Order-8-3 triangular honeycomb explained

bgcolor=#e8dcc3 colspan=2	Order-8-3 triangular honeycomb
Type	Regular honeycomb
Schläfli symbols
Coxeter diagrams
Cells
Faces
Edge figure
Vertex figure
Dual	Self-dual
Coxeter group	[3,8,3]
Properties	Regular

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 .

Geometry

It has three order-8 triangular tiling 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.

Related polytopes and honeycombs

It is a part of a sequence of regular honeycombs with order-8 triangular tiling cells: .

It is a part of a sequence of regular honeycombs with octagonal tiling vertex figures: .

It is a part of a sequence of self-dual regular honeycombs: .

Order-8-4 triangular honeycomb

bgcolor=#e8dcc3 colspan=2	Order-8-4 triangular honeycomb
Type	Regular honeycomb
Schläfli symbols
Coxeter diagrams	=
Cells
Faces
Edge figure
Vertex figure	r
Dual
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 .

It has four order-8 triangular tilings,, 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.

It has a second construction as a uniform honeycomb, Schläfli symbol, 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

bgcolor=#e8dcc3 colspan=2	Order-8-5 triangular honeycomb
Type	Regular honeycomb
Schläfli symbols
Coxeter diagrams
Cells
Faces
Edge figure
Vertex figure
Dual
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 . It has five order-8 triangular tiling,, 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.

Order-8-6 triangular honeycomb

bgcolor=#e8dcc3 colspan=2	Order-8-6 triangular honeycomb
Type	Regular honeycomb
Schläfli symbols
Coxeter diagrams	=
Cells
Faces
Edge figure
Vertex figure
Dual
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 . It has infinitely many order-8 triangular tiling,, 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,, vertex figure.

Order-8-infinite triangular honeycomb

bgcolor=#e8dcc3 colspan=2	Order-8-infinite triangular honeycomb
Type	Regular honeycomb
Schläfli symbols
Coxeter diagrams	=
Cells
Faces
Edge figure
Vertex figure
Dual
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 . It has infinitely many order-8 triangular tiling,, 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,, vertex figure.

It has a second construction as a uniform honeycomb, Schläfli symbol, 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

bgcolor=#e8dcc3 colspan=2	Order-8-3 square honeycomb
Type	Regular honeycomb
Schläfli symbol
Coxeter diagram
Cells
Faces
Vertex figure
Dual
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, with three order-4 octagonal tilings meeting at each edge. The vertex figure of this honeycomb is an octagonal tiling, .

Order-8-3 pentagonal honeycomb

bgcolor=#e8dcc3 colspan=2	Order-8-3 pentagonal honeycomb
Type	Regular honeycomb
Schläfli symbol
Coxeter diagram
Cells
Faces
Vertex figure
Dual
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, with three order-8 pentagonal tilings meeting at each edge. The vertex figure of this honeycomb is an octagonal tiling, .

Order-8-3 hexagonal honeycomb

bgcolor=#e8dcc3 colspan=2	Order-8-3 hexagonal honeycomb
Type	Regular honeycomb
Schläfli symbol
Coxeter diagram
Cells
Faces
Vertex figure
Dual
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, with three order-5 hexagonal tilings meeting at each edge. The vertex figure of this honeycomb is an octagonal tiling, .

Order-8-3 apeirogonal honeycomb

bgcolor=#e8dcc3 colspan=2	Order-8-3 apeirogonal honeycomb
Type	Regular honeycomb
Schläfli symbol
Coxeter diagram
Cells
Faces
Vertex figure
Dual
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, with three order-8 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is an octagonal tiling, .

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.

Order-8-4 square honeycomb

bgcolor=#e8dcc3 colspan=2	Order-8-4 square honeycomb
Type	Regular honeycomb
Schläfli symbol
Coxeter diagrams	=
Cells
Faces
Edge figure
Vertex figure
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 .

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.

Order-8-5 pentagonal honeycomb

bgcolor=#e8dcc3 colspan=2	Order-8-5 pentagonal honeycomb
Type	Regular honeycomb
Schläfli symbol
Coxeter diagrams
Cells
Faces
Edge figure
Vertex figure
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 .

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.

Order-8-6 hexagonal honeycomb

bgcolor=#e8dcc3 colspan=2	Order-8-6 hexagonal honeycomb
Type	Regular honeycomb
Schläfli symbols
Coxeter diagrams	=
Cells
Faces
Edge figure
Vertex figure
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 . It has six order-8 hexagonal tilings,, 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.

It has a second construction as a uniform honeycomb, Schläfli symbol, 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

bgcolor=#e8dcc3 colspan=2	Order-8-infinite apeirogonal honeycomb
Type	Regular honeycomb
Schläfli symbols
Coxeter diagrams	↔
Cells
Faces
Edge figure
Vertex figure
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 . It has infinitely many order-8 apeirogonal tiling 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.

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

See also

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

References

• Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. . (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
• The Beauty of Geometry: Twelve Essays (1999), Dover Publications,, (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
• Jeffrey R. Weeks The Shape of Space, 2nd edition (Chapters 16–17: Geometries on Three-manifolds I, II)
• George Maxwell, Sphere Packings and Hyperbolic Reflection Groups, JOURNAL OF ALGEBRA 79,78-97 (1982) http://www.sciencedirect.com/science/article/pii/0021869382903180
• Hao Chen, Jean-Philippe Labbé, Lorentzian Coxeter groups and Boyd-Maxwell ball packings, (2013)https://arxiv.org/abs/1310.8608
• Visualizing Hyperbolic Honeycombs arXiv:1511.02851 Roice Nelson, Henry Segerman (2015)

External links

• Hyperbolic Catacombs Carousel: honeycomb YouTube, Roice Nelson
• John Baez, Visual insights: Honeycomb (2014/08/01) Honeycomb Meets Plane at Infinity (2014/08/14)
• Danny Calegari, Kleinian, a tool for visualizing Kleinian groups, Geometry and the Imagination 4 March 2014. https://web.archive.org/web/20161109004910/http://math.uchicago.edu/~dannyc/papers/kleinian_mtf_Feb_2014.pdf