Order-4-5 square honeycomb | |
---|---|
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 |
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Vertex figure |
![]() {4,2} ![]() ![]() ![]() ![]() ![]() |
![]() {4,3} ![]() ![]() ![]() ![]() ![]() |
![]() {4,4} ![]() ![]() ![]() ![]() ![]() |
![]() {4,5} ![]() ![]() ![]() ![]() ![]() |
![]() {4,6} ![]() ![]() ![]() ![]() ![]() |
![]() {4,∞} ![]() ![]() ![]() ![]() ![]() |
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 | [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 | |
---|---|
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))].
Order-4-5 square honeycomb | |
---|---|
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 | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
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 |
![]() |
![]() |
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Vertex figure |
![]() {4,2} ![]() ![]() ![]() ![]() ![]() |
![]() {4,3} ![]() ![]() ![]() ![]() ![]() |
![]() {4,4} ![]() ![]() ![]() ![]() ![]() |
![]() {4,5} ![]() ![]() ![]() ![]() ![]() |
![]() {4,6} ![]() ![]() ![]() ![]() ![]() |
![]() {4,∞} ![]() ![]() ![]() ![]() ![]() |
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 | [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 | |
---|---|
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))].