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Order-4-5 square honeycomb

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Order-4-5 square honeycomb
Type Regular honeycomb
Schläfli symbols {4,4,5}
Coxeter diagrams
Cells {4,4} [[File:Uniform tiling 44-t0.svg Faces {4}
Edge figure {5}
Vertex figure {4,5}
Dual {5,4,4}
Coxeter group
Properties Regular

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

Images


Poincaré disk model
Ideal surface

Related polytopes and honeycombs

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

{4,4,p} honeycombs
Space E H
Form Affine Paracompact Noncompact
Name {4,4,2} {4,4,3} {4,4,4} {4,4,5} {4,4,6} ...{4,4,∞}
Coxeter












Image
Vertex
figure
{4,2}
{4,3}
{4,4}
{4,5}
{4,6}
{4,∞}

Order-4-6 square honeycomb

Order-4-6 square honeycomb
Type Regular honeycomb
Schläfli symbols {4,4,6}
{4,(4,3,4)}
Coxeter diagrams
=
Cells {4,4}
Faces {4}
Edge figure {6}
Vertex figure {4,6}
{(4,3,4)}
Dual {6,4,4}
Coxeter group
Properties Regular

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


Poincaré disk model
Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {4,(4,3,4)}, Coxeter diagram, , with alternating types or colors of square tiling cells. In Coxeter notation the half symmetry is = .

Order-4-infinite square honeycomb

Order-4-infinite square honeycomb
Type Regular honeycomb
Schläfli symbols {4,4,∞}
{4,(4,∞,4)}
Coxeter diagrams
=
Cells {4,4}
Faces {4}
Edge figure {∞}
Vertex figure {4,∞}
{(4,∞,4)}
Dual {∞,4,4}
Coxeter group
Properties Regular

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


Poincaré disk model
Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {4,(4,∞,4)}, Coxeter diagram, = , with alternating types or colors of square tiling cells. In Coxeter notation the half symmetry is = .

See also

References

External links

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