Order-4-5 square honeycomb | |
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Type | Regular honeycomb |
Schläfli symbols | {4,4,5} |
Coxeter diagrams | |
Cells | {4,4} |
Faces | {4} |
Edge figure | {5} |
Vertex figure | {4,5} |
Dual | {5,4,4} |
Coxeter group | [4,4,5] |
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.
Poincaré disk model |
Ideal surface |
It a part of a sequence of regular polychora and honeycombs with square tiling cells: {4,4,p}
{4,4,p} honeycombs | |||||||||||
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Space | E3 | H3 | |||||||||
Form | Affine | Paracompact | Noncompact | ||||||||
Name | {4,4,2} | {4,4,3} | {4,4,4} | {4,4,5} | {4,4,6} | ... {4,4,∞} | |||||
Coxeter |
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Image | |||||||||||
Vertex figure |
{4,2} |
{4,3} |
{4,4} |
{4,5} |
{4,6} |
{4,∞} |
Order-4-6 square honeycomb | |
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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 | [4,4,6] [4,((4,3,4))] |
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 [4,4,6,1+] = [4,((4,3,4))].
Order-4-infinite square honeycomb | |
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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 | [∞,4,3] [4,((4,∞,4))] |
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 [4,4,∞,1+] = [4,((4,∞,4))].