![]() | This article may be too technical for most readers to understand.(June 2020) |
Order-4-5 square honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbol | {4,5,4} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{4,5}
![]() |
Faces | {4} |
Edge figure | {4} |
Vertex figure | {5,4} |
Dual | self-dual |
Coxeter group | [4,5,4] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-4 square honeycomb (or 4,5,4 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,5,4}.
All vertices are ultra-ideal (existing beyond the ideal boundary) with four order-5 square tilings existing around each edge and with an order-4 pentagonal tiling vertex figure.
![]() Poincaré disk model |
![]() Ideal surface |
It a part of a sequence of regular polychora and honeycombs {p,5,p}:
{p,5,p} regular honeycombs | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Space | H3 | ||||||||||
Form | Compact | Noncompact | |||||||||
Name | {3,5,3} | {4,5,4} | {5,5,5} | {6,5,6} | {7,5,7} | {8,5,8} | ... {∞,5,∞} | ||||
Image |
![]() |
![]() |
![]() |
![]() |
![]() | ||||||
Cells {p,5} |
![]() {3,5} |
![]() {4,5} |
![]() {5,5} |
![]() {6,5} |
![]() {7,5} |
![]() {8,5} |
![]() {∞,5} | ||||
Vertex figure {5,p} |
![]() {5,3} |
![]() {5,4} |
![]() {5,5} |
![]() {5,6} |
![]() {5,7} |
![]() {5,8} |
![]() {5,∞} |
Order-5-5 pentagonal honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbol | {5,5,5} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{5,5}
![]() |
Faces | {5} |
Edge figure | {5} |
Vertex figure | {5,5} |
Dual | self-dual |
Coxeter group | [5,5,5] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-5 pentagonal honeycomb (or 5,5,5 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {5,5,5}.
All vertices are ultra-ideal (existing beyond the ideal boundary) with five order-5 pentagonal tilings existing around each edge and with an order-5 pentagonal tiling vertex figure.
![]() Poincaré disk model |
![]() Ideal surface |
Order-5-6 hexagonal honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbols | {6,5,6} {6,(5,3,5)} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{6,5}
![]() |
Faces | {6} |
Edge figure | {6} |
Vertex figure |
{5,6}
![]() {(5,3,5)} ![]() |
Dual | self-dual |
Coxeter group | [6,5,6] [6,((5,3,5))] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-6 hexagonal honeycomb (or 6,5,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,5,6}. It has six order-5 hexagonal tilings, {6,5}, 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 pentagonal tiling vertex arrangement.
![]() Poincaré disk model |
![]() Ideal surface |
It has a second construction as a uniform honeycomb,
Schläfli symbol {6,(5,3,5)}, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [6,5,6,1+] = [6,((5,3,5))].
Order-5-7 hexagonal honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbols | {7,5,7} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{7,5}
![]() |
Faces | {6} |
Edge figure | {6} |
Vertex figure |
{5,7}
![]() |
Dual | self-dual |
Coxeter group | [7,5,7] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-7 heptagonal honeycomb (or 7,5,7 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {7,5,7}. It has seven order-5 heptagonal tilings, {7,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many heptagonal tilings existing around each vertex in an order-7 pentagonal tiling vertex arrangement.
![]() Ideal surface |
Order-5-infinite apeirogonal honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbols | {∞,5,∞} {∞,(5,∞,5)} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{∞,5}
![]() |
Faces | {∞} |
Edge figure | {∞} |
Vertex figure |
![]() ![]() |
Dual | self-dual |
Coxeter group | [∞,5,∞] [∞,((5,∞,5))] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-infinite apeirogonal honeycomb (or ∞,5,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {∞,5,∞}. It has infinitely many order-5 apeirogonal tilings {∞,5} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-5 apeirogonal tilings existing around each vertex in an infinite-order pentagonal tiling vertex arrangement.
![]() Poincaré disk model |
![]() Ideal surface |
It has a second construction as a uniform honeycomb,
Schläfli symbol {∞,(5,∞,5)}, Coxeter diagram, , with alternating types or colors of cells.
