Order-7 cubic honeycomb

Order-7 cubic honeycomb

Type

Schläfli symbols

Coxeter diagrams

Cells

Faces

Edge figure

Vertex figure

Dual

Coxeter group

Properties

In the geometry of hyperbolic 3-space, the order-7 cubic honeycomb is a regular space-filling tessellation (or honeycomb). With Schläfli symbol {4,3,7}, it has infinitely many cubes {4,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many cubes existing around each vertex in an order-7 triangular tiling vertex arrangement.

Images

Poincaré disk model


Cell-centered
Hyperbolic_honeycomb_4-3-8_poincare.png


One cell at center


One cell with ideal surface

It is one of a series of regular polytopes and honeycombs with cubic cells: {4,3,p}:

It is a part of a sequence of hyperbolic honeycombs with order-7 triangular tiling vertex figures, {p,3,7}.

{3,3,7}

{4,3,7}

{5,3,7}

{6,3,7}

{7,3,7}

{8,3,7}

{∞,3,7}
Hyperbolic_honeycomb_3-3-7_poincare_cc.png
Hyperbolic_honeycomb_4-3-7_poincare_cc.png
Hyperbolic_honeycomb_5-3-7_poincare_cc.png
Hyperbolic_honeycomb_6-3-7_poincare.png
Hyperbolic_honeycomb_7-3-7_poincare.png
Hyperbolic_honeycomb_8-3-7_poincare.png
Hyperbolic_honeycomb_i-3-7_poincare.png

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Order-8 cubic honeycomb

Order-8 cubic honeycomb

Type

Schläfli symbols

Coxeter diagrams

Cells

Faces

Edge figure

Vertex figure

Dual

Coxeter group

Properties

In the geometry of hyperbolic 3-space, the order-8 cubic honeycomb a regular space-filling tessellation (or honeycomb). With Schläfli symbol {4,3,8}. It has infinitely many cubes {4,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many cubes existing around each vertex in an order-8 triangular tiling vertex arrangement.


Poincaré disk model
Cell-centered


Poincaré disk model

It has a second construction as a uniform honeycomb, Schläfli symbol {4,(3,4,3)}, Coxeter diagram, , with alternating types or colors of cubic cells. {{-}}

Infinite-order cubic honeycomb

Infinite-order cubic honeycomb

Type

Schläfli symbols

Coxeter diagrams

Cells

Faces

Edge figure

Vertex figure

Dual

Coxeter group

Properties

In the geometry of hyperbolic 3-space, the infinite-order cubic honeycomb a regular space-filling tessellation (or honeycomb). With Schläfli symbol {4,3,∞}. It has infinitely many cubes {4,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many cubes existing around each vertex in an infinite-order triangular tiling vertex arrangement.


Poincaré disk model
Cell-centered


Poincaré disk model

It has a second construction as a uniform honeycomb, Schläfli symbol {4,(3,∞,3)}, Coxeter diagram, , with alternating types or colors of cubic cells.

See also

  • Convex uniform honeycombs in hyperbolic space
  • List of regular polytopes
  • Infinite-order hexagonal tiling 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) 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)