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In the geometry of hyperbolic 3-space, the order-7 cubic honeycomb is a regular space-filling tessellation (or honeycomb). With Schläfli symbol {4,3,7}, it has seven cubes {4,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many cubes existing around each vertex in an order-7 triangular tiling vertex arrangement.

Images

Poincaré disk model
Hyperbolic honeycomb 4-3-7 poincare cc.png
Cell-centered		 Hyperbolic honeycomb 4-3-7 poincare.png
Order-7 cubic honeycomb cell.png
One cell at center		 Order-7 cubic honeycomb cell2.png
One cell with ideal surface

Related polytopes and honeycombs

It is one of a series of regular polytopes and honeycombs with cubic cells: {4,3,p}:

{4,3,p} polytopes
Space S3 H3
Form Finite Compact Paracompact Noncompact
Name {4,3,3} {4,3,4} {4,3,5} {4,3,6} {4,3,7} {4,3,8} ... {4,3,∞}
Image Schlegel wireframe 8-cell.png Cubic honeycomb.png H3 435 CC center.png H3 436 CC center.png Hyperbolic honeycomb 4-3-7 poincare.png Hyperbolic honeycomb 4-3-8 poincare.png Hyperbolic honeycomb 4-3-i poincare.png
Vertex
figure		 Tetrahedron.png
{3,3}		 Octahedron.png
{3,4}		 Icosahedron.png
{3,5}		 Uniform tiling 63-t2.svg
{3,6}		 Order-7 triangular tiling.svg
{3,7}		 H2-8-3-primal.svg
{3,8}		 H2 tiling 23i-4.png
{3,∞}

It is a part of a sequence of hyperbolic honeycombs with order-7 triangular tiling vertex figures, {p,3,7}.

{3,3,7} {4,3,7} {5,3,7} {6,3,7} {7,3,7} {8,3,7} {∞,3,7}
Hyperbolic honeycomb 3-3-7 poincare cc.png Hyperbolic honeycomb 4-3-7 poincare cc.png Hyperbolic honeycomb 5-3-7 poincare cc.png Hyperbolic honeycomb 6-3-7 poincare.png Hyperbolic honeycomb 7-3-7 poincare.png Hyperbolic honeycomb 8-3-7 poincare.png Hyperbolic honeycomb i-3-7 poincare.png
Order-8 cubic honeycomb
Type Regular honeycomb
Schläfli symbols {4,3,8}
{4,(3,8,3)}
Coxeter diagrams CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node h0.png = CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel branch.pngCDel label4.png
Cells {4,3} Uniform polyhedron-43-t0.png
Faces {4}
Edge figure {8}
Vertex figure {3,8}, {(3,4,3)}
H2-8-3-primal.svgH2 tiling 334-4.png
Dual {8,3,4}
Coxeter group [4,3,8]
[4,((3,4,3))]
Properties Regular

In the geometry of hyperbolic 3-space, the order-8 cubic honeycomb a regular space-filling tessellation (or honeycomb). With Schläfli symbol {4,3,8}. It has eight cubes {4,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many cubes existing around each vertex in an order-8 triangular tiling vertex arrangement.

Hyperbolic honeycomb 4-3-8 poincare cc.png
Poincaré disk model
Cell-centered		 Hyperbolic honeycomb 4-3-8 poincare.png
Poincaré disk model

It has a second construction as a uniform honeycomb, Schläfli symbol {4,(3,4,3)}, Coxeter diagram, CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel branch.pngCDel label4.png, with alternating types or colors of cubic cells.
Infinite-order cubic honeycomb

Infinite-order cubic honeycomb
Type Regular honeycomb
Schläfli symbols {4,3,∞}
{4,(3,∞,3)}
Coxeter diagrams CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node h0.png = CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel branch.pngCDel labelinfin.png
Cells {4,3} Uniform polyhedron-43-t0.png
Faces {4}
Edge figure {∞}
Vertex figure {3,∞}, {(3,∞,3)}
H2 tiling 23i-4.pngH2 tiling 33i-4.png
Dual {∞,3,4}
Coxeter group [4,3,∞]
[4,((3,∞,3))]
Properties Regular

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

Hyperbolic honeycomb 4-3-i poincare cc.png
Poincaré disk model
Cell-centered		 Hyperbolic honeycomb 4-3-i poincare.png
Poincaré disk model

It has a second construction as a uniform honeycomb, Schläfli symbol {4,(3,∞,3)}, Coxeter diagram, CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel branch.pngCDel labelinfin.png, with alternating types or colors of cubic cells.
See also

Convex uniform honeycombs in hyperbolic space
List of regular polytopes
Infinite-order hexagonal tiling 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]

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