# Order-5-4 square honeycomb

Order-4-5 square honeycomb
Type Regular honeycomb
Schläfli symbol {4,5,4}
Coxeter diagrams       Cells {4,5} Faces {4}
Edge figure {4}
Vertex figure {5,4}
Dual self-dual
Coxeter group [4,5,4]
Properties Regular

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.

## 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 honeycomb
Type Regular honeycomb
Schläfli symbol {5,5,5}
Coxeter diagrams       Cells {5,5} Faces {5}
Edge figure {5}
Vertex figure {5,5}
Dual self-dual
Coxeter group [5,5,5]
Properties Regular

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.

### Order-5-6 hexagonal honeycomb

Order-5-6 hexagonal honeycomb
Type Regular honeycomb
Schläfli 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)} Dual self-dual
Coxeter group [6,5,6]
[6,((5,3,5))]
Properties Regular

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))].

### Order-5-7 heptagonal honeycomb

Order-5-7 hexagonal honeycomb
Type Regular honeycomb
Schläfli symbols {7,5,7}
Coxeter diagrams       Cells {7,5} Faces {6}
Edge figure {6}
Vertex figure {5,7} Dual self-dual
Coxeter group [7,5,7]
Properties Regular

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.

### Order-5-infinite apeirogonal honeycomb

Order-5-infinite apeirogonal honeycomb
Type Regular honeycomb
Schläfli symbols {∞,5,∞}
{∞,(5,∞,5)}
Coxeter diagrams                    Cells {∞,5} Faces {∞}
Edge figure {∞}
Vertex figure {5,∞} {(5,∞,5)}
Dual self-dual
Coxeter group [∞,5,∞]
[∞,((5,∞,5))]
Properties Regular

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.