Order-8-3 triangular honeycombTypeRegular honeycombSchläfli symbols{3,8,3}Coxeter diagramsCells{3,8} Faces{3}Edge figure{3}Vertex figure{8,3} DualSelf-dualCoxeter group[3,8,3]PropertiesRegular In the geometry of hyperbolic 3-space, the order-8-3 triangular honeycomb (or 3,8,3 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,8,3}.
Geometry It has three order-8 triangular tiling {3,8} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an octagonal tiling vertex figure.
Poincaré disk model
Related polytopes and honeycombs It is a part of a sequence of regular honeycombs with order-8 triangular tiling cells: {3,8,p}.
It is a part of a sequence of regular honeycombs with octagonal tiling vertex figures: {p,8,3}.
It is a part of a sequence of self-dual regular honeycombs: {p,8,p}.
Order-8-4 triangular honeycomb Order-8-4 triangular honeycombTypeRegular honeycombSchläfli symbols{3,8,4}Coxeter diagrams = Cells{3,8} Faces{3}Edge figure{4}Vertex figure{8,4} r{8,8} Dual{4,8,3}Coxeter group[3,8,4]PropertiesRegular In the geometry of hyperbolic 3-space, the order-8-4 triangular honeycomb (or 3,8,4 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,8,4}.
It has four order-8 triangular tilings, {3,8}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-8 triangular tilings existing around each vertex in an order-4 hexagonal tiling vertex arrangement.
Poincaré disk model
It has a second construction as a uniform honeycomb, Schläfli symbol {3,81,1}, Coxeter diagram, , with alternating types or colors of order-8 triangular tiling cells. In Coxeter notation the half symmetry is [3,8,4,1+] = [3,81,1].
Order-8-5 triangular honeycomb Order-8-5 triangular honeycombTypeRegular honeycombSchläfli symbols{3,8,5}Coxeter diagramsCells{3,8} Faces{3}Edge figure{5}Vertex figure{8,5} Dual{5,8,3}Coxeter group[3,8,5]PropertiesRegular In the geometry of hyperbolic 3-space, the order-8-3 triangular honeycomb (or 3,8,5 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,8,5}. It has five order-8 triangular tiling, {3,8}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-8 triangular tilings existing around each vertex in an order-5 octagonal tiling vertex figure.
Poincaré disk model
Order-8-6 triangular honeycomb Order-8-6 triangular honeycombTypeRegular honeycombSchläfli symbols{3,8,6}{3,(8,3,8)}Coxeter diagrams = Cells{3,8} Faces{3}Edge figure{6}Vertex figure{8,6} {(8,3,8)} Dual{6,8,3}Coxeter group[3,8,6]PropertiesRegular In the geometry of hyperbolic 3-space, the order-8-6 triangular honeycomb (or 3,8,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,8,6}. It has infinitely many order-8 triangular tiling, {3,8}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-8 triangular tilings existing around each vertex in an order-6 octagonal tiling, {8,6}, vertex figure.
Poincaré disk model
Order-8-infinite triangular honeycomb Order-8-infinite triangular honeycombTypeRegular honeycombSchläfli symbols{3,8,∞}{3,(8,∞,8)}Coxeter diagrams = Cells{3,8} Faces{3}Edge figure{∞}Vertex figure{8,∞} {(8,∞,8)} Dual{∞,8,3}Coxeter group[∞,8,3][3,((8,∞,8))]PropertiesRegular In the geometry of hyperbolic 3-space, the order-8-infinite triangular honeycomb (or 3,8,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,8,∞}. It has infinitely many order-8 triangular tiling, {3,8}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-8 triangular tilings existing around each vertex in an infinite-order octagonal tiling, {8,∞}, vertex figure.
Poincaré disk model
It has a second construction as a uniform honeycomb, Schläfli symbol {3,(8,∞,8)}, Coxeter diagram, = , with alternating types or colors of order-8 triangular tiling cells. In Coxeter notation the half symmetry is [3,8,∞,1+] = [3,((8,∞,8))].
Order-8-3 square honeycomb Order-8-3 square honeycombTypeRegular honeycombSchläfli symbol{4,8,3}Coxeter diagramCells{4,8} Faces{4}Vertex figure{8,3}Dual{3,8,4}Coxeter group[4,8,3]PropertiesRegular In the geometry of hyperbolic 3-space, the order-8-3 square honeycomb (or 4,8,3 honeycomb) a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an octagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
The Schläfli symbol of the order-8-3 square honeycomb is {4,8,3}, with three order-4 octagonal tilings meeting at each edge. The vertex figure of this honeycomb is an octagonal tiling, {8,3}.
Poincaré disk model
Order-8-3 pentagonal honeycomb
Order-8-3 pentagonal honeycombTypeRegular honeycombSchläfli symbol{5,8,3}Coxeter diagramCells{5,8} Faces{5}Vertex figure{8,3}Dual{3,8,5}Coxeter group[5,8,3]PropertiesRegular In the geometry of hyperbolic 3-space, the order-8-3 pentagonal honeycomb (or 5,8,3 honeycomb) a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-8 pentagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
The Schläfli symbol of the order-6-3 pentagonal honeycomb is {5,8,3}, with three order-8 pentagonal tilings meeting at each edge. The vertex figure of this honeycomb is an octagonal tiling, {8,3}.
