Order-8-3 triangular honeycomb

In the geometry of hyperbolic 3-space, the order-8-3 triangular honeycomb (or 3,8,3 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,8,3}.

Order-8-3 triangular honeycomb
Type	Regular honeycomb
Schläfli symbols	{3,8,3}
Coxeter diagrams
Cells	{3,8}
Faces	{3}
Edge figure	{3}
Vertex figure	{8,3}
Dual	Self-dual
Coxeter group	[3,8,3]
Properties	Regular

Geometry

It has three order-8 triangular tiling {3,8} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an octagonal tiling vertex figure.

Poincaré disk model

Related polytopes and honeycombs

It is a part of a sequence of regular honeycombs with order-8 triangular tiling cells: {3,8,p}.

It is a part of a sequence of regular honeycombs with octagonal tiling vertex figures: {p,8,3}.

It is a part of a sequence of self-dual regular honeycombs: {p,8,p}.

Order-8-4 triangular honeycomb

Order-8-4 triangular honeycomb
Type	Regular honeycomb
Schläfli symbols	{3,8,4}
Coxeter diagrams	=
Cells	{3,8}
Faces	{3}
Edge figure	{4}
Vertex figure	{8,4} r{8,8}
Dual	{4,8,3}
Coxeter group	[3,8,4]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-8-4 triangular honeycomb (or 3,8,4 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,8,4}.

It has four order-8 triangular tilings, {3,8}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-8 triangular tilings existing around each vertex in an order-4 hexagonal tiling vertex arrangement.

Poincaré disk model

It has a second construction as a uniform honeycomb, Schläfli symbol {3,8^1,1}, Coxeter diagram, , with alternating types or colors of order-8 triangular tiling cells. In Coxeter notation the half symmetry is [3,8,4,1⁺] = [3,8^1,1].

Order-8-5 triangular honeycomb

Order-8-5 triangular honeycomb
Type	Regular honeycomb
Schläfli symbols	{3,8,5}
Coxeter diagrams
Cells	{3,8}
Faces	{3}
Edge figure	{5}
Vertex figure	{8,5}
Dual	{5,8,3}
Coxeter group	[3,8,5]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-8-3 triangular honeycomb (or 3,8,5 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,8,5}. It has five order-8 triangular tiling, {3,8}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-8 triangular tilings existing around each vertex in an order-5 octagonal tiling vertex figure.

Poincaré disk model

Order-8-6 triangular honeycomb

Order-8-6 triangular honeycomb
Type	Regular honeycomb
Schläfli symbols	{3,8,6} {3,(8,3,8)}
Coxeter diagrams	=
Cells	{3,8}
Faces	{3}
Edge figure	{6}
Vertex figure	{8,6} {(8,3,8)}
Dual	{6,8,3}
Coxeter group	[3,8,6]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-8-6 triangular honeycomb (or 3,8,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,8,6}. It has infinitely many order-8 triangular tiling, {3,8}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-8 triangular tilings existing around each vertex in an order-6 octagonal tiling, {8,6}, vertex figure.

Poincaré disk model

Order-8-infinite triangular honeycomb

Order-8-infinite triangular honeycomb
Type	Regular honeycomb
Schläfli symbols	{3,8,∞} {3,(8,∞,8)}
Coxeter diagrams	=
Cells	{3,8}
Faces	{3}
Edge figure	{∞}
Vertex figure	{8,∞} {(8,∞,8)}
Dual	{∞,8,3}
Coxeter group	[∞,8,3] [3,((8,∞,8))]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-8-infinite triangular honeycomb (or 3,8,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,8,∞}. It has infinitely many order-8 triangular tiling, {3,8}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-8 triangular tilings existing around each vertex in an infinite-order octagonal tiling, {8,∞}, vertex figure.

Poincaré disk model

It has a second construction as a uniform honeycomb, Schläfli symbol {3,(8,∞,8)}, Coxeter diagram, = , with alternating types or colors of order-8 triangular tiling cells. In Coxeter notation the half symmetry is [3,8,∞,1⁺] = [3,((8,∞,8))].

Order-8-3 square honeycomb

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

In the geometry of hyperbolic 3-space, the order-8-3 square honeycomb (or 4,8,3 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 order-8-3 square honeycomb is {4,8,3}, with three order-4 octagonal tilings meeting at each edge. The vertex figure of this honeycomb is an octagonal tiling, {8,3}.

