Order-7 dodecahedral honeycomb

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Order-7 dodecahedral honeycomb
Type Regular honeycomb
Schläfli symbols {5,3,7}
Coxeter diagrams CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 7.pngCDel node.png
Cells {5,3} Uniform polyhedron-53-t0.png
Faces {5}
Edge figure {7}
Vertex figure {3,7}
H2 tiling 237-4.png
Dual {7,3,5}
Coxeter group [5,3,7]
Properties Regular

In the geometry of hyperbolic 3-space, the order-7 dodecahedral honeycomb a regular space-filling tessellation (or honeycomb).

Geometry[edit]

With Schläfli symbol {5,3,7}, it has seven dodecahedra {5,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many dodecahedra existing around each vertex in an order-7 triangular tiling vertex arrangement.

Hyperbolic honeycomb 5-3-7 poincare cc.png
Poincaré disk model
Cell-centered
Hyperbolic honeycomb 5-3-7 poincare.png
Poincaré disk model
H3 537 UHS plane at infinity.png
Ideal surface

Related polytopes and honeycombs[edit]

It a part of a sequence of regular polytopes and honeycombs with dodecahedral cells, {5,3,p}.

It a part of a sequence of honeycombs {5,p,7}.

It a part of a sequence of honeycombs {p,3,7}.

{3,3,7} {4,3,7} {5,3,7} {6,3,7} {7,3,7} {8,3,7} {∞,3,7}
Hyperbolic honeycomb 3-3-7 poincare cc.png Hyperbolic honeycomb 4-3-7 poincare cc.png Hyperbolic honeycomb 5-3-7 poincare cc.png Hyperbolic honeycomb 6-3-7 poincare.png Hyperbolic honeycomb 7-3-7 poincare.png Hyperbolic honeycomb 8-3-7 poincare.png Hyperbolic honeycomb i-3-7 poincare.png

Order-8 dodecahedral honeycomb[edit]

Order-8 dodecahedral honeycomb
Type Regular honeycomb
Schläfli symbols {5,3,8}
{5,(3,4,3)}
Coxeter diagrams CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node h0.png = CDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel branch.pngCDel label4.png
Cells {5,3} Uniform polyhedron-53-t0.png
Faces {5}
Edge figure {8}
Vertex figure {3,8}, {(3,4,3)}
H2 tiling 238-4.pngH2 tiling 334-4.png
Dual {8,3,5}
Coxeter group [5,3,8]
[5,((3,4,3))]
Properties Regular

In the geometry of hyperbolic 3-space, the order-8 dodecahedral honeycomb a regular space-filling tessellation (or honeycomb). With Schläfli symbol {5,3,8}, it has eight dodecahedra {5,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many dodecahedra existing around each vertex in an order-8 triangular tiling vertex arrangement.

Hyperbolic honeycomb 5-3-8 poincare cc.png
Poincaré disk model
Cell-centered
Hyperbolic honeycomb 5-3-8 poincare.png
Poincaré disk model

It has a second construction as a uniform honeycomb, Schläfli symbol {5,(3,4,3)}, Coxeter diagram, CDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel branch.pngCDel label4.png, with alternating types or colors of dodecahedral cells.

Infinite-order dodecahedral honeycomb[edit]

Infinite-order dodecahedral honeycomb
Type Regular honeycomb
Schläfli symbols {5,3,∞}
{5,(3,∞,3)}
Coxeter diagrams CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node h0.png = CDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel branch.pngCDel labelinfin.png
Cells {5,3} Uniform polyhedron-53-t0.png
Faces {5}
Edge figure {∞}
Vertex figure {3,∞}, {(3,∞,3)}
H2 tiling 23i-4.pngH2 tiling 33i-4.png
Dual {∞,3,5}
Coxeter group [5,3,∞]
[5,((3,∞,3))]
Properties Regular

In the geometry of hyperbolic 3-space, the infinite-order dodecahedral honeycomb a regular space-filling tessellation (or honeycomb). With Schläfli symbol {5,3,∞}. It has infinitely many dodecahedra {5,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many dodecahedra existing around each vertex in an infinite-order triangular tiling vertex arrangement.


Hyperbolic honeycomb 5-3-i poincare cc.png
Poincaré disk model
Cell-centered
Hyperbolic honeycomb 5-3-i poincare.png
Poincaré disk model
H3 53i UHS plane at infinity.png
Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {5,(3,∞,3)}, Coxeter diagram, CDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel branch.pngCDel labelinfin.png, with alternating types or colors of dodecahedral cells.

See also[edit]

References[edit]

External links[edit]