Order-5-4 square honeycomb

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}.

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.

It a part of a sequence of regular polychora and honeycombs {p,5,p}:

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.

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.

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))].

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.

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.

It has a second construction as a uniform honeycomb, Schläfli symbol {∞,(5,∞,5)}, Coxeter diagram, , with alternating types or colors of cells.

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

• 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 99-35678,ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
• Jeffrey R. Weeks The Shape of Space, 2nd editionISBN 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) [1]
• Hao Chen, Jean-Philippe Labbé, Lorentzian Coxeter groups and Boyd-Maxwell ball packings, (2013)[2]
• Visualizing Hyperbolic Honeycombs arXiv:1511.02851 Roice Nelson, Henry Segerman (2015)

• 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. [3]