Order-3-7 heptagonal honeycomb

From The Right Wiki
Jump to navigationJump to search
Order-3-7 heptagonal honeycomb
Type Regular honeycomb
Schläfli symbol {7,3,7}
Coxeter diagrams File:CDel node 1.pngFile:CDel 7.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 7.pngFile:CDel node.png
Cells {7,3} File:Heptagonal tiling.svg
Faces {7}
Edge figure {7}
Vertex figure {3,7}
Dual self-dual
Coxeter group [7,3,7]
Properties Regular

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

Geometry

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

File:Hyperbolic honeycomb 7-3-7 poincare.png
Poincaré disk model 		File:H3 737 UHS plane at infinity.png
Ideal surface

Related polytopes and honeycombs

It a part of a sequence of regular polychora and honeycombs {p,3,p}:

{p,3,p} regular honeycombs
Space S3 Euclidean E3 H3
Form Finite Affine Compact Paracompact Noncompact
Name {3,3,3} {4,3,4} {5,3,5} {6,3,6} {7,3,7} {8,3,8} ...{∞,3,∞}
Image File:Stereographic polytope 5cell.png File:Cubic honeycomb.png File:H3 535 CC center.png File:H3 636 FC boundary.png File:Hyperbolic honeycomb 7-3-7 poincare.png File:Hyperbolic honeycomb 8-3-8 poincare.png File:Hyperbolic honeycomb i-3-i poincare.png
Cells File:Tetrahedron.png
{3,3} 		File:Hexahedron.png
{4,3} 		File:Dodecahedron.png
{5,3} 		File:Uniform tiling 63-t0.svg
{6,3} 		File:Heptagonal tiling.svg
{7,3} 		File:H2-8-3-dual.svg
{8,3} 		File:H2-I-3-dual.svg
{∞,3}
Vertex
figure 		File:5-cell verf.svg
{3,3} 		File:Cubic honeycomb verf.svg
{3,4} 		File:Order-5 dodecahedral honeycomb verf.png
{3,5} 		File:Uniform tiling 63-t2.svg
{3,6} 		File:Order-7 triangular tiling.svg
{3,7} 		File:H2-8-3-primal.svg
{3,8} 		File:H2 tiling 23i-4.png
{3,∞}

Order-3-8 octagonal honeycomb

Order-3-8 octagonal honeycomb
Type Regular honeycomb
Schläfli symbols {8,3,8}
{8,(3,4,3)}
Coxeter diagrams File:CDel node 1.pngFile:CDel 8.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 8.pngFile:CDel node.png
File:CDel node 1.pngFile:CDel 8.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 8.pngFile:CDel node h0.png = File:CDel node 1.pngFile:CDel 8.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel branch.pngFile:CDel label4.png
Cells {8,3} File:H2-8-3-dual.svg
Faces {8}
Edge figure {8}
Vertex figure {3,8} File:H2-8-3-primal.svg
{(3,8,3)} File:H2 tiling 338-4.png
Dual self-dual
Coxeter group [8,3,8]
[8,((3,4,3))]
Properties Regular

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

File:Hyperbolic honeycomb 8-3-8 poincare.png
Poincaré disk model

It has a second construction as a uniform honeycomb, Schläfli symbol {8,(3,4,3)}, Coxeter diagram, File:CDel node 1.pngFile:CDel 8.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel branch.pngFile:CDel label4.png, with alternating types or colors of cells. In Coxeter notation the half symmetry is [8,3,8,1+] = [8,((3,4,3))].

Order-3-infinite apeirogonal honeycomb

Order-3-infinite apeirogonal honeycomb
Type Regular honeycomb
Schläfli symbols {∞,3,∞}
{∞,(3,∞,3)}
Coxeter diagrams File:CDel node 1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.png
File:CDel node 1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node h0.png ↔ File:CDel node 1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel branch.pngFile:CDel labelinfin.png
Cells {∞,3} File:H2-I-3-dual.svg
Faces {∞}
Edge figure {∞}
Vertex figure File:H2 tiling 23i-4.png {3,∞}
File:H2 tiling 33i-4.png {(3,∞,3)}
Dual self-dual
Coxeter group [∞,3,∞]
[∞,((3,∞,3))]
Properties Regular

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

File:Hyperbolic honeycomb i-3-i poincare.png
Poincaré disk model 		File:H3 i3i UHS plane at infinity.png
Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {∞,(3,∞,3)}, Coxeter diagram, File:CDel node 1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel branch.pngFile:CDel labelinfin.png, with alternating types or colors of apeirogonal tiling 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

Retrieved from "https://rightwiki.org/index.php?title=Order-3-7_heptagonal_honeycomb&oldid=18304065#Order-3-8_octagonal_honeycomb"