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Order-3-7 hexagonal honeycomb

January 01, 1970

Order-3-7 hexagonal honeycomb

Poincaré disk model
Type Regular honeycomb
Schläfli symbol {6,3,7}
Coxeter diagrams
Cells {6,3}
Faces {6}
Edge figure {7}
Vertex figure {3,7}
Dual {7,3,6}
Coxeter group [6,3,7]
Properties Regular

In the geometry of hyperbolic 3-space, the order-3-7 hexagonal honeycomb or (6,3,7 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,3,7}.

Geometry edit

All vertices are ultra-ideal (existing beyond the ideal boundary) with seven hexagonal tilings existing around each edge and with an order-7 triangular tiling vertex figure.

Ideal surface
 
Rendered intersection of honeycomb with the ideal plane in Poincaré half-space model 		 
Closeup

Related polytopes and honeycombs edit

It a part of a sequence of regular polychora and honeycombs with hexagonal tiling cells.

{6,3,p} honeycombs
Space H3
Form Paracompact Noncompact
Name {6,3,3} {6,3,4} {6,3,5} {6,3,6} {6,3,7} {6,3,8} ... {6,3,∞}
Coxeter
       
        		               
     
        		               
     
        		               
       		       
      
 
Image              
Vertex
figure
{3,p}
      		 
{3,3}
      		 
{3,4}
     
    		 
{3,5}
      		 
{3,6}
     
    		 
{3,7}
      		 
{3,8}
     
     		 
{3,∞}
     
    

Order-3-8 hexagonal honeycomb edit

Order-3-8 hexagonal honeycomb
Type Regular honeycomb
Schläfli symbols {6,3,8}
{6,(3,4,3)}
Coxeter diagrams        
        =       
Cells {6,3}  
Faces {6}
Edge figure {8}
Vertex figure {3,8} {(3,4,3)}
  
Dual {8,3,6}
Coxeter group [6,3,8]
[6,((3,4,3))]
Properties Regular

In the geometry of hyperbolic 3-space, the order-3-8 hexagonal honeycomb or (6,3,8 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,3,8}. It has eight hexagonal tilings, {6,3}, 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-8 triangular tiling vertex arrangement.

It has a second construction as a uniform honeycomb, Schläfli symbol {6,(3,4,3)}, Coxeter diagram,       , with alternating types or colors of tetrahedral cells. In Coxeter notation the half symmetry is [6,3,8,1+] = [6,((3,4,3))].

Order-3-infinite hexagonal honeycomb edit

Order-3-infinite hexagonal honeycomb
Type Regular honeycomb
Schläfli symbols {6,3,∞}
{6,(3,∞,3)}
Coxeter diagrams        
        ↔       
        ↔  
Cells {6,3}  
Faces {6}
Edge figure {∞}
Vertex figure {3,∞}, {(3,∞,3)}
  
Dual {∞,3,6}
Coxeter group [6,3,∞]
[6,((3,∞,3))]
Properties Regular

In the geometry of hyperbolic 3-space, the order-3-infinite hexagonal honeycomb or (6,3,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,3,∞}. It has infinitely many hexagonal tiling {6,3} 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 triangular tiling vertex arrangement.

 
Poincaré disk model 		 
Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {6,(3,∞,3)}, Coxeter diagram,       , with alternating types or colors of hexagonal tiling cells.

See also edit

References edit

  • 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 edit

  • 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.

