Order-3-7 heptagonal honeycomb

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Order-3-7 heptagonal honeycomb
Type Regular honeycomb
Schläfli symbol {7,3,7}
Coxeter diagrams CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.pngCDel 7.pngCDel node.png
Cells {7,3} H2 tiling 237-1.png
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[edit]

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.

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

Related polytopes and honeycombs[edit]

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

Order-3-8 octagonal honeycomb[edit]

Order-3-8 hexagonal honeycomb
Type Regular honeycomb
Schläfli symbols {8,3,8}
{8,(3,4,3)}
Coxeter diagrams CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.png
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node h0.png = CDel node 1.pngCDel 8.pngCDel node.pngCDel split1.pngCDel branch.pngCDel label4.png
Cells {8,3} H2 tiling 238-1.png
Faces {8}
Edge figure {8}
Vertex figure {3,8} H2 tiling 238-4.png
{(3,8,3)} 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.

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, CDel node 1.pngCDel 8.pngCDel node.pngCDel split1.pngCDel branch.pngCDel 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[edit]

Order-3-infinite apeirogonal honeycomb
Type Regular honeycomb
Schläfli symbols {∞,3,∞}
{∞,(3,∞,3)}
Coxeter diagrams CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node h0.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel split1.pngCDel branch.pngCDel labelinfin.png
Cells {∞,3} H2 tiling 23i-1.png
Faces {∞}
Edge figure {∞}
Vertex figure H2 tiling 23i-4.png {3,∞}
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 hexagonal tilings existing around each vertex in an infinite-order triangular tiling vertex arrangement.

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

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

See also[edit]

References[edit]

External links[edit]