Order-7 dodecahedral honeycomb
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In the geometry of hyperbolic 3-space, the order-7 dodecahedral honeycomb is a regular space-filling tessellation (or honeycomb).
| Order-7 dodecahedral honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbols | {5,3,7} |
| Coxeter diagrams |      |
| Cells | {5,3}  |
| Faces | {5} |
| Edge figure | {7} |
| Vertex figure | {3,7} |
| Dual | {7,3,5} |
| Coxeter group | [5,3,7] |
| Properties | Regular |
Geometry
With Schläfli symbol {5,3,7}, it has seven dodecahedra {5,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many dodecahedra existing around each vertex in an order-7 triangular tiling vertex arrangement.
 Poincaré disk model Cell-centered |
 Poincaré disk model |
 Ideal surface |
Related polytopes and honeycombs
Summarize
Perspective
It a part of a sequence of regular polytopes and honeycombs with dodecahedral cells, {5,3,p}.
It a part of a sequence of honeycombs {5,p,7}.
It a part of a sequence of honeycombs {p,3,7}.
Order-8 dodecahedral honeycomb
| Order-8 dodecahedral honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbols | {5,3,8} {5,(3,4,3)} |
| Coxeter diagrams |   =    |
| Cells | {5,3} |
| Faces | {5} |
| Edge figure | {8} |
| Vertex figure | {3,8}, {(3,4,3)}  |
| Dual | {8,3,5} |
| Coxeter group | [5,3,8] [5,((3,4,3))] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-8 dodecahedral honeycomb a regular space-filling tessellation (or honeycomb). With Schläfli symbol {5,3,8}, it has eight dodecahedra {5,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many dodecahedra existing around each vertex in an order-8 triangular tiling vertex arrangement.
 Poincaré disk model Cell-centered |
 Poincaré disk model |
It has a second construction as a uniform honeycomb, Schläfli symbol {5,(3,4,3)}, Coxeter diagram, , with alternating types or colors of dodecahedral cells.
Infinite-order dodecahedral honeycomb
| Infinite-order dodecahedral honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbols | {5,3,∞} {5,(3,∞,3)} |
| Coxeter diagrams |  =  |
| Cells | {5,3} |
| Faces | {5} |
| Edge figure | {∞} |
| Vertex figure | {3,∞}, {(3,∞,3)}  |
| Dual | {∞,3,5} |
| Coxeter group | [5,3,∞] [5,((3,∞,3))] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the infinite-order dodecahedral honeycomb a regular space-filling tessellation (or honeycomb). With Schläfli symbol {5,3,∞}. It has infinitely many dodecahedra {5,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many dodecahedra existing around each vertex in an infinite-order triangular tiling vertex arrangement.
 Poincaré disk model Cell-centered |
 Poincaré disk model |
 Ideal surface |
It has a second construction as a uniform honeycomb, Schläfli symbol {5,(3,∞,3)}, Coxeter diagram, , with alternating types or colors of dodecahedral cells.
See also
References
External links
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