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Order-5-4 square honeycomb
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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}.
 | 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 |
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Geometry
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 |
Related polytopes and honeycombs
Summarize
Perspective
It a part of a sequence of regular polychora and honeycombs {p,5,p}:
Order-5-5 pentagonal honeycomb
| 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
| 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 heptagonal honeycomb
| 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
| Order-5-infinite apeirogonal honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbols | {∞,5,∞} {∞,(5,∞,5)} |
| Coxeter diagrams |  ↔  |
| Cells | {∞,5}  |
| Faces | {∞} |
| Edge figure | {∞} |
| Vertex figure |  {5,∞} {(5,∞,5)} |
| 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.
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See also
References
External links
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