Cubic-square tiling honeycomb

Cubic-square tiling honeycomb
Type	Paracompact uniform honeycomb Semiregular honeycomb
Schläfli symbol	{(4,4,3,4)}, {(4,3,4,4)}
Coxeter diagrams	File:CDel label4.pngFile:CDel branch 10r.pngFile:CDel 4a4b.pngFile:CDel branch.png or File:CDel label4.pngFile:CDel branch 01r.pngFile:CDel 4a4b.pngFile:CDel branch.png File:CDel label4.pngFile:CDel branch 10r.pngFile:CDel 4a4b.pngFile:CDel branch.pngFile:CDel labels.png = File:CDel node 1.pngFile:CDel splitplit1u-44.pngFile:CDel branch3u.pngFile:CDel 4a4buc-cross.pngFile:CDel branch3u 11.pngFile:CDel splitplit2u-44.pngFile:CDel node.png
Cells	{4,3} File:Uniform polyhedron-43-t0.svg {4,4} File:Uniform tiling 44-t0.svg r{4,4} File:Uniform tiling 44-t1-1.svg
Faces	square {4}
Vertex figure	File: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, File:CDel label4.pngFile:CDel branch 10r.pngFile:CDel 4a4b.pngFile:CDel 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 File:CDel node 1.pngFile:CDel splitplit1u-44.pngFile:CDel branch3u.pngFile:CDel 4a4buc-cross.pngFile:CDel branch3u 11.pngFile:CDel splitplit2u-44.pngFile:CDel node.png. Another lower symmetry constructions exists with symmetry [(4,4,(4,3)*)], index 48 and an ideal regular octahedral fundamental domain.

See also

• Convex uniform honeycombs in hyperbolic space
• List of regular polytopes

References

• Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. . (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
• Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II, III, IV, V, p212-213)
• Jeffrey R. Weeks The Shape of Space, 2nd edition (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, (2018) Chapter 13: Hyperbolic Coxeter groups