Cubic-square tiling honeycomb explained

bgcolor=#e7dcc3 colspan=2	Cubic-square tiling honeycomb
Type	Paracompact uniform honeycomb Semiregular honeycomb
Schläfli symbol	,
Coxeter diagrams	or =
Cells
Faces
Vertex figure	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,, and is named by its two regular cells.

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

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 . 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