“Order-4-5 square honeycomb”的意思、由来-开放百科全书

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Order-4-5 square honeycomb

释义

Images
Related polytopes and honeycombs
Order-4-6 square honeycomb Order-4-infinite square honeycomb
See also
References
External links

Order-4-5 square honeycomb
Type	Regular honeycomb
Schläfli symbols	{4,4,5}
Coxeter diagrams	node_1\|4\|node\|4\|node\|5\|node}}
Cells	{4,4}
Faces	{4}
Edge figure	{5}
Vertex figure	{4,5}
Dual	{5,4,4}
Coxeter group	[4,4,5]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-4-5 square honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,4,5}. It has five square tiling {4,4} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many square tiling existing around each vertex in an order-5 square tiling vertex arrangement.

Images

Poincaré disk model	Ideal surface

Related polytopes and honeycombs

It a part of a sequence of regular polychora and honeycombs with square tiling cells: {4,4,p}

Order-4-6 square honeycomb

Order-4-6 square honeycomb
Type	Regular honeycomb
Schläfli symbols	{4,4,6} {4,(4,3,4)}
Coxeter diagrams	node_1\|4\|node\|4\|node\|6\|node}} {{CDD\|node_1\|4\|node\|4\|node\|6\|node_h0}} = {{CDD\|node_1\|4\|node\|split1-44\|branch}}
Cells	{4,4}
Faces	{4}
Edge figure	{6}
Vertex figure	{4,6} {(4,3,4)}
Dual	{6,4,4}
Coxeter group	[4,4,6] [4,((4,3,4))]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-4-6 square honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,4,6}. It has six square tiling, {4,4}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many square tiling existing around each vertex in an order-6 square tiling vertex arrangement.

Poincaré disk model	Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {4,(4,3,4)}, Coxeter diagram, {{CDD|node_1|4|node|split1-44|branch}}, with alternating types or colors of square tiling cells. In Coxeter notation the half symmetry is [4,4,6,1⁺] = [4,((4,3,4))].

Order-4-infinite square honeycomb

Order-4-infinite square honeycomb
Type	Regular honeycomb
Schläfli symbols	{4,4,∞} {4,(4,∞,4)}
Coxeter diagrams	node_1\|4\|node\|4\|node\|infin\|node}} {{CDD\|node_1\|4\|node\|4\|node\|infin\|node_h0}} = {{CDD\|node_1\|4\|node\|split1-44\|branch\|labelinfin}}
Cells	{4,4}
Faces	{4}
Edge figure	{∞}
Vertex figure	{4,∞} {(4,∞,4)}
Dual	{∞,4,4}
Coxeter group	[∞,4,3] [4,((4,∞,4))]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-4-infinite square honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,4,∞}. It has infinitely many square tiling, {4,4}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many square tiling existing around each vertex in an infinite-order square tiling vertex arrangement.

Poincaré disk model	Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {4,(4,∞,4)}, Coxeter diagram, {{CDD|node_1|4|node|4|node|infin|node_h0}} = {{CDD|node_1|4|node|split1-44|branch|labelinfin}}, with alternating types or colors of square tiling cells. In Coxeter notation the half symmetry is [4,4,∞,1⁺] = [4,((4,∞,4))].

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|99035678}}, {{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)
Hao Chen, Jean-Philippe Labbé, Lorentzian Coxeter groups and Boyd-Maxwell ball packings, (2013)[https://arxiv.org/abs/1310.8608]
[https://arxiv.org/abs/1511.02851 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.

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