In hyperbolic geometry, the order-4 dodecahedral honeycomb is one of four compact regular space-filling tessellations of hyperbolic 3-space. With Schläfli symbol {5,3,4}, it has four dodecahedra around each edge, and 8 dodecahedra around each vertex in an octahedral arrangement. Its vertices are constructed from 3 orthogonal axes. Its dual is the order-5 cubic honeycomb.
In hyperbolic geometry, the order-5 dodecahedral honeycomb is one of four compact regular space-filling tessellations in hyperbolic 3-space. With Schläfli symbol {5,3,5}, it has five dodecahedral cells around each edge, and each vertex is surrounded by twenty dodecahedra. Its vertex figure is an icosahedron.
In geometry, the icosahedral honeycomb is one of four compact, regular, space-filling tessellations 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.
In hyperbolic 3-space, the order-6 tetrahedral honeycomb is a paracompact regular space-filling tessellation. It is paracompact because it has vertex figures composed of an infinite number of faces, and has all vertices as ideal points at infinity. With Schläfli symbol {3,3,6}, the order-6 tetrahedral honeycomb has six ideal tetrahedra around each edge. All vertices are ideal, with infinitely many tetrahedra existing around each vertex in a triangular tiling vertex figure.
The order-6 dodecahedral honeycomb is one of 11 paracompact regular honeycombs in hyperbolic 3-space. It is paracompact because it has vertex figures composed of an infinite number of faces, with all vertices as ideal points at infinity. It has Schläfli symbol {5,3,6}, with six ideal dodecahedral cells surrounding each edge of the honeycomb. Each vertex is ideal, and surrounded by infinitely many dodecahedra. The honeycomb has a triangular tiling vertex figure.
The triangular tiling honeycomb is one of 11 paracompact regular space-filling tessellations in hyperbolic 3-space. It is called paracompact because it has infinite cells and vertex figures, with all vertices as ideal points at infinity. It has Schläfli symbol {3,6,3}, being composed of triangular tiling cells. Each edge of the honeycomb is surrounded by three cells, and each vertex is ideal with infinitely many cells meeting there. Its vertex figure is a hexagonal tiling.
The order-4 octahedral honeycomb is a regular paracompact honeycomb in hyperbolic 3-space. It is paracompact because it has infinite vertex figures, with all vertices as ideal points at infinity. Given by Schläfli symbol {3,4,4}, it has four ideal octahedra around each edge, and infinite octahedra around each vertex in a square tiling vertex figure.
In the geometry of hyperbolic 3-space, the cubic-octahedral honeycomb is a compact uniform honeycomb, constructed from cube, octahedron, and cuboctahedron cells, in a rhombicuboctahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.
In the geometry of hyperbolic 3-space, the dodecahedral-icosahedral honeycomb is a uniform honeycomb, constructed from dodecahedron, icosahedron, and icosidodecahedron cells, in a rhombicosidodecahedron vertex figure.
In the geometry of hyperbolic 3-space, the tetrahedron-cube honeycomb is a compact uniform honeycomb, constructed from cube, tetrahedron, and cuboctahedron cells, in a rhombicuboctahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.
In the geometry of hyperbolic 3-space, the tetrahedron-octahedron honeycomb is a compact uniform honeycomb, constructed from octahedron and tetrahedron cells, in a rhombicuboctahedron vertex figure.
In the geometry of hyperbolic 3-space, the tetrahedral-dodecahedral honeycomb is a compact uniform honeycomb, constructed from dodecahedron, tetrahedron, and icosidodecahedron cells, in a rhombitetratetrahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.
In the geometry of hyperbolic 3-space, the tetrahedral-icosahedral honeycomb is a compact uniform honeycomb, constructed from icosahedron, tetrahedron, and octahedron cells, in an icosidodecahedron vertex figure. It has a single-ring Coxeter diagram , and is named by its two regular cells.
In the geometry of hyperbolic 3-space, the octahedron-dodecahedron honeycomb is a compact uniform honeycomb, constructed from dodecahedron, octahedron, and icosidodecahedron cells, in a rhombicuboctahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.
In the geometry of hyperbolic 3-space, the octahedron-hexagonal tiling honeycomb is a paracompact uniform honeycomb, constructed from octahedron, hexagonal tiling, and trihexagonal tiling cells, in a rhombicuboctahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.
In the geometry of hyperbolic 3-space, the hexagonal tiling-triangular tiling honeycomb is a paracompact uniform honeycomb, constructed from triangular tiling, hexagonal tiling, and trihexagonal tiling cells, in a rhombitrihexagonal tiling vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.
In the geometry of hyperbolic 3-space, the cubic-triangular tiling honeycomb is a paracompact uniform honeycomb, constructed from cube, triangular tiling, and cuboctahedron cells, in a rhombitrihexagonal tiling vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.
In the geometry of hyperbolic 3-space, the tetrahedral-triangular tiling honeycomb is a paracompact uniform honeycomb, constructed from triangular tiling, tetrahedron, and octahedron cells, in an icosidodecahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.
In the geometry of hyperbolic 3-space, the cubic-square tiling honeycomb is a paracompact uniform honeycomb, constructed from cube and square tiling cells, in a rhombicuboctahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.
In the geometry of hyperbolic 3-space, the tetrahedral-square tiling honeycomb is a paracompact uniform honeycomb, constructed from tetrahedron, cuboctahedron and square tiling cells, in a rhombicuboctahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.
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