Английская Википедия:Hexagonal tiling-triangular tiling honeycomb

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Hexagonal tiling-triangular tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol {(3,6,3,6)} or {(6,3,6,3)}
Coxeter diagrams Шаблон:CDD or Шаблон:CDD or Шаблон:CDD or Шаблон:CDD
Файл:CDel K6 636 10.png
Cells {3,6} Файл:Uniform tiling 63-t2.png
{6,3} Файл:Uniform tiling 63-t0.png
r{6,3} Файл:Uniform tiling 63-t1.png
Faces triangular {3}
square {4}
hexagon {6}
Vertex figure Файл:Uniform tiling 63-t02.png
rhombitrihexagonal tiling
Coxeter group [(6,3)[2]]
Properties Vertex-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, Шаблон:CDD, and is named by its two regular cells.

Шаблон: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 Файл:CDel K6 636 10.png.

Related honeycombs

The cyclotruncated octahedral-hexagonal tiling honeycomb, Шаблон:CDD has a higher symmetry construction as the order-4 hexagonal tiling.

See also

References

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