It is a part of a sequence of regular polychora and honeycombs with tetrahedral cells, {3,3,p}.
More information {3,3,p} polytopes, Space ...
| {3,3,p} polytopes |
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| Space |
S3 |
H3 |
| Form |
Finite |
Paracompact |
Noncompact |
| Name |
{3,3,3}
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{3,3,4}
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{3,3,5}
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{3,3,6}
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{3,3,7}
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{3,3,8}
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... {3,3,∞}
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| Image |
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Vertex
figure |
{3,3}
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{3,4}
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{3,5}
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{3,6}
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{3,7}
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{3,8}
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{3,∞}
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It is a part of a sequence of hyperbolic honeycombs with order-7 triangular tiling vertex figures, {p,3,7}.
More information {3,3,7}, {4,3,7} ...
| {3,3,7} |
{4,3,7} |
{5,3,7} |
{6,3,7} |
{7,3,7} |
{8,3,7} |
{∞,3,7} |
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It is a part of a sequence of hyperbolic honeycombs, {3,p,7}.
Order-8 tetrahedral honeycomb
| 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.
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
More information Infinite-order tetrahedral honeycomb ...
| 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 |
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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.
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))].
