 | ||
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, {3,5}, around each edge, and 12 icosahedra around each vertex, in a regular dodecahedral, {5,3}, vertex figure.
Contents
- Description
- Related regular honeycombs
- Uniform honeycombs
- Rectified icosahedral honeycomb
- Related honeycomb
- Truncated icosahedral honeycomb
- Bitruncated icosahedral honeycomb
- Cantellated icosahedral honeycomb
- Cantitruncated icosahedral honeycomb
- Runcinated icosahedral honeycomb
- Runcitruncated icosahedral honeycomb
- Omnitruncated icosahedral honeycomb
- Omnisnub icosahedral honeycomb
- Partially diminished icosahedral honeycomb
- References
A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.
Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.
Description
The dihedral angle of a Euclidean icosahedron is 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 exactly 120 degrees, so three of these fit around an edge.
Related regular honeycombs
There are four regular compact honeycombs in 3D hyperbolic space:
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:
Perspective 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 disphenoid vertex figure.
Cantellated icosahedral honeycomb
The cantellated icosahedral honeycomb, t0,2{3,5,3}, , has rhombicosidodecahedron and icosidodecahedron cells, with a triangular prism vertex figure.
Cantitruncated icosahedral honeycomb
The cantitruncated icosahedral honeycomb, t0,1,2{3,5,3}, , has truncated icosidodecahedron, icosidodecahedron, triangular prism and hexagonal 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 prismRuncitruncated icosahedral honeycomb
The runcitruncated icosahedral honeycomb, t0,1,3{3,5,3}, , has truncated icosahedron, rhombicosidodecahedron, hexagonal prism and triangular prism cells, with a square pyramid vertex figure.
Viewed from center of triangular prismOmnitruncated icosahedral honeycomb
The omnitruncated icosahedral honeycomb, t0,1,2,3{3,5,3}, , has truncated icosidodecahedron and pentagonal prism cells, with a tetrahedral vertex figure.
Centered on hexagonal prismOmnisnub 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 can't be made with uniform cells.
Partially diminished icosahedral honeycomb
The partially diminished icosahedral honeycomb or parabidiminished icosahedral honeycomb, pd{3,5,3}, is a nonwythoffian 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.