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In the geometry of hyperbolic 3-space, the order-3-7 hexagonal honeycomb or (6,3,7 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,3,7}.
Order-3-7 hexagonal honeycomb | |
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![]() Poincaré disk model | |
Type | Regular honeycomb |
Schläfli symbol | {6,3,7} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells | {6,3} ![]() |
Faces | {6} |
Edge figure | {7} |
Vertex figure | {3,7} |
Dual | {7,3,6} |
Coxeter group | [6,3,7] |
Properties | Regular |
All vertices are ultra-ideal (existing beyond the ideal boundary) with seven hexagonal tilings existing around each edge and with an order-7 triangular tiling vertex figure.
![]() Rendered intersection of honeycomb with the ideal plane in Poincaré half-space model |
![]() Closeup |
It a part of a sequence of regular polychora and honeycombs with hexagonal tiling cells.
Order-3-8 hexagonal honeycomb | |
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Type | Regular honeycomb |
Schläfli symbols | {6,3,8} {6,(3,4,3)} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells | {6,3} ![]() |
Faces | {6} |
Edge figure | {8} |
Vertex figure | {3,8} {(3,4,3)}![]() ![]() |
Dual | {8,3,6} |
Coxeter group | [6,3,8] [6,((3,4,3))] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-3-8 hexagonal honeycomb or (6,3,8 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,3,8}. It has eight hexagonal tilings, {6,3}, 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-8 triangular tiling vertex arrangement.
![]() Poincaré disk model |
It has a second construction as a uniform honeycomb, Schläfli symbol {6,(3,4,3)}, Coxeter diagram, , with alternating types or colors of tetrahedral cells. In Coxeter notation the half symmetry is [6,3,8,1+] = [6,((3,4,3))].
Order-3-infinite hexagonal honeycomb | |
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Type | Regular honeycomb |
Schläfli symbols | {6,3,∞} {6,(3,∞,3)} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells | {6,3} ![]() |
Faces | {6} |
Edge figure | {∞} |
Vertex figure | {3,∞}, {(3,∞,3)}![]() ![]() |
Dual | {∞,3,6} |
Coxeter group | [6,3,∞] [6,((3,∞,3))] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-3-infinite hexagonal honeycomb or (6,3,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,3,∞}. It has infinitely many hexagonal tiling {6,3} 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 triangular tiling vertex arrangement.
![]() Poincaré disk model |
![]() Ideal surface |
It has a second construction as a uniform honeycomb, Schläfli symbol {6,(3,∞,3)}, Coxeter diagram, , with alternating types or colors of hexagonal tiling cells.
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