January 01, 1970
order, hexagonal, honeycomb, poincaré, disk, model, type, regular, honeycomb, schläfli, symbol, coxeter, diagrams, cells, faces, edge, figure, vertex, figure, dual, coxeter, group, properties, regular, geometry, hyperbolic, space, order, hexagonal, honeycomb, . Order 3 7 hexagonal honeycomb Poincare disk model Type Regular honeycomb Schlafli symbol 6 3 7 Coxeter diagrams Cells 6 3 Faces 6 Edge figure 7 Vertex figure 3 7 Dual 7 3 6 Coxeter group 6 3 7 Properties Regular In the geometry of hyperbolic 3 space the order 3 7 hexagonal honeycomb or 6 3 7 honeycomb a regular space filling tessellation or honeycomb with Schlafli symbol 6 3 7 Contents 1 Geometry 2 Related polytopes and honeycombs 2 1 Order 3 8 hexagonal honeycomb 2 2 Order 3 infinite hexagonal honeycomb 3 See also 4 References 5 External linksGeometry editAll vertices are ultra ideal existing beyond the ideal boundary with seven hexagonal tilings existing around each edge and with an order 7 triangular tiling vertex figure Ideal surface nbsp Rendered intersection of honeycomb with the ideal plane in Poincare half space model nbsp CloseupRelated polytopes and honeycombs editIt a part of a sequence of regular polychora and honeycombs with hexagonal tiling cells 6 3 p honeycombs vte Space H3 Form Paracompact Noncompact Name 6 3 3 6 3 4 6 3 5 6 3 6 6 3 7 6 3 8 6 3 Coxeter nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Image nbsp nbsp nbsp nbsp nbsp nbsp nbsp Vertexfigure 3 p nbsp nbsp nbsp nbsp nbsp nbsp 3 3 nbsp nbsp nbsp nbsp nbsp nbsp 3 4 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 3 5 nbsp nbsp nbsp nbsp nbsp nbsp 3 6 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 3 7 nbsp nbsp nbsp nbsp nbsp nbsp 3 8 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 3 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Order 3 8 hexagonal honeycomb edit Order 3 8 hexagonal honeycomb Type Regular honeycomb Schlafli symbols 6 3 8 6 3 4 3 Coxeter diagrams nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Cells 6 3 nbsp Faces 6 Edge figure 8 Vertex figure 3 8 3 4 3 nbsp nbsp Dual 8 3 6 Coxeter group 6 3 8 6 3 4 3 Properties Regular In the geometry of hyperbolic 3 space the order 3 8 hexagonal honeycomb or 6 3 8 honeycomb is a regular space filling tessellation or honeycomb with Schlafli symbol 6 3 8 It has eight hexagonal tilings 6 3 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 8 triangular tiling vertex arrangement nbsp Poincare disk model It has a second construction as a uniform honeycomb Schlafli symbol 6 3 4 3 Coxeter diagram nbsp nbsp nbsp nbsp nbsp nbsp with alternating types or colors of tetrahedral cells In Coxeter notation the half symmetry is 6 3 8 1 6 3 4 3 Order 3 infinite hexagonal honeycomb edit Order 3 infinite hexagonal honeycomb Type Regular honeycomb Schlafli symbols 6 3 6 3 3 Coxeter diagrams nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Cells 6 3 nbsp Faces 6 Edge figure Vertex figure 3 3 3 nbsp nbsp Dual 3 6 Coxeter group 6 3 6 3 3 Properties Regular In the geometry of hyperbolic 3 space the order 3 infinite hexagonal honeycomb or 6 3 honeycomb is a regular space filling tessellation or honeycomb with Schlafli symbol 6 3 It has infinitely many hexagonal tiling 6 3 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 triangular tiling vertex arrangement nbsp Poincare disk model nbsp Ideal surface It has a second construction as a uniform honeycomb Schlafli symbol 6 3 3 Coxeter diagram nbsp nbsp nbsp nbsp nbsp nbsp with alternating types or colors of hexagonal tiling cells See also editConvex uniform honeycombs in hyperbolic space List of regular polytopes Infinite order dodecahedral honeycombReferences editCoxeter 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 Labbe Lorentzian Coxeter groups and Boyd Maxwell ball packings 2013 2 Visualizing Hyperbolic Honeycombs arXiv 1511 02851 Roice Nelson Henry Segerman 2015 External links editJohn 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 Retrieved from https en wikipedia org w index php title Order 3 7 hexagonal honeycomb amp oldid 1196139781 Order 3 8 hexagonal honeycomb, wikipedia, wiki, book, books, library,

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