Order-6-4 triangular honeycomb
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In the geometry of hyperbolic 3-space, the order-6-4 triangular honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,6,4}.
Order-6-4 triangular honeycomb | |
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Type | Regular honeycomb |
Schläfli symbols | {3,6,4} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells | {3,6} ![]() |
Faces | {3} |
Edge figure | {4} |
Vertex figure | {6,4} ![]() r{6,6} ![]() |
Dual | {4,6,3} |
Coxeter group | [3,6,4] |
Properties | Regular |
Geometry
It has four triangular tiling {3,6} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an order-4 hexagonal tiling vertex arrangement.
![]() Poincaré disk model |
![]() Ideal surface |
It has a second construction as a uniform honeycomb, Schläfli symbol {3,61,1}, Coxeter diagram, , with alternating types or colors of triangular tiling cells. In Coxeter notation the half symmetry is [3,6,4,1+] = [3,61,1].
Related polytopes and honeycombs
Summarize
Perspective
It a part of a sequence of regular polychora and honeycombs with triangular tiling cells: {3,6,p}
Order-6-5 triangular honeycomb
Order-6-5 triangular honeycomb | |
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Type | Regular honeycomb |
Schläfli symbol | {3,6,5} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells | {3,6} ![]() |
Faces | {3} |
Edge figure | {5} |
Vertex figure | {6,5} ![]() |
Dual | {5,6,3} |
Coxeter group | [3,6,5] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-6-3 triangular honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,6,5}. It has five triangular tiling, {3,6}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an order-5 hexagonal tiling vertex arrangement.
![]() Poincaré disk model |
![]() Ideal surface |
Order-6-6 triangular honeycomb
Order-6-6 triangular honeycomb | |
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Type | Regular honeycomb |
Schläfli symbols | {3,6,6} {3,(6,3,6)} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells | {3,6} ![]() |
Faces | {3} |
Edge figure | {6} |
Vertex figure | {6,6} ![]() {(6,3,6)} ![]() |
Dual | {6,6,3} |
Coxeter group | [3,6,6] [3,((6,3,6))] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-6-6 triangular honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,6,6}. It has infinitely many triangular tiling, {3,6}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an order-6 triangular tiling vertex arrangement.
![]() Poincaré disk model |
![]() Ideal surface |
It has a second construction as a uniform honeycomb, Schläfli symbol {3,(6,3,6)}, Coxeter diagram, =
, with alternating types or colors of triangular tiling cells. In Coxeter notation the half symmetry is [3,6,6,1+] = [3,((6,3,6))].
Order-6-infinite triangular honeycomb
Order-6-infinite triangular honeycomb | |
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Type | Regular honeycomb |
Schläfli symbols | {3,6,∞} {3,(6,∞,6)} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells | {3,6} ![]() |
Faces | {3} |
Edge figure | {∞} |
Vertex figure | {6,∞} ![]() {(6,∞,6)} ![]() |
Dual | {∞,6,3} |
Coxeter group | [∞,6,3] [3,((6,∞,6))] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-6-infinite triangular honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,6,∞}. It has infinitely many triangular tiling, {3,6}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular 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,(6,∞,6)}, Coxeter diagram, =
, with alternating types or colors of triangular tiling cells. In Coxeter notation the half symmetry is [3,6,∞,1+] = [3,((6,∞,6))].
See also
References
External links
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