Order-7-3 triangular honeycombTypeRegular honeycombSchläfli symbols{3,7,3}Coxeter diagramsCells{3,7} Faces{3}Edge figure{3}Vertex figure{7,3} DualSelf-dualCoxeter group[3,7,3]PropertiesRegular In the geometry of hyperbolic 3-space, the order-7-3 triangular honeycomb (or 3,7,3 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,7,3}.
Geometry It has three order-7 triangular tiling {3,7} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in a heptagonal tiling vertex figure.
Poincaré disk modelIdeal surfaceUpper half space model with selective cells shownHyperbolic Catacombs Roice Nelson and Henry Segerman, 2014
Related polytopes and honeycombs It a part of a sequence of self-dual regular honeycombs: {p,7,p}.
It is a part of a sequence of regular honeycombs with order-7 triangular tiling cells: {3,7,p}.
It is a part of a sequence of regular honeycombs with heptagonal tiling vertex figures: {p,7,3}.
Order-7-4 triangular honeycomb Order-7-4 triangular honeycombTypeRegular honeycombSchläfli symbols{3,7,4}Coxeter diagrams = Cells{3,7} Faces{3}Edge figure{4}Vertex figure{7,4} r{7,7} Dual{4,7,3}Coxeter group[3,7,4]PropertiesRegular In the geometry of hyperbolic 3-space, the order-7-4 triangular honeycomb (or 3,7,4 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,7,4}.
It has four order-7 triangular tilings, {3,7}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-7 triangular tilings existing around each vertex in an order-4 heptagonal tiling vertex arrangement.
Poincaré disk modelIdeal surface
It has a second construction as a uniform honeycomb, Schläfli symbol {3,71,1}, Coxeter diagram, , with alternating types or colors of order-7 triangular tiling cells. In Coxeter notation the half symmetry is [3,7,4,1+] = [3,71,1].
Order-7-5 triangular honeycomb Order-7-5 triangular honeycombTypeRegular honeycombSchläfli symbols{3,7,5}Coxeter diagramsCells{3,7} Faces{3}Edge figure{5}Vertex figure{7,5} Dual{5,7,3}Coxeter group[3,7,5]PropertiesRegular In the geometry of hyperbolic 3-space, the order-7-3 triangular honeycomb (or 3,7,5 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,7,5}. It has five order-7 triangular tiling, {3,7}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-7 triangular tilings existing around each vertex in an order-5 heptagonal tiling vertex figure.
Poincaré disk modelIdeal surface
Order-7-6 triangular honeycomb Order-7-6 triangular honeycombTypeRegular honeycombSchläfli symbols{3,7,6}{3,(7,3,7)}Coxeter diagrams = Cells{3,7} Faces{3}Edge figure{6}Vertex figure{7,6} {(7,3,7)} Dual{6,7,3}Coxeter group[3,7,6]PropertiesRegular In the geometry of hyperbolic 3-space, the order-7-6 triangular honeycomb (or 3,7,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,7,6}. It has infinitely many order-7 triangular tiling, {3,7}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-7 triangular tilings existing around each vertex in an order-6 heptagonal tiling, {7,6}, vertex figure.
Poincaré disk modelIdeal surface
Order-7-infinite triangular honeycomb Order-7-infinite triangular honeycombTypeRegular honeycombSchläfli symbols{3,7,∞}{3,(7,∞,7)}Coxeter diagrams = Cells{3,7} Faces{3}Edge figure{∞}Vertex figure{7,∞} {(7,∞,7)} Dual{∞,7,3}Coxeter group[∞,7,3][3,((7,∞,7))]PropertiesRegular In the geometry of hyperbolic 3-space, the order-7-infinite triangular honeycomb (or 3,7,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,7,∞}. It has infinitely many order-7 triangular tiling, {3,7}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-7 triangular tilings existing around each vertex in an infinite-order heptagonal tiling, {7,∞}, vertex figure.
Poincaré disk modelIdeal surface
It has a second construction as a uniform honeycomb, Schläfli symbol {3,(7,∞,7)}, Coxeter diagram, = , with alternating types or colors of order-7 triangular tiling cells. In Coxeter notation the half symmetry is [3,7,∞,1+] = [3,((7,∞,7))].
Order-7-3 square honeycomb Order-7-3 square honeycombTypeRegular honeycombSchläfli symbol{4,7,3}Coxeter diagramCells{4,7} Faces{4}Vertex figure{7,3}Dual{3,7,4}Coxeter group[4,7,3]PropertiesRegular In the geometry of hyperbolic 3-space, the order-7-3 square honeycomb (or 4,7,3 honeycomb) a regular space-filling tessellation (or honeycomb). Each infinite cell consists of a heptagonal 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-7-3 square honeycomb is {4,7,3}, with three order-4 heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is a heptagonal tiling, {7,3}.