![]() | This article may be too technical for most readers to understand.(June 2020) |
Order-4-5 square honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbol | {4,5,4} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{4,5}
![]() |
Faces | {4} |
Edge figure | {4} |
Vertex figure | {5,4} |
Dual | self-dual |
Coxeter group | [4,5,4] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-4 square honeycomb (or 4,5,4 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,5,4}.
All vertices are ultra-ideal (existing beyond the ideal boundary) with four order-5 square tilings existing around each edge and with an order-4 pentagonal tiling vertex figure.
![]() Poincaré disk model |
![]() Ideal surface |
It a part of a sequence of regular polychora and honeycombs {p,5,p}:
{p,5,p} regular honeycombs | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Space | H3 | ||||||||||
Form | Compact | Noncompact | |||||||||
Name | {3,5,3} | {4,5,4} | {5,5,5} | {6,5,6} | {7,5,7} | {8,5,8} | ... {∞,5,∞} | ||||
Image |
![]() |
![]() |
![]() |
![]() |
![]() | ||||||
Cells {p,5} |
![]() {3,5} |
![]() {4,5} |
![]() {5,5} |
![]() {6,5} |
![]() {7,5} |
![]() {8,5} |
![]() {∞,5} | ||||
Vertex figure {5,p} |
![]() {5,3} |
![]() {5,4} |
![]() {5,5} |
![]() {5,6} |
![]() {5,7} |
![]() {5,8} |
![]() {5,∞} |
Order-5-5 pentagonal honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbol | {5,5,5} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{5,5}
![]() |
Faces | {5} |
Edge figure | {5} |
Vertex figure | {5,5} |
Dual | self-dual |
Coxeter group | [5,5,5] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-5 pentagonal honeycomb (or 5,5,5 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {5,5,5}.
All vertices are ultra-ideal (existing beyond the ideal boundary) with five order-5 pentagonal tilings existing around each edge and with an order-5 pentagonal tiling vertex figure.
![]() Poincaré disk model |
![]() Ideal surface |
Order-5-6 hexagonal honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbols | {6,5,6} {6,(5,3,5)} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{6,5}
![]() |
Faces | {6} |
Edge figure | {6} |
Vertex figure |
{5,6}
![]() {(5,3,5)} ![]() |
Dual | self-dual |
Coxeter group | [6,5,6] [6,((5,3,5))] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-6 hexagonal honeycomb (or 6,5,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,5,6}. It has six order-5 hexagonal tilings, {6,5}, 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 pentagonal tiling vertex arrangement.
![]() Poincaré disk model |
![]() Ideal surface |
It has a second construction as a uniform honeycomb,
Schläfli symbol {6,(5,3,5)}, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [6,5,6,1+] = [6,((5,3,5))].
Order-5-7 hexagonal honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbols | {7,5,7} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{7,5}
![]() |
Faces | {6} |
Edge figure | {6} |
Vertex figure |
{5,7}
![]() |
Dual | self-dual |
Coxeter group | [7,5,7] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-7 heptagonal honeycomb (or 7,5,7 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {7,5,7}. It has seven order-5 heptagonal tilings, {7,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many heptagonal tilings existing around each vertex in an order-7 pentagonal tiling vertex arrangement.
![]() Ideal surface |
Order-5-infinite apeirogonal honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbols | {∞,5,∞} {∞,(5,∞,5)} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{∞,5}
![]() |
Faces | {∞} |
Edge figure | {∞} |
Vertex figure |
![]() ![]() |
Dual | self-dual |
Coxeter group | [∞,5,∞] [∞,((5,∞,5))] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-infinite apeirogonal honeycomb (or ∞,5,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {∞,5,∞}. It has infinitely many order-5 apeirogonal tilings {∞,5} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-5 apeirogonal tilings existing around each vertex in an infinite-order pentagonal tiling vertex arrangement.
![]() Poincaré disk model |
![]() Ideal surface |
It has a second construction as a uniform honeycomb,
Schläfli symbol {∞,(5,∞,5)}, Coxeter diagram, , with alternating types or colors of cells.