Poincaré disk model
Order-8-3 hexagonal honeycomb Order-8-3 hexagonal honeycombTypeRegular honeycombSchläfli symbol{6,8,3}Coxeter diagramCells{6,8} Faces{6}Vertex figure{8,3}Dual{3,8,6}Coxeter group[6,8,3]PropertiesRegular In the geometry of hyperbolic 3-space, the order-8-3 hexagonal honeycomb (or 6,8,3 honeycomb) a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-6 hexagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
The Schläfli symbol of the order-8-3 hexagonal honeycomb is {6,8,3}, with three order-5 hexagonal tilings meeting at each edge. The vertex figure of this honeycomb is an octagonal tiling, {8,3}.
Poincaré disk model
Order-8-3 apeirogonal honeycomb Order-8-3 apeirogonal honeycombTypeRegular honeycombSchläfli symbol{∞,8,3}Coxeter diagramCells{∞,8} FacesApeirogon {∞}Vertex figure{8,3}Dual{3,8,∞}Coxeter group[∞,8,3]PropertiesRegular In the geometry of hyperbolic 3-space, the order-8-3 apeirogonal honeycomb (or ∞,8,3 honeycomb) a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-8 apeirogonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
The Schläfli symbol of the apeirogonal tiling honeycomb is {∞,8,3}, with three order-8 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is an octagonal tiling, {8,3}.
The "ideal surface" projection below is a plane-at-infinity, in the Poincaré half-space model of H3. It shows an Apollonian gasket pattern of circles inside a largest circle.
Poincaré disk model
Order-8-4 square honeycomb Order-8-4 square honeycombTypeRegular honeycombSchläfli symbol{4,8,4}Coxeter diagrams = Cells{4,8} Faces{4}Edge figure{4}Vertex figure{8,4}Dualself-dualCoxeter group[4,8,4]PropertiesRegular In the geometry of hyperbolic 3-space, the order-8-4 square honeycomb (or 4,8,4 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,8,4}.
All vertices are ultra-ideal (existing beyond the ideal boundary) with four order-5 square tilings existing around each edge and with an order-4 octagonal tiling vertex figure.
Poincaré disk model
Order-8-5 pentagonal honeycomb Order-8-5 pentagonal honeycombTypeRegular honeycombSchläfli symbol{5,8,5}Coxeter diagramsCells{5,8} Faces{5}Edge figure{5}Vertex figure{8,5}Dualself-dualCoxeter group[5,8,5]PropertiesRegular In the geometry of hyperbolic 3-space, the order-8-5 pentagonal honeycomb (or 5,8,5 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {5,8,5}.
All vertices are ultra-ideal (existing beyond the ideal boundary) with five order-8 pentagonal tilings existing around each edge and with an order-5 pentagonal tiling vertex figure.
Poincaré disk model
Order-8-6 hexagonal honeycomb Order-8-6 hexagonal honeycombTypeRegular honeycombSchläfli symbols{6,8,6}{6,(8,3,8)}Coxeter diagrams = Cells{6,8} Faces{6}Edge figure{6}Vertex figure{8,6} {(5,3,5)} Dualself-dualCoxeter group[6,8,6][6,((8,3,8))]PropertiesRegular In the geometry of hyperbolic 3-space, the order-8-6 hexagonal honeycomb (or 6,8,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,8,6}. It has six order-8 hexagonal tilings, {6,8}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many hexagonal tilings existing around each vertex in an order-6 octagonal tiling vertex arrangement.
Poincaré disk model
It has a second construction as a uniform honeycomb, Schläfli symbol {6,(8,3,8)}, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [6,8,6,1+] = [6,((8,3,8))].
Order-8-infinite apeirogonal honeycomb Order-8-infinite apeirogonal honeycombTypeRegular honeycombSchläfli symbols{∞,8,∞}{∞,(8,∞,8)}Coxeter diagrams ↔ Cells{∞,8} Faces{∞}Edge figure{∞}Vertex figure {8,∞} {(8,∞,8)}Dualself-dualCoxeter group[∞,8,∞][∞,((8,∞,8))]PropertiesRegular In the geometry of hyperbolic 3-space, the order-8-infinite apeirogonal honeycomb (or ∞,8,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {∞,8,∞}. It has infinitely many order-8 apeirogonal tiling {∞,8} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-8 apeirogonal tilings existing around each vertex in an infinite-order octagonal tiling vertex figure.
Poincaré disk model
It has a second construction as a uniform honeycomb, Schläfli symbol {∞,(8,∞,8)}, Coxeter diagram, , with alternating types or colors of cells.
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 Hyperbolic Catacombs Carousel: {3,7,3} honeycomb YouTube, Roice Nelson 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.