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