Order-3-6 heptagonal honeycomb

In the geometry of hyperbolic 3-space, the order-3-6 heptagonal 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.

Order-3-6 heptagonal honeycomb
TypeRegular honeycomb
Schläfli symbol{7,3,6}
{7,3[3]}
Coxeter diagram
=
Cells{7,3}
Faces{7}
Vertex figure{3,6}
Dual{6,3,7}
Coxeter group[7,3,6]
[7,3[3]]
PropertiesRegular

Geometry

The Schläfli symbol of the order-3-6 heptagonal honeycomb is {7,3,6}, with six heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is an triangular tiling, {3,6}.

It has a quasiregular construction, , which can be seen as alternately colored cells.


Poincaré disk model
Ideal surface

It is a part of a series of regular polytopes and honeycombs with {p,3,6} Schläfli symbol, and triangular tiling vertex figures.

Hyperbolic uniform honeycombs: {p,3,6} and {p,3[3]}
Form Paracompact Noncompact
Name {3,3,6}
{3,3[3]}		 {4,3,6}
{4,3[3]}		 {5,3,6}
{5,3[3]}		 {6,3,6}
{6,3[3]}		 {7,3,6}
{7,3[3]}		 {8,3,6}
{8,3[3]}		 ... {∞,3,6}
{∞,3[3]}






Image
Cells
{3,3}
{4,3}
{5,3}
{6,3}
{7,3}
{8,3}
{∞,3}

Order-3-6 octagonal honeycomb
Order-3-6 octagonal honeycomb
TypeRegular honeycomb
Schläfli symbol{8,3,6}
{8,3[3]}
Coxeter diagram
=
Cells{8,3}
FacesOctagon {8}
Vertex figuretriangular tiling {3,6}
Dual{6,3,8}
Coxeter group[8,3,6]
[8,3[3]]
PropertiesRegular

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

It has a quasiregular construction, , which can be seen as alternately colored cells.


Poincaré disk model

Order-3-6 apeirogonal honeycomb
Order-3-6 apeirogonal honeycomb
TypeRegular honeycomb
Schläfli symbol{∞,3,6}
{∞,3[3]}
Coxeter diagram
=
Cells{∞,3}
FacesApeirogon {∞}
Vertex figuretriangular tiling {3,6}
Dual{6,3,∞}
Coxeter group[∞,3,6]
[∞,3[3]]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-3-6 apeirogonal 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-3-6 apeirogonal honeycomb is {∞,3,6}, with six order-3 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is a triangular tiling, {3,6}.


Poincaré disk model
Ideal surface

It has a quasiregular construction, , which can be seen as alternately colored cells.

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

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)
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