Hyperbolic great dodecahedral honeycomb
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| Great dodecahedral honeycomb | |
|---|---|
| (No image) | |
| Schläfli symbol | {5,3,5} |
| Type | regular hyperbolic honeycomb |
| Cells | dodecahedron {5,3} |
| Faces | pentagon {5} |
| Edge figure | pentagon {5} |
| Vertex figure | icosahedron {3,5} |
| Cells/edge | {5,3}5 |
| Cells/vertex | {5,3}20 |
| Euler characteristic | 0 |
| Dual | Self-dual |
| Symmetry group | group [5,3,5] |
| Properties | Regular |
The great dodecahedral honeycomb is one of four regular space-filling tessellations (or honeycombs) in hyperbolic 3-space.
Five dodecahedra exist on each edge, and 20 dodecahedra around each vertex.
The dihedral angle of a dodecahedron is ~116.6°, so no more than three of them can fit around an edge in Euclidean 3-space. In curved space, however, the dihedral angle depends on the size of the figure; a hyperbolic regular dodecahedron can be made with dihedral angles as small as one-sixth of a circle (if its vertices are at infinity).
There is another honeycomb in hyperbolic 3-space called the small dodecahedral honeycomb which has only 4 dodecahedra per edge. These honeycombs are also related to the 120-cell which can be considered as a honeycomb in positively-curved space (the surface of a 4 dimensional sphere), with 3 dodecahedra on each edge.
The bitruncated form, t1,2{5,3,5}, of this honeycomb has all truncated icosahedron cells.
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