Order-7 cubic honeycomb
TypeRegular honeycomb
Schläfli symbols{4,3,7}
Coxeter diagrams
Cells{4,3}
Faces{4}
Edge figure{7}
Vertex figure{3,7}
Dual{7,3,4}
Coxeter group[4,3,7]
PropertiesRegular

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 seven 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

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}:

{4,3,p} polytopes
Space S3 H3
Form Finite Compact Paracompact Noncompact
Name {4,3,3} {4,3,4} {4,3,5} {4,3,6} {4,3,7} {4,3,8} ... {4,3,∞}
Image
Vertex
figure
{3,3}
{3,4}
{3,5}
{3,6}
{3,7}
{3,8}
{3,∞}

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}

Order-8 cubic honeycomb

Order-8 cubic honeycomb
TypeRegular honeycomb
Schläfli symbols{4,3,8}
{4,(3,8,3)}
Coxeter diagrams
=
Cells{4,3}
Faces{4}
Edge figure{8}
Vertex figure{3,8}, {(3,4,3)}
Dual{8,3,4}
Coxeter group[4,3,8]
[4,((3,4,3))]
PropertiesRegular

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 eight 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
TypeRegular honeycomb
Schläfli symbols{4,3,∞}
{4,(3,∞,3)}
Coxeter diagrams
=
Cells{4,3}
Faces{4}
Edge figure{∞}
Vertex figure{3,∞}, {(3,∞,3)}
Dual{∞,3,4}
Coxeter group[4,3,∞]
[4,((3,∞,3))]
PropertiesRegular

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

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)
    • Hao Chen, Jean-Philippe Labbé, Lorentzian Coxeter groups and Boyd-Maxwell ball packings, (2013)
    • Visualizing Hyperbolic Honeycombs arXiv:1511.02851 Roice Nelson, Henry Segerman (2015)
    This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.