Order-3-7 heptagonal honeycomb

All vertices are ultra-ideal (existing beyond the ideal boundary) with seven heptagonal tilings existing around each edge and with an order-7 triangular tiling vertex figure.

It a part of a sequence of regular polychora and honeycombs {p,3,p}:

{{Symmetric2 tessellations}}

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

It has a second construction as a uniform honeycomb, Schläfli symbol {8,(3,4,3)}, Coxeter diagram, {{CDD|node_1|8|node|split1|branch|label4}}, with alternating types or colors of cells. In Coxeter notation the half symmetry is [8,3,8,1⁺] = [8,((3,4,3))].

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

It has a second construction as a uniform honeycomb, Schläfli symbol {∞,(3,∞,3)}, Coxeter diagram, {{CDD|node_1|infin|node|split1|branch|labelinfin}}, with alternating types or colors of apeirogonal tiling cells.

• Convex uniform honeycombs in hyperbolic space
• List of regular polytopes
• Infinite-order dodecahedral honeycomb

{{reflist}}

• 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|99035678}}, {{ISBN|0-486-40919-8}} (Chapter 10, [http://www.mathunion.org/ICM/ICM1954.3/Main/icm1954.3.0155.0169.ocr.pdf 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) [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]
• [https://arxiv.org/abs/1511.02851 Visualizing Hyperbolic Honeycombs arXiv:1511.02851] Roice Nelson, Henry Segerman (2015)

• John Baez, Visual insights: [http://blogs.ams.org/visualinsight/2014/08/01/733-honeycomb/ {7,3,3} Honeycomb] (2014/08/01) [http://blogs.ams.org/visualinsight/2014/08/14/733-honeycomb-meets-plane-at-infinity/ {7,3,3} Honeycomb Meets Plane at Infinity] (2014/08/14)
• Danny Calegari, [http://lamington.wordpress.com/2014/03/04/kleinian-a-tool-for-visualizing-kleinian-groups/Kleinian 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]

Category:Heptagonal tilings

Category:Infinite-order tilings

Category:3-honeycombs

Category:Regular 3-honeycombs