part of a series of regular polytopes and honeycombs with {p,3,3} Schläfli symbol, and tetrahedral vertex figures:
It is a part of a series of regular honeycombs, {7,3,p}.
It is a part of a series of regular honeycombs, with {7,p,3}.
Octagonal tiling honeycomb
| Octagonal tiling honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {8,3,3} t{8,4,3} 2t{4,8,4} t{4[3,3]} |
| Coxeter diagram |       (all 4s) |
| Cells | {8,3}  |
| Faces | Octagon {8} |
| Vertex figure | tetrahedron {3,3} |
| Dual | {3,3,8} |
| Coxeter group | [8,3,3] |
| Properties | Regular |
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}.
 Poincaré disk model (vertex centered) |
 Direct subgroups of [8,3,3] |
Apeirogonal tiling honeycomb
| Apeirogonal tiling honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {∞,3,3} t{∞,3,3} 2t{∞,∞,∞} t{∞[3,3]} |
| Coxeter diagram |    (all ∞) |
| Cells | {∞,3}  |
| Faces | Apeirogon {∞} |
| Vertex figure | tetrahedron {3,3} |
| Dual | {3,3,∞} |
| Coxeter group | [∞,3,3] |
| Properties | Regular |
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.
 Poincaré disk model (vertex centered) |
 Ideal surface |
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