Order-3-7 hexagonal honeycomb

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Order-3-7 hexagonal honeycomb
Hyperbolic honeycomb 6-3-7 poincare.png
Poincaré disk model
Type Regular honeycomb
Schläfli symbol {6,3,7}
Coxeter diagrams CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 7.pngCDel node.png
Cells {6,3} Uniform tiling 63-t0.png
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

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
H3 637 UHS plane at infinity view 1.png
Rendered intersection of honeycomb with the ideal plane in Poincaré half-space model
H3 637 UHS plane at infinity view 2.png
Closeup

Related polytopes and honeycombs

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

Order-3-8 hexagonal honeycomb

Order-3-8 hexagonal honeycomb
Type Regular honeycomb
Schläfli symbols {6,3,8}
{6,(3,4,3)}
Coxeter diagrams CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node h0.png = CDel node 1.pngCDel 6.pngCDel node.pngCDel split1.pngCDel branch.pngCDel label4.png
Cells {6,3} Uniform tiling 63-t0.png
Faces {6}
Edge figure {8}
Vertex figure {3,8} {(3,4,3)}
H2-8-3-primal.svgUniform tiling 433-t2.png
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.

Hyperbolic honeycomb 6-3-8 poincare.png
Poincaré disk model

It has a second construction as a uniform honeycomb, Schläfli symbol {6,(3,4,3)}, Coxeter diagram, CDel node 1.pngCDel 6.pngCDel node.pngCDel split1.pngCDel branch.pngCDel label4.png, 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

Order-3-infinite hexagonal honeycomb
Type Regular honeycomb
Schläfli symbols {6,3,∞}
{6,(3,∞,3)}
Coxeter diagrams CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node h0.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel split1.pngCDel branch.pngCDel labelinfin.png
CDel node 1.pngCDel 6.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel infin.pngCDel node.pngCDD 6-3star-infin.png
Cells {6,3} Uniform tiling 63-t0.png
Faces {6}
Edge figure {∞}
Vertex figure {3,∞}, {(3,∞,3)}
H2 tiling 23i-4.pngH2 tiling 33i-4.png
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.

Hyperbolic honeycomb 6-3-i poincare.png
Poincaré disk model
H3 63i UHS plane at infinity.png
Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {6,(3,∞,3)}, Coxeter diagram, CDel node 1.pngCDel 6.pngCDel node.pngCDel split1.pngCDel branch.pngCDel labelinfin.png, with alternating types or colors of hexagonal tiling cells.

See also

  • Convex uniform honeycombs in hyperbolic space
  • List of regular polytopes
  • Infinite-order dodecahedral honeycomb

References

External links