Order-7 cubic honeycomb – Wikipedia

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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.

Related polytopes and honeycombs[edit]

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[edit]

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.

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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[edit]

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.

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[edit]

References[edit]

  • 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[edit]

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