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Hexagonal tiling-triangular tiling honeycomb

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!bgcolor=#e7dcc3 colspan=2|Hexagonal tiling-triangular tiling honeycomb

bgcolor=#e7dcc3|TypeParacompact uniform honeycomb
bgcolor=#e7dcc3|Schläfli symbol{(3,6,3,6)} or {(6,3,6,3)}
bgcolor=#e7dcc3|Coxeter diagrams{{CDD|label6|branch_10r|3ab|branch|label6}} or {{CDD|label6|branch_01r|3ab|branch|label6}} or {{CDD|label6|branch|3ab|branch_10l|label6}} or {{CDD|label6|branch|3ab|branch_01l|label6}}
File:CDel K6 636 10.png
bgcolor=#e7dcc3|Cells{3,6} 40px
{6,3} 40px
r{6,3} 40px
bgcolor=#e7dcc3|Facestriangular {3}
square {4}
hexagon {6}
bgcolor=#e7dcc3|Vertex figure80px
rhombitrihexagonal tiling
bgcolor=#e7dcc3|Coxeter group[(6,3)[2]]
bgcolor=#e7dcc3|PropertiesVertex-uniform, edge-uniform

In the geometry of hyperbolic 3-space, the hexagonal tiling-triangular tiling honeycomb is a paracompact uniform honeycomb, constructed from triangular tiling, hexagonal tiling, and trihexagonal tiling cells, in a rhombitrihexagonal tiling vertex figure. It has a single-ring Coxeter diagram, {{CDD|label6|branch_10r|3ab|branch|label6}}, and is named by its two regular cells.

{{Honeycomb}}

Symmetry

A lower symmetry form, index 6, of this honeycomb can be constructed with [(6,3,6,3*)] symmetry, represented by a cube fundamental domain, and an octahedral Coxeter diagram File:CDel K6 636 10.png.

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Related honeycombs

The cyclotruncated octahedral-hexagonal tiling honeycomb, {{CDD|label6|branch_10r|3ab|branch_10l|label6}} has a higher symmetry construction as the order-4 hexagonal tiling.

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, (2018) Chapter 13: Hyperbolic Coxeter groups

Category:Hexagonal tilings

Category:3-honeycombs