- Source: Order-3-7 heptagonal honeycomb
In the geometry of hyperbolic 3-space, the 3/info/order" target="_blank">order-3-7 heptagonal honeycomb a regular space-filling tessellation (or honeycomb) with Schläfli symbol {7,3,7}.
Geometry
All vertices are ultra-ideal (existing beyond the ideal boundary) with seven heptagonal tilings existing around each edge and with an 3/info/order" target="_blank">order-7 triangular tiling vertex figure.
Related polytopes and honeycombs
It a part of a sequence of regular polychora and honeycombs {p,3,p}:
= 3/info/order" target="_blank">Order-3-8 octagonal honeycomb
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In the geometry of hyperbolic 3-space, the 3/info/order" target="_blank">order-3-8 octagonal honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {8,3,8}. It has eight octagonal tilings, {8,3}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many octagonal tilings existing around each vertex in an 3/info/order" target="_blank">order-8 triangular tiling vertex arrangement.
It has a second construction as a uniform honeycomb, Schläfli symbol {8,(3,4,3)}, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [8,3,8,1+] = [8,((3,4,3))].
= 3/info/order" target="_blank">Order-3-infinite apeirogonal honeycomb
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In the geometry of hyperbolic 3-space, the 3/info/order" target="_blank">order-3-infinite apeirogonal honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {∞,3,∞}. It has infinitely many 3/info/order" target="_blank">order-3 apeirogonal tiling {∞,3} around each edge. All vertices are ultra-ideal (Existing beyond the ideal boundary) with infinitely many apeirogonal tilings existing around each vertex in an infinite-3/info/order" target="_blank">order triangular tiling vertex arrangement.
It has a second construction as a uniform honeycomb, Schläfli symbol {∞,(3,∞,3)}, Coxeter diagram, , with alternating types or colors of apeirogonal tiling cells.
See also
Convex uniform honeycombs in hyperbolic space
List of regular polytopes
Infinite-3/info/order" target="_blank">order dodecahedral honeycomb
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
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. [3]
Kata Kunci Pencarian:
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