- Source: Order-7-3 triangular honeycomb
In the geometry of hyperbolic 3-space, the 3/info/order" target="_blank">order-3/info/7" target="_blank">7-3 triangular honeycomb (or 3,3/info/7" target="_blank">7,3 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,3/info/7" target="_blank">7,3}.
Geometry
It has three 3/info/order" target="_blank">order-3/info/7" target="_blank">7 triangular tiling {3,3/info/7" target="_blank">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.
Related polytopes and honeycombs
It a part of a sequence of self-dual regular honeycombs: {p,3/info/7" target="_blank">7,p}.
It is a part of a sequence of regular honeycombs with 3/info/order" target="_blank">order-3/info/7" target="_blank">7 triangular tiling cells: {3,3/info/7" target="_blank">7,p}.
It isa part of a sequence of regular honeycombs with heptagonal tiling vertex figures: {p,3/info/7" target="_blank">7,3}.
= 3/info/order" target="_blank">Order-3/info/7" target="_blank">7-4 triangular honeycomb
=In the geometry of hyperbolic 3-space, the 3/info/order" target="_blank">order-3/info/7" target="_blank">7-4 triangular honeycomb (or 3,3/info/7" target="_blank">7,4 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,3/info/7" target="_blank">7,4}.
It has four 3/info/order" target="_blank">order-3/info/7" target="_blank">7 triangular tilings, {3,3/info/7" target="_blank">7}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many 3/info/order" target="_blank">order-3/info/7" target="_blank">7 triangular tilings existing around each vertex in an 3/info/order" target="_blank">order-4 hexagonal tiling vertex arrangement.
It has a second construction as a uniform honeycomb, Schläfli symbol {3,71,1}, Coxeter diagram, , with alternating types or colors of 3/info/order" target="_blank">order-3/info/7" target="_blank">7 triangular tiling cells. In Coxeter notation the half symmetry is [3,3/info/7" target="_blank">7,4,1+] = [3,71,1].
= 3/info/order" target="_blank">Order-3/info/7" target="_blank">7-5 triangular honeycomb
=In the geometry of hyperbolic 3-space, the 3/info/order" target="_blank">order-3/info/7" target="_blank">7-3 triangular honeycomb (or 3,3/info/7" target="_blank">7,5 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,3/info/7" target="_blank">7,5}. It has five 3/info/order" target="_blank">order-3/info/7" target="_blank">7 triangular tiling, {3,3/info/7" target="_blank">7}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many 3/info/order" target="_blank">order-3/info/7" target="_blank">7 triangular tilings existing around each vertex in an 3/info/order" target="_blank">order-5 heptagonal tiling vertex figure.
= 3/info/order" target="_blank">Order-3/info/7" target="_blank">7-6 triangular honeycomb
=In the geometry of hyperbolic 3-space, the 3/info/order" target="_blank">order-3/info/7" target="_blank">7-6 triangular honeycomb (or 3,3/info/7" target="_blank">7,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,3/info/7" target="_blank">7,6}. It has infinitely many 3/info/order" target="_blank">order-3/info/7" target="_blank">7 triangular tiling, {3,3/info/7" target="_blank">7}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many 3/info/order" target="_blank">order-3/info/7" target="_blank">7 triangular tilings existing around each vertex in an 3/info/order" target="_blank">order-6 heptagonal tiling, {3/info/7" target="_blank">7,6}, vertex figure.
= 3/info/order" target="_blank">Order-3/info/7" target="_blank">7-infinite triangular honeycomb
=In the geometry of hyperbolic 3-space, the 3/info/order" target="_blank">order-3/info/7" target="_blank">7-infinite triangular honeycomb (or 3,3/info/7" target="_blank">7,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,3/info/7" target="_blank">7,∞}. It has infinitely many 3/info/order" target="_blank">order-3/info/7" target="_blank">7 triangular tiling, {3,3/info/7" target="_blank">7}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many 3/info/order" target="_blank">order-3/info/7" target="_blank">7 triangular tilings existing around each vertex in an infinite-3/info/order" target="_blank">order heptagonal tiling, {3/info/7" target="_blank">7,∞}, vertex figure.
