Order-5 octahedral honeycomb

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Short description: Tesselation in regular space
Order-5 octahedral honeycomb
Type Regular honeycomb
Schläfli symbols {3,4,5}
Coxeter diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node.png
Cells {3,4} Uniform polyhedron-34-t0.png
Faces {3}
Edge figure {5}
Vertex figure {4,5} H2-5-4-primal.svg
Dual {5,4,3}
Coxeter group [3,4,5]
Properties Regular

In the geometry of hyperbolic 3-space, the order-5 octahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,4,5}. It has five octahedra {3,4} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many octahedra existing around each vertex in an order-5 square tiling vertex arrangement.

Images

Hyperbolic honeycomb 3-4-5 poincare cc.png
Poincaré disk model
(cell centered) 		H3 345 UHS plane at infinity.png
Ideal surface

Related polytopes and honeycombs

It a part of a sequence of regular polychora and honeycombs with octahedral cells: {3,4,p}

Order-6 octahedral honeycomb

Order-6 octahedral honeycomb
Type Regular honeycomb
Schläfli symbols {3,4,6}
{3,(3,4,3)}
Coxeter diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node h0.png = CDel node 1.pngCDel 3.pngCDel node.pngCDel split1-44.pngCDel branch.png
Cells {3,4} Uniform polyhedron-34-t0.png
Faces {3}
Edge figure {6}
Vertex figure {4,6} H2 tiling 246-4.png
{(4,3,4)} Uniform tiling 443-t1.png
Dual {6,4,3}
Coxeter group [3,4,6]
[3,((4,3,4))]
Properties Regular

In the geometry of hyperbolic 3-space, the order-6 octahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,4,6}. It has six octahedra, {3,4}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many octahedra existing around each vertex in an order-6 square tiling vertex arrangement.

Hyperbolic honeycomb 3-4-6 poincare cc.png
Poincaré disk model
(cell centered) 		H3 346 UHS plane at infinity.png
Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {3,(4,3,4)}, Coxeter diagram, CDel node 1.pngCDel 3.pngCDel node.pngCDel split1-44.pngCDel branch.png, with alternating types or colors of octahedral cells. In Coxeter notation the half symmetry is [3,4,6,1+] = [3,((4,3,4))].

Order-7 octahedral honeycomb

Order-7 octahedral honeycomb
Type Regular honeycomb
Schläfli symbols {3,4,7}
Coxeter diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 7.pngCDel node.png
Cells {3,4} Uniform polyhedron-34-t0.png
Faces {3}
Edge figure {7}
Vertex figure {4,7} H2 tiling 247-4.png
Dual {7,4,3}
Coxeter group [3,4,7]
Properties Regular

In the geometry of hyperbolic 3-space, the order-7 octahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,4,7}. It has seven octahedra, {3,4}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many octahedra existing around each vertex in an order-7 square tiling vertex arrangement.

Hyperbolic honeycomb 3-4-7 poincare cc.png
Poincaré disk model
(cell centered) 		H3 347 UHS plane at infinity.png
Ideal surface

Order-8 octahedral honeycomb

Order-8 octahedral honeycomb
Type Regular honeycomb
Schläfli symbols {3,4,8}
Coxeter diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node.png
Cells {3,4} Uniform polyhedron-34-t0.png
Faces {3}
Edge figure {8}
Vertex figure {4,8} H2 tiling 248-4.png
Dual {8,4,3}
Coxeter group [3,4,8]
Properties Regular

In the geometry of hyperbolic 3-space, the order-8 octahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,4,8}. It has eight octahedra, {3,4}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many octahedra existing around each vertex in an order-8 square tiling vertex arrangement.

Hyperbolic honeycomb 3-4-8 poincare cc.png
Poincaré disk model
(cell centered)

Infinite-order octahedral honeycomb

Infinite-order octahedral honeycomb
Type Regular honeycomb
Schläfli symbols {3,4,∞}
{3,(4,∞,4)}
Coxeter diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel infin.pngCDel node h0.png = CDel node 1.pngCDel 3.pngCDel node.pngCDel split1-44.pngCDel branch.pngCDel labelinfin.png
Cells {3,4} Uniform polyhedron-34-t0.png
Faces {3}
Edge figure {∞}
Vertex figure {4,∞} H2 tiling 24i-4.png
{(4,∞,4)} H2 tiling 44i-4.png
Dual {∞,4,3}
Coxeter group [∞,4,3]
[3,((4,∞,4))]
Properties Regular

In the geometry of hyperbolic 3-space, the infinite-order octahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,4,∞}. It has infinitely many octahedra, {3,4}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many octahedra existing around each vertex in an infinite-order square tiling vertex arrangement.

Hyperbolic honeycomb 3-4-i poincare cc.png
Poincaré disk model
(cell centered) 		H3 34i UHS plane at infinity.png
Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {3,(4,∞,4)}, Coxeter diagram, CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel infin.pngCDel node h0.png = CDel node 1.pngCDel 3.pngCDel node.pngCDel split1-44.pngCDel branch.pngCDel labelinfin.png, with alternating types or colors of octahedral cells. In Coxeter notation the half symmetry is [3,4,∞,1+] = [3,((4,∞,4))].

See also

  • Convex uniform honeycombs in hyperbolic space
  • List of regular polytopes

References

External links



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