Order-4 icosahedral honeycomb
In the geometry of hyperbolic 3-space, the order-4 icosahedral honeycomb is a regular space-filling tessellation with Schläfli symbol.
Geometry
It has four icosahedra around each edge. All vertices are ultra-ideal with infinitely many icosahedra existing around each vertex in an order-4 pentagonal tiling vertex arrangement.Poincaré disk model | Ideal surface |
It has a second construction as a uniform honeycomb, Schläfli symbol, Coxeter diagram,, with alternating types or colors of icosahedral cells. In Coxeter notation the half symmetry is = .
Related polytopes and honeycombs
It a part of a sequence of regular polychora and honeycombs with icosahedral cells:Order-5 icosahedral honeycomb
In the geometry of hyperbolic 3-space, the order-5 icosahedral honeycomb is a regular space-filling tessellation with Schläfli symbol. It has five icosahedra,, around each edge. All vertices are ultra-ideal with infinitely many icosahedra existing around each vertex in an order-5 pentagonal tiling vertex arrangement.Poincaré disk model | Ideal surface |
Order-6 icosahedral honeycomb
In the geometry of hyperbolic 3-space, the order-6 icosahedral honeycomb is a regular space-filling tessellation with Schläfli symbol. It has six icosahedra,, around each edge. All vertices are ultra-ideal with infinitely many icosahedra existing around each vertex in an order-6 pentagonal tiling vertex arrangement.Poincaré disk model | Ideal surface |
Order-7 icosahedral honeycomb
In the geometry of hyperbolic 3-space, the order-7 icosahedral honeycomb is a regular space-filling tessellation with Schläfli symbol. It has seven icosahedra,, around each edge. All vertices are ultra-ideal with infinitely many icosahedra existing around each vertex in an order-7 pentagonal tiling vertex arrangement.Poincaré disk model | Ideal surface |
Order-8 icosahedral honeycomb
In the geometry of hyperbolic 3-space, the order-8 icosahedral honeycomb is a regular space-filling tessellation with Schläfli symbol. It has eight icosahedra,, around each edge. All vertices are ultra-ideal with infinitely many icosahedra existing around each vertex in an order-8 pentagonal tiling vertex arrangement.Poincaré disk model |
Infinite-order icosahedral honeycomb
In the geometry of hyperbolic 3-space, the infinite-order icosahedral honeycomb is a regular space-filling tessellation with Schläfli symbol. It has infinitely many icosahedra,, around each edge. All vertices are ultra-ideal with infinitely many icosahedra existing around each vertex in an infinite-order triangular tiling vertex arrangement.Poincaré disk model | Ideal surface |
It has a second construction as a uniform honeycomb, Schläfli symbol, Coxeter diagram, =, with alternating types or colors of icosahedral cells. In Coxeter notation the half symmetry is = .