It has a second construction as a uniform honeycomb, Schläfli symbol {3,(3/info/7" target="_blank">7,∞,3/info/7" target="_blank">7)}, Coxeter diagram, = , with alternating types or colors of 3/info/order" target="_blank">order-3/info/7" target="_blank">7 triangular tiling cells. In Coxeter notation the half symmetry is [3,3/info/7" target="_blank">7,∞,1+] = [3,((3/info/7" target="_blank">7,∞,3/info/7" target="_blank">7))].
= 3/info/order" target="_blank">Order-3/info/7" target="_blank">7-3 square honeycomb
=In the geometry of hyperbolic 3-space, the 3/info/order" target="_blank">order-3/info/7" target="_blank">7-3 square honeycomb (or 4,3/info/7" target="_blank">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 3/info/order" target="_blank">order-3/info/7" target="_blank">7-3 square honeycomb is {4,3/info/7" target="_blank">7,3}, with three 3/info/order" target="_blank">order-4 heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is a heptagonal tiling, {3/info/7" target="_blank">7,3}.
= 3/info/order" target="_blank">Order-3/info/7" target="_blank">7-3 pentagonal honeycomb
=In the geometry of hyperbolic 3-space, the 3/info/order" target="_blank">order-3/info/7" target="_blank">7-3 pentagonal honeycomb (or 5,3/info/7" target="_blank">7,3 honeycomb) a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an 3/info/order" target="_blank">order-3/info/7" target="_blank">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 3/info/order" target="_blank">order-6-3 pentagonal honeycomb is {5,3/info/7" target="_blank">7,3}, with three 3/info/order" target="_blank">order-3/info/7" target="_blank">7 pentagonal tilings meeting at each edge. The vertex figure of this honeycomb is a heptagonal tiling, {3/info/7" target="_blank">7,3}.
= 3/info/order" target="_blank">Order-3/info/7" target="_blank">7-3 hexagonal honeycomb
=In the geometry of hyperbolic 3-space, the 3/info/order" target="_blank">order-3/info/7" target="_blank">7-3 hexagonal honeycomb (or 6,3/info/7" target="_blank">7,3 honeycomb) a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an 3/info/order" target="_blank">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 3/info/order" target="_blank">order-3/info/7" target="_blank">7-3 hexagonal honeycomb is {6,3/info/7" target="_blank">7,3}, with three 3/info/order" target="_blank">order-5 hexagonal tilings meeting at each edge. The vertex figure of this honeycomb is a heptagonal tiling, {3/info/7" target="_blank">7,3}.
= 3/info/order" target="_blank">Order-3/info/7" target="_blank">7-3 apeirogonal honeycomb
=In the geometry of hyperbolic 3-space, the 3/info/order" target="_blank">order-3/info/7" target="_blank">7-3 apeirogonal honeycomb (or ∞,3/info/7" target="_blank">7,3 honeycomb) a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an 3/info/order" target="_blank">order-3/info/7" target="_blank">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 {∞,3/info/7" target="_blank">7,3}, with three 3/info/order" target="_blank">order-3/info/7" target="_blank">7 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is a heptagonal tiling, {3/info/7" target="_blank">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.
= 3/info/order" target="_blank">Order-3/info/7" target="_blank">7-4 square honeycomb
=In the geometry of hyperbolic 3-space, the 3/info/order" target="_blank">order-3/info/7" target="_blank">7-4 square honeycomb (or 4,3/info/7" target="_blank">7,4 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,3/info/7" target="_blank">7,4}.
