Order-infinite-3 triangular honeycomb

In the geometry of hyperbolic 3-space, the order-infinite-3 triangular honeycomb (or 3,∞,3 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,∞,3}.

Order-infinite-3 triangular honeycomb
Type	Regular honeycomb
Schläfli symbols	{3,∞,3}
Coxeter diagrams
Cells	{3,∞}
Faces	{3}
Edge figure	{3}
Vertex figure	{∞,3}
Dual	Self-dual
Coxeter group	[3,∞,3]
Properties	Regular

Geometry

It has three Infinite-order triangular tiling {3,∞} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an order-3 apeirogonal tiling vertex figure.

Poincaré disk model	Ideal surface

Related polytopes and honeycombs

It is a part of a sequence of regular honeycombs with Infinite-order triangular tiling cells: {3,∞,p}.

It is a part of a sequence of regular honeycombs with order-3 apeirogonal tiling vertex figures: {p,∞,3}.

It is a part of a sequence of self-dual regular honeycombs: {p,∞,p}.

Order-infinite-4 triangular honeycomb

Order-infinite-4 triangular honeycomb
Type	Regular honeycomb
Schläfli symbols	{3,∞,4} {3,∞^1,1}
Coxeter diagrams	=
Cells	{3,∞}
Faces	{3}
Edge figure	{4}
Vertex figure	{∞,4} r{∞,∞}
Dual	{4,∞,3}
Coxeter group	[3,∞,4] [3,∞^1,1]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-infinite-4 triangular honeycomb (or 3,∞,4 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,∞,4}.

It has four infinite-order triangular tilings, {3,∞}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many infinite-order triangular tilings existing around each vertex in an order-4 apeirogonal tiling vertex figure.

Poincaré disk model	Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {3,∞^1,1}, Coxeter diagram, , with alternating types or colors of infinite-order triangular tiling cells. In Coxeter notation the half symmetry is [3,∞,4,1⁺] = [3,∞^1,1].

Order-infinite-5 triangular honeycomb

Order-infinite-5 triangular honeycomb
Type	Regular honeycomb
Schläfli symbols	{3,∞,5}
Coxeter diagrams
Cells	{3,∞}
Faces	{3}
Edge figure	{5}
Vertex figure	{∞,5}
Dual	{5,∞,3}
Coxeter group	[3,∞,5]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-infinite-3 triangular honeycomb (or 3,∞,5 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,∞,5}. It has five infinite-order triangular tiling, {3,∞}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many infinite-order triangular tilings existing around each vertex in an order-5 apeirogonal tiling vertex figure.

Poincaré disk model	Ideal surface

Order-infinite-6 triangular honeycomb

Order-infinite-6 triangular honeycomb
Type	Regular honeycomb
Schläfli symbols	{3,∞,6} {3,(∞,3,∞)}
Coxeter diagrams	=
Cells	{3,∞}
Faces	{3}
Edge figure	{6}
Vertex figure	{∞,6} {(∞,3,∞)}
Dual	{6,∞,3}
Coxeter group	[3,∞,6]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-infinite-6 triangular honeycomb (or 3,∞,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,∞,6}. It has infinitely many infinite-order triangular tiling, {3,∞}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many infinite-order triangular tilings existing around each vertex in an order-6 apeirogonal tiling, {∞,6}, vertex figure.

Poincaré disk model	Ideal surface

Order-infinite-7 triangular honeycomb

Order-infinite-7 triangular honeycomb
Type	Regular honeycomb
Schläfli symbols	{3,∞,7}
Coxeter diagrams
Cells	{3,∞}
Faces	{3}
Edge figure	{7}
Vertex figure	{∞,7}
Dual	{7,∞,3}
Coxeter group	[3,∞,7]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-infinite-7 triangular honeycomb (or 3,∞,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,∞,7}. It has infinitely many infinite-order triangular tiling, {3,∞}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many infinite-order triangular tilings existing around each vertex in an order-7 apeirogonal tiling, {∞,7}, vertex figure.

Ideal surface

Order-infinite-infinite triangular honeycomb

Order-infinite-infinite triangular honeycomb
Type	Regular honeycomb
Schläfli symbols	{3,∞,∞} {3,(∞,∞,∞)}
Coxeter diagrams	=
Cells	{3,∞}
Faces	{3}
Edge figure	{∞}
Vertex figure	{∞,∞} {(∞,∞,∞)}
Dual	{∞,∞,3}
Coxeter group	[∞,∞,3] [3,((∞,∞,∞))]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-infinite-infinite triangular honeycomb (or 3,∞,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,∞,∞}. It has infinitely many infinite-order triangular tiling, {3,∞}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many infinite-order triangular tilings existing around each vertex in an infinite-order apeirogonal tiling, {∞,∞}, vertex figure.

Poincaré disk model	Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {3,(∞,∞,∞)}, Coxeter diagram, = , with alternating types or colors of infinite-order triangular tiling cells. In Coxeter notation the half symmetry is [3,∞,∞,1⁺] = [3,((∞,∞,∞))].

Order-infinite-3 square honeycomb

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

In the geometry of hyperbolic 3-space, the order-infinite-3 square honeycomb (or 4,∞,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.

