order-6 dodecahedral honeycomb
{{short description|Regular geometrical object in hyperbolic space}}
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!bgcolor=#e7dcc3 colspan=2|Order-6 dodecahedral honeycomb | |
| colspan=2 align=center|320px Perspective projection view within Poincaré disk model | |
| bgcolor=#e7dcc3|Type | Hyperbolic regular honeycomb Paracompact uniform honeycomb |
| bgcolor=#e7dcc3|Schläfli symbol | {5,3,6} {5,3[3]} |
| bgcolor=#e7dcc3|Coxeter diagram | {{CDD|node_1|5|node|3|node|6|node}} {{CDD|node_1|5|node|3|node|6|node_h0}} ↔ {{CDD|node_1|5|node|split1|branch}} |
| bgcolor=#e7dcc3|Cells | {5,3} 40px |
| bgcolor=#e7dcc3|Faces | pentagon {5} |
| bgcolor=#e7dcc3|Edge figure | hexagon {6} |
| bgcolor=#e7dcc3|Vertex figure | 80px 80px triangular tiling |
| bgcolor=#e7dcc3|Dual | Order-5 hexagonal tiling honeycomb |
| bgcolor=#e7dcc3|Coxeter group | , [5,3,6] , [5,3[3]] |
| bgcolor=#e7dcc3|Properties | Regular, quasiregular |
The order-6 dodecahedral honeycomb is one of 11 paracompact regular honeycombs in hyperbolic 3-space. It is paracompact because it has vertex figures composed of an infinite number of faces, with all vertices as ideal points at infinity. It has Schläfli symbol {5,3,6}, with six ideal dodecahedral cells surrounding each edge of the honeycomb. Each vertex is ideal, and surrounded by infinitely many dodecahedra. The honeycomb has a triangular tiling vertex figure.
{{Honeycomb}}
Symmetry
A half symmetry construction exists as {{CDD|node_1|5|node|split1|branch}} with alternately colored dodecahedral cells.
Images
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The order-6 dodecahedral honeycomb is similar to the 2D hyperbolic infinite-order pentagonal tiling, {5,∞}, with pentagonal faces, and with vertices on the ideal surface.
: 240px
Related polytopes and honeycombs
The order-6 dodecahedral honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact.
{{Regular_paracompact_H3_honeycombs}}
There are 15 uniform honeycombs in the [5,3,6] Coxeter group family, including this regular form, and its regular dual, the order-5 hexagonal tiling honeycomb.
{{635 family}}
The order-6 dodecahedral honeycomb is part of a sequence of regular polychora and honeycombs with triangular tiling vertex figures:
{{Triangular tiling vertex figure tessellations small}}
It is also part of a sequence of regular polytopes and honeycombs with dodecahedral cells:
{{Dodecahedral_tessellations_small}}
= Rectified order-6 dodecahedral honeycomb =
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!bgcolor=#e7dcc3 colspan=2|Rectified order-6 dodecahedral honeycomb
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