Order-6 tetrahedral honeycomb#Related polytopes and honeycombs
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!bgcolor=#e7dcc3 colspan=2|Order-6 tetrahedral honeycomb | |
| bgcolor=#ffffff align=center colspan=2|320px Perspective projection view within Poincaré disk model | |
| bgcolor=#e7dcc3|Type | Hyperbolic regular honeycomb Paracompact uniform honeycomb |
| bgcolor=#e7dcc3|Schläfli symbols | {3,3,6} {3,3[3]} |
| bgcolor=#e7dcc3|Coxeter diagrams | {{CDD|node_1|3|node|3|node|6|node}} {{CDD|node_1|3|node|3|node|6|node_h0}} ↔ {{CDD|node_1|3|node|split1|branch}} |
| bgcolor=#e7dcc3|Cells | {3,3} 40px |
| bgcolor=#e7dcc3|Faces | triangle {3} |
| bgcolor=#e7dcc3|Edge figure | hexagon {6} |
| bgcolor=#e7dcc3|Vertex figure | 80px 80px triangular tiling |
| bgcolor=#e7dcc3|Dual | Hexagonal tiling honeycomb |
| bgcolor=#e7dcc3|Coxeter groups | , [3,3,6] , [3,3[3]] |
| bgcolor=#e7dcc3|Properties | Regular, quasiregular |
In hyperbolic 3-space, the order-6 tetrahedral honeycomb is a paracompact regular space-filling tessellation (or honeycomb). It is paracompact because it has vertex figures composed of an infinite number of faces, and has all vertices as ideal points at infinity. With Schläfli symbol {3,3,6}, the order-6 tetrahedral honeycomb has six ideal tetrahedra around each edge. All vertices are ideal, with infinitely many tetrahedra existing around each vertex in a triangular tiling vertex figure.Coxeter The Beauty of Geometry, 1999, Chapter 10, Table III
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Symmetry constructions
File:Hyperbolic subgroup tree 336-direct.png]]
The order-6 tetrahedral honeycomb has a second construction as a uniform honeycomb, with Schläfli symbol {3,3[3]}. This construction contains alternating types, or colors, of tetrahedral cells. In Coxeter notation, this half symmetry is represented as [3,3,6,1+] ↔ [3,((3,3,3))], or [3,3[3]]: {{CDD|node_c1|3|node_c2|3|node_c3|6|node_h0}} ↔ {{CDD|node_c1|3|node_c2|split1|branch_c3}}.
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Related polytopes and honeycombs
The order-6 tetrahedral honeycomb is analogous to the two-dimensional infinite-order triangular tiling, {3,∞}. Both tessellations are regular, and only contain triangles and ideal vertices.
The order-6 tetrahedral honeycomb is also a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact.
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This honeycomb is one of 15 uniform paracompact honeycombs in the [6,3,3] Coxeter group, along with its dual, the hexagonal tiling honeycomb.
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The order-6 tetrahedral honeycomb is part of a sequence of regular polychora and honeycombs with tetrahedral cells.
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It is also part of a sequence of honeycombs with triangular tiling vertex figures.
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= Rectified order-6 tetrahedral honeycomb =
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!bgcolor=#e7dcc3 colspan=2|Rectified order-6 tetrahedral honeycomb
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