icosahedral honeycomb#Truncated icosahedral honeycomb
{{short description|Regular tiling of hyperbolic 3-space}}
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!bgcolor=#e7dcc3 colspan=2|Icosahedral honeycomb | |
| bgcolor=#ffffff align=center colspan=2|320px viewpoint cell centered | |
| bgcolor=#e7dcc3|Type | Hyperbolic regular honeycomb Uniform hyperbolic honeycomb |
| bgcolor=#e7dcc3|Schläfli symbol | {{math|{3,5,3} }} |
| bgcolor=#e7dcc3|Coxeter diagram | {{CDD|node_1|3|node|5|node|3|node}} |
| bgcolor=#e7dcc3|Cells | {{math|{5,3} }} (regular icosahedron) 40px |
| bgcolor=#e7dcc3|Faces | {{math|{3} }} (triangle) |
| bgcolor=#e7dcc3|Edge figure | {{math|{3} }} (triangle) |
| bgcolor=#e7dcc3|Vertex figure | 80px dodecahedron |
| bgcolor=#e7dcc3|Dual | Self-dual |
| bgcolor=#e7dcc3|Coxeter group | {{math|{{overline|J}}{{sub|3}}, [3,5,3]}} |
| bgcolor=#e7dcc3|Properties | Regular |
In geometry, the icosahedral honeycomb is one of four compact, regular, space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli symbol {{math|{3,5,3},}} there are three icosahedra around each edge, and 12 icosahedra around each vertex, in a regular dodecahedral vertex figure. It is analogous to the 24-cell and the 5-cell.
{{Honeycomb}}
Description
The dihedral angle of a regular icosahedron is around 138.2°, so it is impossible to fit three icosahedra around an edge in Euclidean 3-space. However, in hyperbolic space, properly scaled icosahedra can have dihedral angles of exactly 120 degrees, so three of those can fit around an edge.
Related regular honeycombs
There are four regular compact honeycombs in 3D hyperbolic space:
{{Regular compact H3 honeycombs}}
Related regular polytopes and honeycombs
It is a member of a sequence of regular polychora and honeycombs {3,p,3} with deltrahedral cells:
{{Symmetric_tessellations}}
It is also a member of a sequence of regular polychora and honeycombs {p,5,p}, with vertex figures composed of pentagons:
{{Symmetric4 tessellations}}
Uniform honeycombs
There are nine uniform honeycombs in the [3,5,3] Coxeter group family, including this regular form as well as the bitruncated form, t1,2{3,5,3}, {{CDD|node|3|node_1|5|node_1|3|node}}, also called truncated dodecahedral honeycomb, each of whose cells are truncated dodecahedra.
{{353 family}}
= Rectified icosahedral honeycomb=
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!bgcolor=#e7dcc3 colspan=2|Rectified icosahedral honeycomb
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