In geometry, the icosahedral honeycomb is one of four compact, regular, space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli symbol 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.
The dihedral angle of a regular icosahedron is around 138.2ð, 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.
There are four regular compact honeycombs in 3D hyperbolic space:
It is a member of a sequence of regular polychora and honeycombs {3,p,3} with deltrahedral cells:
It is also a member of a sequence of regular polychora and honeycombs {p,5,p}, with vertex figures composed of pentagons:
There are nine uniform honeycombs in the [3,5,3] Coxeter group family, including this regular form as well as the bitruncated form, t<sub>1,2</sub>{3,5,3}, , also called truncated dodecahedral honeycomb, each of whose cells are truncated dodecahedra.
The rectified icosahedral honeycomb, t<sub>1</sub>{3,5,3}, , has alternating dodecahedron and icosidodecahedron cells, with a triangular prism vertex figure:
There are four rectified compact regular honeycombs:
The truncated icosahedral honeycomb, t<sub>0,1</sub>{3,5,3}, , has alternating dodecahedron and truncated icosahedron cells, with a triangular pyramid vertex figure.
The bitruncated icosahedral honeycomb, t<sub>1,2</sub>{3,5,3}, , has truncated dodecahedron cells with a tetragonal disphenoid vertex figure.
The cantellated icosahedral honeycomb, t<sub>0,2</sub>{3,5,3}, , has rhombicosidodecahedron, icosidodecahedron, and triangular prism cells, with a wedge vertex figure.
The cantitruncated icosahedral honeycomb, t<sub>0,1,2</sub>{3,5,3}, , has truncated icosidodecahedron, truncated dodecahedron, and triangular prism cells, with a mirrored sphenoid vertex figure.
The runcinated icosahedral honeycomb, t<sub>0,3</sub>{3,5,3}, , has icosahedron and triangular prism cells, with a pentagonal antiprism vertex figure.
The runcitruncated icosahedral honeycomb, t<sub>0,1,3</sub>{3,5,3}, , has truncated icosahedron, rhombicosidodecahedron, hexagonal prism, and triangular prism cells, with an isosceles-trapezoidal pyramid vertex figure.
The runcicantellated icosahedral honeycomb is equivalent to the runcitruncated icosahedral honeycomb.
The omnitruncated icosahedral honeycomb, t<sub>0,1,2,3</sub>{3,5,3}, , has truncated icosidodecahedron and hexagonal prism cells, with a phyllic disphenoid vertex figure.
The omnisnub icosahedral honeycomb, h(t<sub>0,1,2,3</sub>{3,5,3}), , has snub dodecahedron, octahedron, and tetrahedron cells, with an irregular vertex figure. It is vertex-transitive, but cannot be made with uniform cells.
The partially diminished icosahedral honeycomb or parabidiminished icosahedral honeycomb, pd{3,5,3}, is a non-Wythoffian uniform honeycomb with dodecahedron and pentagonal antiprism cells, with a tetrahedrally diminished dodecahedron vertex figure. The icosahedral cells of the {3,5,3} are diminished at opposite vertices (parabidiminished), leaving a pentagonal antiprism (parabidiminished icosahedron) core, and creating new dodecahedron cells above and below.