Order-5-4 square honeycomb
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 .
It a part of a sequence of regular polychora and honeycombs {p ,5,p }:
{p ,5,p } regular honeycombs
Space
H3
Form
Compact
Noncompact
Name
{3,5,3}
{4,5,4}
{5,5,5}
{6,5,6}
{7,5,7}
{8,5,8}
...{∞,5,∞}
Image
Cellsp ,5}
{3,5}
{4,5}
{5,5}
{6,5}
{7,5}
{8,5}
{∞,5}
Vertexp }
{5,3}
{5,4}
{5,5}
{5,6}
{5,7}
{5,8}
{5,∞}
Order-5-5 pentagonal honeycomb
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}
Coxeter diagrams
Cells
{6,5}
Faces
{6}
Edge figure
{6}
Vertex figure
{5,6}
Dual
self-dual
Coxeter group [6,5,6]
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, + ] = [6,((5,3,5))].
Order-5-7 heptagonal honeycomb
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,∞}
Coxeter diagrams
Cells
{∞,5}
Faces
{∞}
Edge figure
{∞}
Vertex figure
{5,∞}
Dual
self-dual
Coxeter group [∞,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,
See also
References
Coxeter , Regular Polytopes ISBN 0-486-61480-8 The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678 , ISBN 0-486-40919-8 Regular Honeycombs in Hyperbolic Space Archived 2016-06-10 at the Wayback Machine ) Table IIIJeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 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
