Order-7 cubic honeycomb | |
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Type | Regular honeycomb |
Schläfli symbols | {4,3,7} |
Coxeter diagrams | |
Cells | {4,3} |
Faces | {4} |
Edge figure | {7} |
Vertex figure | {3,7} |
Dual | {7,3,4} |
Coxeter group | [4,3,7] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-7 cubic honeycomb is a regular space-filling tessellation (or honeycomb). With Schläfli symbol {4,3,7}, it has seven cubes {4,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many cubes existing around each vertex in an order-7 triangular tiling vertex arrangement.
Cell-centered |
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One cell at center |
One cell with ideal surface |
It is one of a series of regular polytopes and honeycombs with cubic cells: {4,3,p}:
{4,3,p} polytopes | |||||||
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Space | S3 | H3 | |||||
Form | Finite | Compact | Paracompact | Noncompact | |||
Name | {4,3,3} | {4,3,4} | {4,3,5} | {4,3,6} | {4,3,7} | {4,3,8} | ... {4,3,∞} |
Image | |||||||
Vertex figure |
{3,3} |
{3,4} |
{3,5} |
{3,6} |
{3,7} |
{3,8} |
{3,∞} |
It is a part of a sequence of hyperbolic honeycombs with order-7 triangular tiling vertex figures, {p,3,7}.
{3,3,7} | {4,3,7} | {5,3,7} | {6,3,7} | {7,3,7} | {8,3,7} | {∞,3,7} |
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Order-8 cubic honeycomb | |
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Type | Regular honeycomb |
Schläfli symbols | {4,3,8} {4,(3,8,3)} |
Coxeter diagrams | = |
Cells | {4,3} |
Faces | {4} |
Edge figure | {8} |
Vertex figure | {3,8}, {(3,4,3)} |
Dual | {8,3,4} |
Coxeter group | [4,3,8] [4,((3,4,3))] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-8 cubic honeycomb a regular space-filling tessellation (or honeycomb). With Schläfli symbol {4,3,8}. It has eight cubes {4,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many cubes 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 {4,(3,4,3)}, Coxeter diagram, , with alternating types or colors of cubic cells.
Infinite-order cubic honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbols | {4,3,∞} {4,(3,∞,3)} |
Coxeter diagrams | = |
Cells | {4,3} |
Faces | {4} |
Edge figure | {∞} |
Vertex figure | {3,∞}, {(3,∞,3)} |
Dual | {∞,3,4} |
Coxeter group | [4,3,∞] [4,((3,∞,3))] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the infinite-order cubic honeycomb a regular space-filling tessellation (or honeycomb). With Schläfli symbol {4,3,∞}. It has infinitely many cubes {4,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many cubes existing around each vertex in an infinite-order triangular tiling vertex arrangement.
Poincaré disk model Cell-centered |
Poincaré disk model |
It has a second construction as a uniform honeycomb, Schläfli symbol {4,(3,∞,3)}, Coxeter diagram, , with alternating types or colors of cubic cells.