Poincaré disk modelIdeal surface
Order-7-3 pentagonal honeycomb
Order-7-3 pentagonal honeycombTypeRegular honeycombSchläfli symbol{5,7,3}Coxeter diagramCells{5,7} Faces{5}Vertex figure{7,3}Dual{3,7,5}Coxeter group[5,7,3]PropertiesRegular In the geometry of hyperbolic 3-space, the order-7-3 pentagonal honeycomb (or 5,7,3 honeycomb) a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-7 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 {5,7,3}, with three order-7 pentagonal tilings meeting at each edge. The vertex figure of this honeycomb is a heptagonal tiling, {7,3}.
Poincaré disk modelIdeal surface
Order-7-3 hexagonal honeycomb Order-7-3 hexagonal honeycombTypeRegular honeycombSchläfli symbol{6,7,3}Coxeter diagramCells{6,7} Faces{6}Vertex figure{7,3}Dual{3,7,6}Coxeter group[6,7,3]PropertiesRegular In the geometry of hyperbolic 3-space, the order-7-3 hexagonal honeycomb (or 6,7,3 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-7-3 hexagonal honeycomb is {6,7,3}, with three order-5 hexagonal tilings meeting at each edge. The vertex figure of this honeycomb is a heptagonal tiling, {7,3}.
Poincaré disk modelIdeal surface
Order-7-3 apeirogonal honeycomb Order-7-3 apeirogonal honeycombTypeRegular honeycombSchläfli symbol{∞,7,3}Coxeter diagramCells{∞,7} FacesApeirogon {∞}Vertex figure{7,3}Dual{3,7,∞}Coxeter group[∞,7,3]PropertiesRegular In the geometry of hyperbolic 3-space, the order-7-3 apeirogonal honeycomb (or ∞,7,3 honeycomb) a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-7 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 {∞,7,3}, with three order-7 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is a heptagonal tiling, {7,3}.
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 modelIdeal surface
Order-7-4 square honeycomb Order-7-4 square honeycombTypeRegular honeycombSchläfli symbol{4,7,4}Coxeter diagrams = Cells{4,7} Faces{4}Edge figure{4}Vertex figure{7,4}Dualself-dualCoxeter group[4,7,4]PropertiesRegular In the geometry of hyperbolic 3-space, the order-7-4 square honeycomb (or 4,7,4 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,7,4}.
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 heptagonal tiling vertex figure.
Poincaré disk modelIdeal surface
Order-7-5 pentagonal honeycomb Order-7-5 pentagonal honeycombTypeRegular honeycombSchläfli symbol{5,7,5}Coxeter diagramsCells{5,7} Faces{5}Edge figure{5}Vertex figure{7,5}Dualself-dualCoxeter group[5,7,5]PropertiesRegular In the geometry of hyperbolic 3-space, the order-7-5 pentagonal honeycomb (or 5,7,5 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {5,7,5}.
All vertices are ultra-ideal (existing beyond the ideal boundary) with five order-7 pentagonal tilings existing around each edge and with an order-5 heptagonal tiling vertex figure.
Poincaré disk modelIdeal surface
Order-7-6 hexagonal honeycomb Order-7-6 hexagonal honeycombTypeRegular honeycombSchläfli symbols{6,7,6}{6,(7,3,7)}Coxeter diagrams = Cells{6,7} Faces{6}Edge figure{6}Vertex figure{7,6} {(5,3,5)} Dualself-dualCoxeter group[6,7,6][6,((7,3,7))]PropertiesRegular In the geometry of hyperbolic 3-space, the order-7-6 hexagonal honeycomb (or 6,7,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,7,6}. It has six order-7 hexagonal tilings, {6,7}, 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 heptagonal tiling vertex arrangement.
Poincaré disk modelIdeal surface
It has a second construction as a uniform honeycomb, Schläfli symbol {6,(7,3,7)}, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [6,7,6,1+] = [6,((7,3,7))].
Order-7-infinite apeirogonal honeycomb Order-7-infinite apeirogonal honeycombTypeRegular honeycombSchläfli symbols{∞,7,∞}{∞,(7,∞,7)}Coxeter diagrams ↔ Cells{∞,7} Faces{∞}Edge figure{∞}Vertex figure {7,∞} {(7,∞,7)}Dualself-dualCoxeter group[∞,7,∞][∞,((7,∞,7))]PropertiesRegular In the geometry of hyperbolic 3-space, the order-7-infinite apeirogonal honeycomb (or ∞,7,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {∞,7,∞}. It has infinitely many order-7 apeirogonal tiling {∞,7} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-7 apeirogonal tilings existing around each vertex in an infinite-order heptagonal tiling vertex figure.
Poincaré disk modelIdeal surface
It has a second construction as a uniform honeycomb, Schläfli symbol {∞,(7,∞,7)}, 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.