Icosahedral honeycomb The icosahedral honeycomb is one of four compact regular space-filling tessellations 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 .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. There are four regular compact honeycombs in 3D hyperbolic space:It is a member of a sequence of regular polychora and honeycombs with deltrahedral cells: It is also a member of a sequence of regular polychora and honeycombs, with vertex figures composed of pentagons:Uniform honeycombs There are nine uniform honeycombs in the Coxeter group family , including this regular form as well as the bitruncated form, t1,2 , , also called truncated dodecahedral honeycomb , each of whose cells are truncated dodecahedra .Rectified icosahedral honeycomb The rectified icosahedral honeycomb , t1 ,, has alternating dodecahedron and icosidodecahedron cells, with a triangular prism vertex figure:Related honeycomb There are four rectified compact regular honeycombs:Truncated icosahedral honeycomb The truncated icosahedral honeycomb , t0,1 ,, has alternating dodecahedron and truncated icosahedron cells, with a triangular pyramid vertex figure.Related honeycombs Bitruncated icosahedral honeycomb The bitruncated icosahedral honeycomb , t1,2 ,, has truncated dodecahedron cells with a tetragonal disphenoid vertex figure.Related honeycombs Cantellated icosahedral honeycomb The cantellated icosahedral honeycomb , t0,2 ,, has rhombicosidodecahedron , icosidodecahedron, and triangular prism cells, with a wedge vertex figure.Related honeycombs Cantitruncated icosahedral honeycomb The cantitruncated icosahedral honeycomb , t0,1,2 ,, has truncated icosidodecahedron , truncated dodecahedron, and triangular prism cells, with a mirrored sphenoid vertex figure.Related honeycombs Runcinated icosahedral honeycomb The runcinated icosahedral honeycomb , t0,3 ,, has icosahedron and triangular prism cells, with a pentagonal antiprism vertex figure.Related honeycombs Runcitruncated icosahedral honeycomb The runcitruncated icosahedral honeycomb , t0,1,3 ,, 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.Related honeycombs Omnitruncated icosahedral honeycomb The omnitruncated icosahedral honeycomb , t0,1,2,3 ,, has truncated icosidodecahedron and hexagonal prism cells, with a phyllic disphenoid vertex figure.Related honeycombs Omnisnub icosahedral honeycomb The omnisnub icosahedral honeycomb , h,, has snub dodecahedron , octahedron , and tetrahedron cells, with an irregular vertex figure. It is vertex-transitive , but cannot be made with uniform cells.Partially diminished icosahedral honeycomb The partially diminished icosahedral honeycomb or parabidiminished icosahedral honeycomb , pd, is a non-Wythoffian uniform honeycomb with dodecahedron and pentagonal antiprism cells, with a tetrahedrally diminished dodecahedron vertex figure. The icosahedral cells of the are diminished at opposite vertices, leaving a pentagonal antiprism core , and creating new dodecahedron cells above and below.
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