Cubic-square tiling honeycomb

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Cubic-square tiling honeycomb
Type Paracompact uniform honeycomb
Semiregular honeycomb
Schläfli symbol {(4,4,3,4)}, {(4,3,4,4)}
Coxeter diagrams CDel label4.pngCDel branch 10r.pngCDel 4a4b.pngCDel branch.png or CDel label4.pngCDel branch 01r.pngCDel 4a4b.pngCDel branch.png
CDel node 1.pngCDel splitplit1u-44.pngCDel branch3u.pngCDel 4a4buc-cross.pngCDel branch3u 11.pngCDel splitplit2u-44.pngCDel node.png
Cells {4,3} Uniform polyhedron-43-t0.png
{4,4} Uniform tiling 44-t0.png
r{4,4} Uniform tiling 44-t1.png
Faces square {4}
Vertex figure Uniform polyhedron-43-t02.png
Rhombicuboctahedron
Coxeter group [(4,4,4,3)]
Properties Vertex-transitive, edge-transitive

In the geometry of hyperbolic 3-space, the cubic-square tiling honeycomb is a paracompact uniform honeycomb, constructed from cube and square tiling cells, in a rhombicuboctahedron vertex figure. It has a single-ring Coxeter diagram, CDel label4.pngCDel branch 10r.pngCDel 4a4b.pngCDel branch.png, and is named by its two regular cells.

A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.

Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.

It represents a semiregular honeycomb as defined by all regular cells, although from the Wythoff construction, rectified square tiling r{4,4}, becomes the regular square tiling {4,4}.

Symmetry

A lower symmetry form, index 6, of this honeycomb can be constructed with [(4,4,4,3*)] symmetry, represented by a trigonal trapezohedron fundamental domain, and Coxeter diagram CDel node 1.pngCDel splitplit1u-44.pngCDel branch3u.pngCDel 4a4buc-cross.pngCDel branch3u 11.pngCDel splitplit2u-44.pngCDel node.png.

See also

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)
  • Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 ISBN 0-486-40919-8 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II,III,IV,V, p212-213)
  • Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 (Chapter 16-17: Geometries on Three-manifolds I,II)
  • Norman Johnson Uniform Polytopes, Manuscript
    • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
    • N.W. Johnson: Geometries and Transformations, (2015) Chapter 13: Hyperbolic Coxeter groups