Order-4-3 pentagonal honeycomb

Order-4-3 pentagonal honeycomb
Type	Regular honeycomb
Schläfli symbol	{5,4,3}
Coxeter diagram
Cells	{5,4}
Faces	{5}
Vertex figure	{4,3}
Dual	{3,4,5}
Coxeter group	[5,4,3]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-4-3 pentagonal honeycomb or 5,4,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each infinite cell is an order-4 pentagonal 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-4-3 pentagonal honeycomb is {5,4,3}, with three order-4 pentagonal tilings meeting at each edge. The vertex figure of this honeycomb is a cube, {4,3}.

Poincaré disk model (Vertex centered)

Ideal surface

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

Order-4-3 hexagonal honeycomb
Type	Regular honeycomb
Schläfli symbol	{6,4,3}
Coxeter diagram
Cells	{6,4}
Faces	{6}
Vertex figure	{4,3}
Dual	{3,4,6}
Coxeter group	[6,4,3]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-4-3 hexagonal honeycomb or 6,4,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-4 hexagonal 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-4-3 hexagonal honeycomb is {6,4,3}, with three order-4 hexagonal tilings meeting at each edge. The vertex figure of this honeycomb is a cube, {4,3}.

Poincaré disk model (Vertex centered)

Ideal surface

Order-4-3 heptagonal honeycomb
Type	Regular honeycomb
Schläfli symbol	{7,4,3}
Coxeter diagram
Cells	{7,4}
Faces	{7}
Vertex figure	{4,3}
Dual	{3,4,7}
Coxeter group	[7,4,3]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-4-3 heptagonal honeycomb or 7,4,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-4 heptagonal 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-4-3 heptagonal honeycomb is {7,4,3}, with three order-4 heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is a cube, {4,3}.

Poincaré disk model (Vertex centered)

Ideal surface

Order-4-3 octagonal honeycomb
Type	Regular honeycomb
Schläfli symbol	{8,4,3}
Coxeter diagram
Cells	{8,4}
Faces	{8}
Vertex figure	{4,3}
Dual	{3,4,8}
Coxeter group	[8,4,3]
Properties	Regular

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

Poincaré disk model (Vertex centered)

Order-4-3 apeirogonal honeycomb
Type	Regular honeycomb
Schläfli symbol	{∞,4,3}
Coxeter diagram
Cells	{∞,4}
Faces	Apeirogon {∞}
Vertex figure	{4,3}
Dual	{3,4,∞}
Coxeter group	[∞,4,3]
Properties	Regular

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

The "ideal surface" projection below is a plane-at-infinity, in the Poincaré half-space model of H3. It shows an Apollonian gasket pattern of circles inside a largest circle.

Poincaré disk model (Vertex centered)

Ideal surface

• Convex uniform honeycombs in hyperbolic space
• List of regular polytopes

• Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
• The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
• Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 (Chapters 16–17: Geometries on Three-manifolds I, II)
• George Maxwell, Sphere Packings and Hyperbolic Reflection Groups, JOURNAL OF ALGEBRA 79,78-97 (1982) [1]
• Hao Chen, Jean-Philippe Labbé, Lorentzian Coxeter groups and Boyd-Maxwell ball packings, (2013)[2]
• Visualizing Hyperbolic Honeycombs arXiv:1511.02851 Roice Nelson, Henry Segerman (2015)

• John Baez, Visual insights: {7,3,3} Honeycomb (2014/08/01) {7,3,3} Honeycomb Meets Plane at Infinity (2014/08/14)
• Danny Calegari, Kleinian, a tool for visualizing Kleinian groups, Geometry and the Imagination 4 March 2014. [3]