Heptagonal tiling honeycomb

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:

{p,3,3} honeycombs
Space		S3			H3
Form		Finite			Paracompact	Noncompact
Name		{3,3,3}	{4,3,3}	{5,3,3}	{6,3,3}	{7,3,3}	{8,3,3}	... {∞,3,3}
Image
Coxeter diagrams	1
	4
	6
	12
	24
Cells {p,3}		{3,3}	{4,3}	{5,3}	{6,3}	{7,3}	{8,3}	{∞,3}

It is a part of a series of regular honeycombs, {7,3,p}.

{7,3,3}	{7,3,4}	{7,3,5}	{7,3,6}	{7,3,7}	{7,3,8}	...{7,3,∞}

It is a part of a series of regular honeycombs, with {7,p,3}.

{7,3,3}	{7,4,3}	{7,5,3}...

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
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

• 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 Archived 2016-06-10 at the Wayback Machine) 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]