All vertices are ultra-ideal (existing beyond the ideal boundary) with four 3/info/order" target="_blank">order-5 square tilings existing around each edge and with an 3/info/order" target="_blank">order-4 heptagonal tiling vertex figure.
= 3/info/order" target="_blank">Order-3/info/7" target="_blank">7-5 pentagonal honeycomb
=In the geometry of hyperbolic 3-space, the 3/info/order" target="_blank">order-3/info/7" target="_blank">7-5 pentagonal honeycomb (or 5,3/info/7" target="_blank">7,5 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {5,3/info/7" target="_blank">7,5}.
All vertices are ultra-ideal (existing beyond the ideal boundary) with five 3/info/order" target="_blank">order-3/info/7" target="_blank">7 pentagonal tilings existing around each edge and with an 3/info/order" target="_blank">order-5 heptagonal tiling vertex figure.
= 3/info/order" target="_blank">Order-3/info/7" target="_blank">7-6 hexagonal honeycomb
=In the geometry of hyperbolic 3-space, the 3/info/order" target="_blank">order-3/info/7" target="_blank">7-6 hexagonal honeycomb (or 6,3/info/7" target="_blank">7,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,3/info/7" target="_blank">7,6}. It has six 3/info/order" target="_blank">order-3/info/7" target="_blank">7 hexagonal tilings, {6,3/info/7" target="_blank">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 3/info/order" target="_blank">order-6 heptagonal tiling vertex arrangement.
It has a second construction as a uniform honeycomb, Schläfli symbol {6,(3/info/7" target="_blank">7,3,3/info/7" target="_blank">7)}, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [6,3/info/7" target="_blank">7,6,1+] = [6,((3/info/7" target="_blank">7,3,3/info/7" target="_blank">7))].
= 3/info/order" target="_blank">Order-3/info/7" target="_blank">7-infinite apeirogonal honeycomb
=In the geometry of hyperbolic 3-space, the 3/info/order" target="_blank">order-3/info/7" target="_blank">7-infinite apeirogonal honeycomb (or ∞,3/info/7" target="_blank">7,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {∞,3/info/7" target="_blank">7,∞}. It has infinitely many 3/info/order" target="_blank">order-3/info/7" target="_blank">7 apeirogonal tiling {∞,3/info/7" target="_blank">7} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many 3/info/order" target="_blank">order-3/info/7" target="_blank">7 apeirogonal tilings existing around each vertex in an infinite-3/info/order" target="_blank">order heptagonal tiling vertex figure.
It has a second construction as a uniform honeycomb, Schläfli symbol {∞,(3/info/7" target="_blank">7,∞,3/info/7" target="_blank">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. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 (Chapters 16–17: Geometries on Three-manifolds I, II)
George Maxwell, Sphere Packings and Hyperbolic Reflection Groups, JOURNAL OF ALGEBRA 79,78-97 (1982) [1]
Hao Chen, Jean-Philippe Labbé, Lorentzian Coxeter groups and Boyd-Maxwell ball packings, (2013)[2]
Visualizing Hyperbolic Honeycombs arXiv:1511.02851 Roice Nelson, Henry Segerman (2015)
External links
Hyperbolic Catacombs Carousel: {3,3/info/7" target="_blank">7,3} honeycomb YouTube, Roice Nelson
John Baez, Visual insights: {3/info/7" target="_blank">7,3,3} Honeycomb (2014/08/01) {3/info/7" target="_blank">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. [3]
Kata Kunci Pencarian:
- Daftar bentuk matematika
- Order-7-3 triangular honeycomb
- Order-infinite-3 triangular honeycomb
- Order-7 tetrahedral honeycomb
- Order-3-7 hexagonal honeycomb
- Order-3-7 heptagonal honeycomb
- Order-6 triangular hosohedral honeycomb
- Order-7 triangular tiling
- Order-7 dodecahedral honeycomb
- Order-7 cubic honeycomb
- Tetrahedral-octahedral honeycomb