Order-4-5 pentagonal honeycomb
In the geometry of hyperbolic 3-space , the order-4-5 pentagonal honeycomb a regular space-filling tessellation (or honeycomb ) with Schläfli symbol {5,4,5}.
Geometry
All vertices are ultra-ideal (existing beyond the ideal boundary) with five order-4 pentagonal tilings existing around each edge and with an order-5 square tiling vertex figure .
Related polytopes and honeycombs
It a part of a sequence of regular polychora and honeycombs {p ,4,p }:
{p ,4,p } regular honeycombs
Space
S3
Euclidean E3
H3
Form
Finite
Paracompact
Noncompact
Name
{3,4,3}
{4,4,4}
{5,4,5}
{6,4,6}
{7,4,7}
{8,4,8}
...{∞,4,∞}
Image
Cells {p ,4}
{3,4}
{4,4}
{5,4}
{6,4}
{7,4}
{8,4}
{∞,4}
Vertex figure {4,p }
{4,3}
{4,4}
{4,5}
{4,6}
{4,7}
{4,8}
{4,∞}
Order-4-6 hexagonal honeycomb
Order-4-6 hexagonal honeycomb
Type
Regular honeycomb
Schläfli symbols
{6,4,6} {6,(4,3,4)}
Coxeter diagrams
=
Cells
{6,4}
Faces
{6}
Edge figure
{6}
Vertex figure
{4,6} {(4,3,4)}
Dual
self-dual
Coxeter group
[6,4,6] [6,((4,3,4))]
Properties
Regular
In the geometry of hyperbolic 3-space , the order-4-6 hexagonal honeycomb is a regular space-filling tessellation (or honeycomb ) with Schläfli symbol {6,3,6}. It has six order-4 hexagonal tilings , {6,4}, 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 square tiling vertex arrangement .
It has a second construction as a uniform honeycomb, Schläfli symbol {6,(4,3,4)}, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [6,4,6,1+ ] = [6,((4,3,4))].
Order-4-infinite apeirogonal honeycomb
Order-4-infinite apeirogonal honeycomb
Type
Regular honeycomb
Schläfli symbols
{∞,4,∞} {∞,(4,∞,4)}
Coxeter diagrams
↔
Cells
{∞,4}
Faces
{∞}
Edge figure
{∞}
Vertex figure
{4,∞} {(4,∞,4)}
Dual
self-dual
Coxeter group
[∞,4,∞] [∞,((4,∞,4))]
Properties
Regular
In the geometry of hyperbolic 3-space , the order-4-infinite apeirogonal honeycomb is a regular space-filling tessellation (or honeycomb ) with Schläfli symbol {∞,4,∞}. It has infinitely many order-4 apeirogonal tiling {∞,4} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many hexagonal tilings existing around each vertex in an infinite-order square tiling vertex arrangement .
It has a second construction as a uniform honeycomb, Schläfli symbol {∞,(4,∞,4)}, Coxeter diagram, , with alternating types or colors of 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) [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