Icosahedral honeycomb

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.

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:

Related regular polytopes and honeycombs
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:

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},, also called truncated dodecahedral honeycomb, each of whose cells are truncated dodecahedra.

Rectified icosahedral honeycomb
The rectified icosahedral honeycomb, t1{3,5,3},, has alternating dodecahedron and icosidodecahedron cells, with a triangular prism vertex figure:
 * H3 353 CC center 0100.pngRectified icosahedral honeycomb.pngPerspective projections from center of Poincaré disk model

Related honeycomb
There are four rectified compact regular honeycombs:

Truncated icosahedral honeycomb
The truncated icosahedral honeycomb, t0,1{3,5,3},, has alternating dodecahedron and truncated icosahedron cells, with a triangular pyramid vertex figure.



Bitruncated icosahedral honeycomb
The bitruncated icosahedral honeycomb, t1,2{3,5,3},, has truncated dodecahedron cells with a tetragonal disphenoid vertex figure.



Cantellated icosahedral honeycomb
The cantellated icosahedral honeycomb, t0,2{3,5,3},, has rhombicosidodecahedron, icosidodecahedron, and triangular prism cells, with a wedge vertex figure.



Cantitruncated icosahedral honeycomb
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.



Runcinated icosahedral honeycomb
The runcinated icosahedral honeycomb, t0,3{3,5,3},, has icosahedron and triangular prism cells, with a pentagonal antiprism vertex figure.


 * Viewed from center of triangular prism

Runcitruncated icosahedral honeycomb
The runcitruncated icosahedral honeycomb, t0,1,3{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.


 * Viewed from center of triangular prism

Omnitruncated icosahedral honeycomb
The omnitruncated icosahedral honeycomb, t0,1,2,3{3,5,3},, has truncated icosidodecahedron and hexagonal prism cells, with a phyllic disphenoid vertex figure.


 * Centered on hexagonal prism

Omnisnub icosahedral honeycomb
The omnisnub icosahedral honeycomb, h(t0,1,2,3{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.

Partially diminished icosahedral honeycomb
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.