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.
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.
Related polytopes and honeycombs
It is a part of a series of regular polytopes and honeycombs with { p,3,6} Schläfli symbol, and triangular tiling vertex figures.
Order-3-6 octagonal honeycomb
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.
Order-3-6 apeirogonal honeycomb
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}.
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. . (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) 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
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.
Honeycombs (geometry) Heptagonal tilings Isogonal 3-honeycombs Isochoric 3-honeycombs Order-3-n 3-honeycombs Order-n-6 3-honeycombs Regular 3-honeycombs