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.

gollark: There's actually another similar thing, Pascal's *Mugging*, in which someone comes up to you and says "give me £100 or I will eternally torture you after you die".

gollark: But there are an infinitely large number of possible gods, and some do weirder things like "punish/reward entirely at random", "have no interest whatsoever in humanity", "punish people who believe in other gods", and all that, and Pascal's Wager just *ignores* those.

gollark: Pascal's Wager might work if the only options are "no god" or "one god, and it's the one you believe in, and they'll reward you if you believe and punish you otherwise".

gollark: Also, I should be specific, "a god and associated religious claims", not just "a god".

gollark: Pascal's Wager is really stupid.

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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