Order-6-4 square honeycomb
It a part of a sequence of regular polychora and honeycombs {p ,6,p }:
Order-6-6 hexagonal honeycomb
Order-5-6 hexagonal honeycomb
Type Regular honeycomb
Schläfli symbols {6,6,6}
Coxeter diagrams
Cells {6,6}
Faces {6}
Edge figure {6}
Vertex figure {6,6}
Dual self-dual
Coxeter group [6,5,6]
Properties Regular
In the geometry of hyperbolic 3-space , the order-6-6 hexagonal honeycomb (or 6,6,6 honeycomb ) is a regular space-filling tessellation (or honeycomb ) with Schläfli symbol {6,6,6}. It has six order-6 hexagonal tilings , {6,6}, 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 hexagonal tiling vertex arrangement .
It has a second construction as a uniform honeycomb, Schläfli symbol {6,(6,3,6)}, Coxeter diagram, + ] = [6,((6,3,6))].
Order-6-infinite apeirogonal honeycomb
Order-6-infinite apeirogonal honeycomb
Type Regular honeycomb
Schläfli symbols {∞,6,∞}
Coxeter diagrams
Cells {∞,6}
Faces {∞}
Edge figure {∞}
Vertex figure {6,∞}
Dual self-dual
Coxeter group [∞,6,∞]
Properties Regular
In the geometry of hyperbolic 3-space , the order-6-infinite apeirogonal honeycomb (or ∞,6,∞ honeycomb ) is a regular space-filling tessellation (or honeycomb ) with Schläfli symbol {∞,6,∞}. It has infinitely many order-6 apeirogonal tiling {∞,6} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-6 apeirogonal tilings existing around each vertex in an infinite-order square tiling vertex arrangement .
It has a second construction as a uniform honeycomb, Schläfli symbol {∞,(6,∞,6)}, Coxeter diagram,
References
Coxeter , Regular Polytopes ISBN 0-486-61480-8 294–296)The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678 , ISBN 0-486-40919-8 Regular Honeycombs in Hyperbolic Space Archived 2016-06-10 at the Wayback Machine Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 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)
