Order-6-4 square honeycomb

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Order-4-6 square honeycomb
Type Regular honeycomb
Schläfli symbol {4,6,4}
Coxeter diagrams CDel node 1.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png
Cells {4,6} H2 tiling 246-4.png
Faces {4}
Edge figure {4}
Vertex figure {6,4}
Dual self-dual
Coxeter group [4,6,4]
Properties Regular

In the geometry of hyperbolic 3-space, the order-6-4 square honeycomb (or 4,6,4 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,6,4}.

Geometry

All vertices are ultra-ideal (existing beyond the ideal boundary) with four order-6 square tilings existing around each edge and with an order-4 hexagonal tiling vertex figure.

Hyperbolic honeycomb 4-6-4 poincare.png
Poincaré disk model 		H3 464 UHS plane at infinity.png
Ideal surface

Related polytopes and honeycombs

It a part of a sequence of regular polychora and honeycombs {p,6,p}:

Order-6-5 hexagonal honeycomb

Order-6-5 pentagonal honeycomb
Type Regular honeycomb
Schläfli symbol {5,6,5}
Coxeter diagrams CDel node 1.pngCDel 5.pngCDel node.pngCDel 6.pngCDel node.pngCDel 5.pngCDel node.png
Cells {5,6} H2 tiling 256-4.png
Faces {5}
Edge figure {5}
Vertex figure {6,5}
Dual self-dual
Coxeter group [5,6,5]
Properties Regular

In the geometry of hyperbolic 3-space, the order-6-5 pentagonal honeycomb (or 5,6,5 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {5,6,5}.

All vertices are ultra-ideal (existing beyond the ideal boundary) with five order-6 pentagonal tilings existing around each edge and with an order-5 hexagonal tiling vertex figure.

Hyperbolic honeycomb 5-6-5 poincare.png
Poincaré disk model 		H3 565 UHS plane at infinity.png
Ideal surface

Order-6-6 hexagonal honeycomb

Order-5-6 hexagonal honeycomb
Type Regular honeycomb
Schläfli symbols {6,6,6}
{6,(6,3,6)}
Coxeter diagrams CDel node 1.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node h0.png = CDel node 1.pngCDel 6.pngCDel node.pngCDel split1-66.pngCDel branch.png
Cells {6,6} H2 tiling 266-1.png
Faces {6}
Edge figure {6}
Vertex figure {6,6} H2 tiling 266-4.png
{(6,3,6)} H2 tiling 366-1.png
Dual self-dual
Coxeter group [6,5,6]
[6,((6,3,6))]
Properties Regular

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

Hyperbolic honeycomb 6-6-6 poincare.png
Poincaré disk model 		H3 666 UHS plane at infinity.png
Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {6,(6,3,6)}, Coxeter diagram, CDel node 1.pngCDel 6.pngCDel node.pngCDel split1-55.pngCDel branch.png, with alternating types or colors of cells. In Coxeter notation the half symmetry is [6,6,6,1+] = [6,((6,3,6))].

Order-6-infinite apeirogonal honeycomb

Order-6-infinite apeirogonal honeycomb
Type Regular honeycomb
Schläfli symbols {∞,6,∞}
{∞,(6,∞,6)}
Coxeter diagrams CDel node 1.pngCDel infin.pngCDel node.pngCDel 6.pngCDel node.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel infin.pngCDel node.pngCDel 6.pngCDel node.pngCDel infin.pngCDel node h0.png ↔ CDel node 1.pngCDel infin.pngCDel node.pngCDel split1-66.pngCDel branch.pngCDel labelinfin.png
Cells {∞,6} H2 tiling 26i-1.png
Faces {∞}
Edge figure {∞}
Vertex figure H2 tiling 26i-4.png {6,∞}
H2 tiling 66i-4.png {(6,∞,6)}
Dual self-dual
Coxeter group [∞,6,∞]
[∞,((6,∞,6))]
Properties Regular

In the geometry of hyperbolic 3-space, the order-6-infinite apeirogonal honeycomb (or ∞,6,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {∞,6,∞}. It has infinitely many order-6 apeirogonal tiling {∞,6} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-6 apeirogonal tilings existing around each vertex in an infinite-order square tiling vertex arrangement.

Hyperbolic honeycomb i-6-i poincare.png
Poincaré disk model 		H3 i6i UHS plane at infinity.png
Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {∞,(6,∞,6)}, Coxeter diagram, CDel node 1.pngCDel infin.pngCDel node.pngCDel split1-66.pngCDel branch.pngCDel labelinfin.png, with alternating types or colors of cells.

See also

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

References

External links



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