Order-4-5 square honeycombTypeRegular honeycombSchläfli symbol{4,5,4}Coxeter diagramsCells{4,5} Faces{4}Edge figure{4}Vertex figure{5,4}Dualself-dualCoxeter group[4,5,4]PropertiesRegular In the geometry of hyperbolic 3-space, the order-5-4 square honeycomb (or 4,5,4 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,5,4}.
Geometry 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 pentagonal tiling vertex figure.
Poincaré disk modelIdeal surface
Related polytopes and honeycombs It a part of a sequence of regular polychora and honeycombs {p,5,p}:
Order-5-5 pentagonal honeycomb Order-5-5 pentagonal honeycombTypeRegular honeycombSchläfli symbol{5,5,5}Coxeter diagramsCells{5,5} Faces{5}Edge figure{5}Vertex figure{5,5}Dualself-dualCoxeter group[5,5,5]PropertiesRegular In the geometry of hyperbolic 3-space, the order-5-5 pentagonal honeycomb (or 5,5,5 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {5,5,5}.
All vertices are ultra-ideal (existing beyond the ideal boundary) with five order-5 pentagonal tilings existing around each edge and with an order-5 pentagonal tiling vertex figure.
Poincaré disk modelIdeal surface
Order-5-6 hexagonal honeycomb Order-5-6 hexagonal honeycombTypeRegular honeycombSchläfli symbols{6,5,6}{6,(5,3,5)}Coxeter diagrams = Cells{6,5} Faces{6}Edge figure{6}Vertex figure{5,6} {(5,3,5)} Dualself-dualCoxeter group[6,5,6][6,((5,3,5))]PropertiesRegular In the geometry of hyperbolic 3-space, the order-5-6 hexagonal honeycomb (or 6,5,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,5,6}. It has six order-5 hexagonal tilings, {6,5}, 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 pentagonal tiling vertex arrangement.
Poincaré disk modelIdeal surface
It has a second construction as a uniform honeycomb, Schläfli symbol {6,(5,3,5)}, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [6,5,6,1+] = [6,((5,3,5))].
Order-5-7 heptagonal honeycomb Order-5-7 hexagonal honeycombTypeRegular honeycombSchläfli symbols{7,5,7}Coxeter diagramsCells{7,5} Faces{6}Edge figure{6}Vertex figure{5,7} Dualself-dualCoxeter group[7,5,7]PropertiesRegular In the geometry of hyperbolic 3-space, the order-5-7 heptagonal honeycomb (or 7,5,7 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {7,5,7}. It has seven order-5 heptagonal tilings, {7,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many heptagonal tilings existing around each vertex in an order-7 pentagonal tiling vertex arrangement.
Ideal surface
Order-5-infinite apeirogonal honeycomb Order-5-infinite apeirogonal honeycombTypeRegular honeycombSchläfli symbols{∞,5,∞}{∞,(5,∞,5)}Coxeter diagrams ↔ Cells{∞,5} Faces{∞}Edge figure{∞}Vertex figure {5,∞} {(5,∞,5)}Dualself-dualCoxeter group[∞,5,∞][∞,((5,∞,5))]PropertiesRegular In the geometry of hyperbolic 3-space, the order-5-infinite apeirogonal honeycomb (or ∞,5,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {∞,5,∞}. It has infinitely many order-5 apeirogonal tilings {∞,5} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-5 apeirogonal tilings existing around each vertex in an infinite-order pentagonal tiling vertex arrangement.
Poincaré disk modelIdeal surface
It has a second construction as a uniform honeycomb, Schläfli symbol {∞,(5,∞,5)}, Coxeter diagram, , with alternating types or colors of cells.
See also Convex uniform honeycombs in hyperbolic space List of regular polytopes Infinite-order dodecahedral honeycomb
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.