Concept
In geometry, the icosahedral honeycomb is one of four compact, regular, space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli symbol {3,5,3}, there are three icosahedra around each edge, and 12 icosahedra around each vertex, in a regular dodecahedral vertex figure. 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. 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, t1,2{3,5,3}, , also called truncated dodecahedral honeycomb, each of whose cells are truncated dodecahedra. The rectified icosahedral honeycomb, t1{3,5,3}, , has alternating dodecahedron and icosidodecahedron cells, with a triangular prism vertex figure: Perspective projections from center of Poincaré disk model There are four rectified compact regular honeycombs: The truncated icosahedral honeycomb, t0,1{3,5,3}, , has alternating dodecahedron and truncated icosahedron cells, with a triangular pyramid vertex figure. The bitruncated icosahedral honeycomb, t1,2{3,5,3}, , has truncated dodecahedron cells with a tetragonal disphenoid vertex figure. The cantellated icosahedral honeycomb, t0,2{3,5,3}, , has rhombicosidodecahedron, icosidodecahedron, and triangular prism cells, with a wedge vertex figure. The cantitruncated icosahedral honeycomb, t0,1,2{3,5,3}, , has truncated icosidodecahedron, truncated dodecahedron, and triangular prism cells, with a mirrored sphenoid vertex figure.
