Regular space-filling tessellation
In the geometry of hyperbolic 3-space, the order-6-4 square honeycomb (or 4,6,4 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,6,4}.
Geometry.
All vertices are ultra-ideal (existing beyond the ideal boundary) with four order-6 square tilings existing around each edge and with an order-4 hexagonal tiling vertex figure.
Related polytopes and honeycombs.
It a part of a sequence of regular polychora and honeycombs {"p",6,"p"}:
Order-6-5 hexagonal honeycomb.
In the geometry of hyperbolic 3-space, the order-6-5 pentagonal honeycomb (or 5,6,5 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {5,6,5}.
All vertices are ultra-ideal (existing beyond the ideal boundary) with five order-6 pentagonal tilings existing around each edge and with an order-5 hexagonal tiling vertex figure.
Order-6-6 hexagonal honeycomb.
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, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [6,6,6,1+] = [6,((6,3,6))].
Order-6-infinite apeirogonal honeycomb.
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, , with alternating types or colors of cells.
References.
 ;templatestyles src="Reflist/styles.css" / ;