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