# Heptagonal tiling honeycomb

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

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 an tetrahedron, {3,3}.

 Poincaré disk model (vertex centered) Ideal surface

## 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:

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}
Coxeter diagram

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 a 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)

### Apeirogonal tiling honeycomb

Apeirogonal tiling honeycomb
Type Regular honeycomb
Schläfli symbol {∞,3,3}
t{∞,3,3}
2t{∞,∞,∞}
Coxeter diagram

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 a 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 a Apollonian gasket pattern of circles inside a largest circle.

 Poincaré disk model (vertex centered) Ideal surface