In the geometry of hyperbolic 3-space, the order-4-5 pentagonal honeycomb a regular space-filling tessellation (or honeycomb) with Schläfli symbol { 5,4,5}.
Geometry
All vertices are ultra-ideal (existing beyond the ideal boundary) with five order-4 pentagonal tilings existing around each edge and with an order-5 square tiling vertex figure.
Related polytopes and honeycombs
It a part of a sequence of regular polychora and honeycombs { p,4,p}:
Order-4-6 hexagonal honeycomb
In the geometry of hyperbolic 3-space, the order-4-6 hexagonal honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol { 6,3,6}. It has six order-4 hexagonal tilings, { 6,4}, 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 square tiling vertex arrangement.
It has a second construction as a uniform honeycomb, Schläfli symbol { 6,(4,3,4)}, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [6,4,6,1+] = [6,((4,3,4))].
Order-4-infinite apeirogonal honeycomb
In the geometry of hyperbolic 3-space, the order-4-infinite apeirogonal honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol { ∞,4,∞}. It has infinitely many order-4 apeirogonal tiling { ∞,4} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many hexagonal tilings existing around each vertex in an infinite-order square tiling vertex arrangement.
It has a second construction as a uniform honeycomb, Schläfli symbol { ∞,(4,∞,4)}, 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: { 5,4,3} Honeycomb (2014/08/01) { 5,4,3} 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) Pentagonal tilings Infinite-order tilings Isogonal 3-honeycombs Isochoric 3-honeycombs Order-4-n 3-honeycombs Order-n-5 3-honeycombs Regular 3-honeycombs
