Order-4-5 pentagonal honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbol | {5,4,5} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{5,4}
![]() |
Faces | {5} |
Edge figure | {5} |
Vertex figure | {4,5} |
Dual | self-dual |
Coxeter group | [5,4,5] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-4-5 pentagonal honeycomb a regular space-filling tessellation (or honeycomb) with Schläfli symbol {5,4,5}.
All vertices are ultra-ideal (existing beyond the ideal boundary) with five order-4 pentagonal tilings existing around each edge and with an order-5 square tiling vertex figure.
![]() Poincaré disk model |
![]() Ideal surface |
It a part of a sequence of regular polychora and honeycombs {p,4,p}:
{p,4,p} regular honeycombs | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Space | S3 | Euclidean E3 | H3 | ||||||||
Form | Finite | Paracompact | Noncompact | ||||||||
Name | {3,4,3} | {4,4,4} | {5,4,5} | {6,4,6} | {7,4,7} | {8,4,8} | ... {∞,4,∞} | ||||
Image |
![]() |
![]() |
![]() |
![]() |
![]() | ||||||
Cells {p,4} |
![]() {3,4} |
![]() {4,4} |
![]() {5,4} |
![]() {6,4} |
![]() {7,4} |
![]() {8,4} |
![]() {∞,4} | ||||
Vertex figure {4,p} |
![]() {4,3} |
![]() {4,4} |
![]() {4,5} |
![]() {4,6} |
![]() {4,7} |
![]() {4,8} |
![]() {4,∞} |
Order-4-6 hexagonal honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbols | {6,4,6} {6,(4,3,4)} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{6,4}
![]() |
Faces | {6} |
Edge figure | {6} |
Vertex figure |
{4,6}
![]() {(4,3,4)} ![]() |
Dual | self-dual |
Coxeter group | [6,4,6] [6,((4,3,4))] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-4-6 hexagonal honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,3,6}. It has six order-4 hexagonal tilings, {6,4}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many hexagonal tilings 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 {6,(4,3,4)}, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [6,4,6,1+] = [6,((4,3,4))].
Order-4-infinite apeirogonal honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbols | {∞,4,∞} {∞,(4,∞,4)} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{∞,4}
![]() |
Faces | {∞} |
Edge figure | {∞} |
Vertex figure |
![]() ![]() |
Dual | self-dual |
Coxeter group | [∞,4,∞] [∞,((4,∞,4))] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-4-infinite apeirogonal honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {∞,4,∞}. It has infinitely many order-4 apeirogonal tiling {∞,4} 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 square tiling vertex arrangement.
![]() Poincaré disk model |
![]() Ideal surface |
It has a second construction as a uniform honeycomb,
Schläfli symbol {∞,(4,∞,4)}, Coxeter diagram, , with alternating types or colors of cells.
Order-4-5 pentagonal honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbol | {5,4,5} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{5,4}
![]() |
Faces | {5} |
Edge figure | {5} |
Vertex figure | {4,5} |
Dual | self-dual |
Coxeter group | [5,4,5] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-4-5 pentagonal honeycomb a regular space-filling tessellation (or honeycomb) with Schläfli symbol {5,4,5}.
All vertices are ultra-ideal (existing beyond the ideal boundary) with five order-4 pentagonal tilings existing around each edge and with an order-5 square tiling vertex figure.
![]() Poincaré disk model |
![]() Ideal surface |
It a part of a sequence of regular polychora and honeycombs {p,4,p}:
{p,4,p} regular honeycombs | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Space | S3 | Euclidean E3 | H3 | ||||||||
Form | Finite | Paracompact | Noncompact | ||||||||
Name | {3,4,3} | {4,4,4} | {5,4,5} | {6,4,6} | {7,4,7} | {8,4,8} | ... {∞,4,∞} | ||||
Image |
![]() |
![]() |
![]() |
![]() |
![]() | ||||||
Cells {p,4} |
![]() {3,4} |
![]() {4,4} |
![]() {5,4} |
![]() {6,4} |
![]() {7,4} |
![]() {8,4} |
![]() {∞,4} | ||||
Vertex figure {4,p} |
![]() {4,3} |
![]() {4,4} |
![]() {4,5} |
![]() {4,6} |
![]() {4,7} |
![]() {4,8} |
![]() {4,∞} |
Order-4-6 hexagonal honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbols | {6,4,6} {6,(4,3,4)} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{6,4}
![]() |
Faces | {6} |
Edge figure | {6} |
Vertex figure |
{4,6}
![]() {(4,3,4)} ![]() |
Dual | self-dual |
Coxeter group | [6,4,6] [6,((4,3,4))] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-4-6 hexagonal honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,3,6}. It has six order-4 hexagonal tilings, {6,4}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many hexagonal tilings 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 {6,(4,3,4)}, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [6,4,6,1+] = [6,((4,3,4))].
Order-4-infinite apeirogonal honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbols | {∞,4,∞} {∞,(4,∞,4)} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{∞,4}
![]() |
Faces | {∞} |
Edge figure | {∞} |
Vertex figure |
![]() ![]() |
Dual | self-dual |
Coxeter group | [∞,4,∞] [∞,((4,∞,4))] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-4-infinite apeirogonal honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {∞,4,∞}. It has infinitely many order-4 apeirogonal tiling {∞,4} 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 square tiling vertex arrangement.
![]() Poincaré disk model |
![]() Ideal surface |
It has a second construction as a uniform honeycomb,
Schläfli symbol {∞,(4,∞,4)}, Coxeter diagram, , with alternating types or colors of cells.