Order-7 dodecahedral honeycombTypeRegular honeycombSchläfli symbols{5,3,7}Coxeter diagramsCells{5,3} Faces{5}Edge figure{7}Vertex figure{3,7}Dual{7,3,5}Coxeter group[5,3,7]PropertiesRegular In the geometry of hyperbolic 3-space, the order-7 dodecahedral honeycomb is a regular space-filling tessellation (or honeycomb).
Geometry With Schläfli symbol {5,3,7}, it has seven dodecahedra {5,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many dodecahedra existing around each vertex in an order-7 triangular tiling vertex arrangement.
Poincaré disk modelCell-centeredPoincaré disk modelIdeal surface
Related polytopes and honeycombs It a part of a sequence of regular polytopes and honeycombs with dodecahedral cells, {5,3,p}.
It a part of a sequence of honeycombs {5,p,7}.
It a part of a sequence of honeycombs {p,3,7}. {3,3,7}{4,3,7}{5,3,7}{6,3,7}{7,3,7}{8,3,7}{∞,3,7}
Order-8 dodecahedral honeycomb Order-8 dodecahedral honeycombTypeRegular honeycombSchläfli symbols{5,3,8}{5,(3,4,3)}Coxeter diagrams = Cells{5,3} Faces{5}Edge figure{8}Vertex figure{3,8}, {(3,4,3)}Dual{8,3,5}Coxeter group[5,3,8][5,((3,4,3))]PropertiesRegular In the geometry of hyperbolic 3-space, the order-8 dodecahedral honeycomb a regular space-filling tessellation (or honeycomb). With Schläfli symbol {5,3,8}, it has eight dodecahedra {5,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many dodecahedra existing around each vertex in an order-8 triangular tiling vertex arrangement.
Poincaré disk modelCell-centeredPoincaré disk model
It has a second construction as a uniform honeycomb, Schläfli symbol {5,(3,4,3)}, Coxeter diagram, , with alternating types or colors of dodecahedral cells.
Infinite-order dodecahedral honeycomb Infinite-order dodecahedral honeycombTypeRegular honeycombSchläfli symbols{5,3,∞}{5,(3,∞,3)}Coxeter diagrams = Cells{5,3} Faces{5}Edge figure{∞}Vertex figure{3,∞}, {(3,∞,3)}Dual{∞,3,5}Coxeter group[5,3,∞][5,((3,∞,3))]PropertiesRegular In the geometry of hyperbolic 3-space, the infinite-order dodecahedral honeycomb a regular space-filling tessellation (or honeycomb). With Schläfli symbol {5,3,∞}. It has infinitely many dodecahedra {5,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many dodecahedra existing around each vertex in an infinite-order triangular tiling vertex arrangement.
Poincaré disk modelCell-centeredPoincaré disk modelIdeal surface It has a second construction as a uniform honeycomb, Schläfli symbol {5,(3,∞,3)}, Coxeter diagram, , with alternating types or colors of dodecahedral cells.
See also Convex uniform honeycombs in hyperbolic space List of regular polytopes Infinite-order hexagonal tiling honeycomb
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. {5,3,∞} Honeycomb in H^3 YouTube rotation of Poincare sphere