# 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.

Poincaré disk model |
Ideal 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 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.

Poincaré disk model |
Ideal surface |

### 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.

Poincaré disk model |
Ideal 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 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.

Ideal surface |

### 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.

Poincaré disk model |
Ideal 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. 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 edition*ISBN: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)

## 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. [3]

Original source: https://en.wikipedia.org/wiki/Order-5-4 square honeycomb.
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