Poincaré disk model

Order-8-3 pentagonal honeycomb

Order-8-3 pentagonal honeycomb
Type	Regular honeycomb
Schläfli symbol	{5,8,3}
Coxeter diagram
Cells	{5,8}
Faces	{5}
Vertex figure	{8,3}
Dual	{3,8,5}
Coxeter group	[5,8,3]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-8-3 pentagonal honeycomb (or 5,8,3 honeycomb) a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-8 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 {5,8,3}, with three order-8 pentagonal tilings meeting at each edge. The vertex figure of this honeycomb is an octagonal tiling, {8,3}.

Poincaré disk model

Order-8-3 hexagonal honeycomb

Order-8-3 hexagonal honeycomb
Type	Regular honeycomb
Schläfli symbol	{6,8,3}
Coxeter diagram
Cells	{6,8}
Faces	{6}
Vertex figure	{8,3}
Dual	{3,8,6}
Coxeter group	[6,8,3]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-8-3 hexagonal honeycomb (or 6,8,3 honeycomb) a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-6 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-8-3 hexagonal honeycomb is {6,8,3}, with three order-5 hexagonal tilings meeting at each edge. The vertex figure of this honeycomb is an octagonal tiling, {8,3}.

Poincaré disk model

Order-8-3 apeirogonal honeycomb

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

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

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

Order-8-4 square honeycomb

Order-8-4 square honeycomb
Type	Regular honeycomb
Schläfli symbol	{4,8,4}
Coxeter diagrams	=
Cells	{4,8}
Faces	{4}
Edge figure	{4}
Vertex figure	{8,4}
Dual	self-dual
Coxeter group	[4,8,4]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-8-4 square honeycomb (or 4,8,4 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,8,4}.

All vertices are ultra-ideal (existing beyond the ideal boundary) with four order-5 square tilings existing around each edge and with an order-4 octagonal tiling vertex figure.

Poincaré disk model

Order-8-5 pentagonal honeycomb

Order-8-5 pentagonal honeycomb
Type	Regular honeycomb
Schläfli symbol	{5,8,5}
Coxeter diagrams
Cells	{5,8}
Faces	{5}
Edge figure	{5}
Vertex figure	{8,5}
Dual	self-dual
Coxeter group	[5,8,5]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-8-5 pentagonal honeycomb (or 5,8,5 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {5,8,5}.

All vertices are ultra-ideal (existing beyond the ideal boundary) with five order-8 pentagonal tilings existing around each edge and with an order-5 pentagonal tiling vertex figure.

Poincaré disk model

Order-8-6 hexagonal honeycomb

Order-8-6 hexagonal honeycomb
Type	Regular honeycomb
Schläfli symbols	{6,8,6} {6,(8,3,8)}
Coxeter diagrams	=
Cells	{6,8}
Faces	{6}
Edge figure	{6}
Vertex figure	{8,6} {(5,3,5)}
Dual	self-dual
Coxeter group	[6,8,6] [6,((8,3,8))]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-8-6 hexagonal honeycomb (or 6,8,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,8,6}. It has six order-8 hexagonal tilings, {6,8}, 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 octagonal tiling vertex arrangement.

Poincaré disk model

It has a second construction as a uniform honeycomb, Schläfli symbol {6,(8,3,8)}, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [6,8,6,1⁺] = [6,((8,3,8))].

Order-8-infinite apeirogonal honeycomb

Order-8-infinite apeirogonal honeycomb
Type	Regular honeycomb
Schläfli symbols	{∞,8,∞} {∞,(8,∞,8)}
Coxeter diagrams	↔
Cells	{∞,8}
Faces	{∞}
Edge figure	{∞}
Vertex figure	{8,∞} {(8,∞,8)}
Dual	self-dual
Coxeter group	[∞,8,∞] [∞,((8,∞,8))]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-8-infinite apeirogonal honeycomb (or ∞,8,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {∞,8,∞}. It has infinitely many order-8 apeirogonal tiling {∞,8} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-8 apeirogonal tilings existing around each vertex in an infinite-order octagonal tiling vertex figure.

Poincaré disk model

It has a second construction as a uniform honeycomb, Schläfli symbol {∞,(8,∞,8)}, Coxeter diagram, , with alternating types or colors of cells.

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: {3,7,3} 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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