The Schläfli symbol of the order-infinite-3 square honeycomb is {4,∞,3}, with three infinite-order square tilings meeting at each edge. The vertex figure of this honeycomb is an order-3 apeirogonal tiling, {∞,3}.

Poincaré disk model	Ideal surface

Order-infinite-3 pentagonal honeycomb

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

In the geometry of hyperbolic 3-space, the order-infinite-3 pentagonal honeycomb (or 5,∞,3 honeycomb) a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an infinite-order 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-6-3 pentagonal honeycomb is {5,∞,3}, with three infinite-order pentagonal tilings meeting at each edge. The vertex figure of this honeycomb is a heptagonal tiling, {∞,3}.

Poincaré disk model	Ideal surface

Order-infinite-3 hexagonal honeycomb

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

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

Poincaré disk model	Ideal surface

Order-infinite-3 heptagonal honeycomb

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

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

Ideal surface

Order-infinite-3 apeirogonal honeycomb

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

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

Order-infinite-4 square honeycomb

Order-infinite-4 square honeycomb
Type	Regular honeycomb
Schläfli symbol	{4,∞,4}
Coxeter diagrams	=
Cells	{4,∞}
Faces	{4}
Edge figure	{4}
Vertex figure	{∞,4} {∞,∞}
Dual	self-dual
Coxeter group	[4,∞,4]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-infinite-4 square honeycomb (or 4,∞,4 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,∞,4}.

All vertices are ultra-ideal (existing beyond the ideal boundary) with four infinite-order square tilings existing around each edge and with an order-4 apeirogonal tiling vertex figure.

Poincaré disk model	Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {4,∞^1,1}, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [4,∞,4,1⁺] = [4,∞^1,1].

Order-infinite-5 pentagonal honeycomb

Order-infinite-5 pentagonal honeycomb
Type	Regular honeycomb
Schläfli symbol	{5,∞,5}
Coxeter diagrams
Cells	{5,∞}
Faces	{5}
Edge figure	{5}
Vertex figure	{∞,5}
Dual	self-dual
Coxeter group	[5,∞,5]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-infinite-5 pentagonal honeycomb (or 5,∞,5 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {5,∞,5}.

All vertices are ultra-ideal (existing beyond the ideal boundary) with five infinite-order pentagonal tilings existing around each edge and with an order-5 apeirogonal tiling vertex figure.

Poincaré disk model	Ideal surface

Order-infinite-6 hexagonal honeycomb

Order-infinite-6 hexagonal honeycomb
Type	Regular honeycomb
Schläfli symbols	{6,∞,6} {6,(∞,3,∞)}
Coxeter diagrams	=
Cells	{6,∞}
Faces	{6}
Edge figure	{6}
Vertex figure	{∞,6} {(5,3,5)}
Dual	self-dual
Coxeter group	[6,∞,6] [6,((∞,3,∞))]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-infinite-6 hexagonal honeycomb (or 6,∞,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,∞,6}. It has six infinite-order hexagonal tilings, {6,∞}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many hexagonal tilings existing around each vertex in an order-6 apeirogonal tiling vertex figure.

Poincaré disk model	Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {6,(∞,3,∞)}, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [6,∞,6,1⁺] = [6,((∞,3,∞))].

Order-infinite-7 heptagonal honeycomb

Order-infinite-7 heptagonal honeycomb
Type	Regular honeycomb
Schläfli symbols	{7,∞,7}
Coxeter diagrams
Cells	{7,∞}
Faces	{7}
Edge figure	{7}
Vertex figure	{∞,7}
Dual	self-dual
Coxeter group	[7,∞,7]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-infinite-7 heptagonal honeycomb (or 7,∞,7 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {7,∞,7}. It has seven infinite-order heptagonal tilings, {7,∞}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many heptagonal tilings existing around each vertex in an order-7 apeirogonal tiling vertex figure.

Ideal surface

Order-infinite-infinite apeirogonal honeycomb

Order-infinite-infinite apeirogonal honeycomb
Type	Regular honeycomb
Schläfli symbols	{∞,∞,∞} {∞,(∞,∞,∞)}
Coxeter diagrams	↔
Cells	{∞,∞}
Faces	{∞}
Edge figure	{∞}
Vertex figure	{∞,∞} {(∞,∞,∞)}
Dual	self-dual
Coxeter group	[∞,∞,∞] [∞,((∞,∞,∞))]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-infinite-infinite apeirogonal honeycomb (or ∞,∞,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {∞,∞,∞}. It has infinitely many infinite-order apeirogonal tiling {∞,∞} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many infinite-order apeirogonal tilings existing around each vertex in an infinite-order apeirogonal tiling vertex figure.

Poincaré disk model	Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {∞,(∞,∞,∞)}, Coxeter diagram, , with alternating types or colors of cells.

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References

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)
Hao Chen, Jean-Philippe Labbé, Lorentzian Coxeter groups and Boyd-Maxwell ball packings, (2013)
Visualizing Hyperbolic Honeycombs arXiv:1511.02851 Roice Nelson, Henry Segerman (2015)

External links

Hyperbolic Catacombs Carousel: {3,∞,3} honeycomb YouTube, Roice Nelson
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.

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