Order-4 icosahedral honeycombTypeRegular honeycombSchläfli symbols{3,5,4}Coxeter diagramsCells{3,5} Faces{3}Edge figure{4}Vertex figure{5,4} Dual{4,5,3}Coxeter group[3,5,4]PropertiesRegular In the geometry of hyperbolic 3-space, the order-4 icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,4}.
Geometry It has four icosahedra {3,5} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many icosahedra existing around each vertex in an order-4 pentagonal tiling vertex arrangement.
Poincaré disk model(Cell centered)Ideal surface
It has a second construction as a uniform honeycomb, Schläfli symbol {3,51,1}, Coxeter diagram, , with alternating types or colors of icosahedral cells. In Coxeter notation the half symmetry is [3,5,4,1+] = [3,51,1].
Related polytopes and honeycombs
It a part of a sequence of regular polychora and honeycombs with icosahedral cells: {3,5,p}
Order-5 icosahedral honeycomb Order-5 icosahedral honeycombTypeRegular honeycombSchläfli symbols{3,5,5}Coxeter diagramsCells{3,5} Faces{3}Edge figure{5}Vertex figure{5,5} Dual{5,5,3}Coxeter group[3,5,5]PropertiesRegular In the geometry of hyperbolic 3-space, the order-5 icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,5}. It has five icosahedra, {3,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many icosahedra existing around each vertex in an order-5 pentagonal tiling vertex arrangement.
Poincaré disk model(Cell centered)Ideal surface
Order-6 icosahedral honeycomb Order-6 icosahedral honeycombTypeRegular honeycombSchläfli symbols{3,5,6}{3,(5,∞,5)}Coxeter diagrams = Cells{3,5} Faces{3}Edge figure{6}Vertex figure{5,6} Dual{6,5,3}Coxeter group[3,5,6]PropertiesRegular In the geometry of hyperbolic 3-space, the order-6 icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,6}. It has six icosahedra, {3,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many icosahedra existing around each vertex in an order-6 pentagonal tiling vertex arrangement.
Poincaré disk model(Cell centered)Ideal surface
Order-7 icosahedral honeycomb Order-7 icosahedral honeycombTypeRegular honeycombSchläfli symbols{3,5,7}Coxeter diagramsCells{3,5} Faces{3}Edge figure{7}Vertex figure{5,7} Dual{7,5,3}Coxeter group[3,5,7]PropertiesRegular In the geometry of hyperbolic 3-space, the order-7 icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,7}. It has seven icosahedra, {3,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many icosahedra existing around each vertex in an order-7 pentagonal tiling vertex arrangement.
Poincaré disk model(Cell centered)Ideal surface
Order-8 icosahedral honeycomb Order-8 icosahedral honeycombTypeRegular honeycombSchläfli symbols{3,5,8}Coxeter diagramsCells{3,5} Faces{3}Edge figure{8}Vertex figure{5,8} Dual{8,5,3}Coxeter group[3,5,8]PropertiesRegular In the geometry of hyperbolic 3-space, the order-8 icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,8}. It has eight icosahedra, {3,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many icosahedra existing around each vertex in an order-8 pentagonal tiling vertex arrangement.
Poincaré disk model(Cell centered)
Infinite-order icosahedral honeycomb Infinite-order icosahedral honeycombTypeRegular honeycombSchläfli symbols{3,5,∞}{3,(5,∞,5)}Coxeter diagrams = Cells{3,5} Faces{3}Edge figure{∞}Vertex figure{5,∞} {(5,∞,5)} Dual{∞,5,3}Coxeter group[∞,5,3][3,((5,∞,5))]PropertiesRegular In the geometry of hyperbolic 3-space, the infinite-order icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,∞}. It has infinitely many icosahedra, {3,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many icosahedra existing around each vertex in an infinite-order triangular tiling vertex arrangement.
Poincaré disk model(Cell centered)Ideal surface
It has a second construction as a uniform honeycomb, Schläfli symbol {3,(5,∞,5)}, Coxeter diagram, = , with alternating types or colors of icosahedral cells. In Coxeter notation the half symmetry is [3,5,∞,1+] = [3,((5,∞,5))].
See also Convex uniform honeycombs in hyperbolic space List of regular polytopes
References
Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. . (Tables I and II: Regular polytopes and honeycombs, pp. 294–296) The Beauty of Geometry: Twelve Essays (1999), Dover Publications, , (Chapter 10, Regular Honeycombs in Hyperbolic Space ) Table III Jeffrey R. Weeks The Shape of Space, 2nd edition (Chapters 16–17: Geometries on Three-manifolds I, II) George Maxwell, Sphere Packings and Hyperbolic Reflection Groups, JOURNAL OF ALGEBRA 79,78-97 (1982) Hao Chen, Jean-Philippe Labbé, Lorentzian Coxeter groups and Boyd-Maxwell ball packings, (2013) Visualizing Hyperbolic Honeycombs arXiv:1511.02851 Roice Nelson, Henry Segerman (2015)
External links John Baez, Visual insights: {7,3,3} Honeycomb (2014/08/01) {7,3,3} Honeycomb Meets Plane at Infinity (2014/08/14) Danny Calegari, Kleinian, a tool for visualizing Kleinian groups, Geometry and the Imagination 4 March 2014.