• Source: Hexagonal tiling-triangular tiling honeycomb

      • In the geometry of hyperbolic 3-space, the hexagonal tiling" target="_blank">tiling-triangular tiling" target="_blank">tiling honeycomb is a paracompact uniform honeycomb, constructed from triangular tiling" target="_blank">tiling, hexagonal tiling" target="_blank">tiling, and trihexagonal tiling" target="_blank">tiling cells, in a rhombitrihexagonal tiling" target="_blank">tiling vertex figure. It has a single-ring Coxeter diagram, , 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" target="_blank">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.


      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 .


      Related honeycombs


      The cyclotruncated octahedral-hexagonal tiling" target="_blank">tiling honeycomb, has a higher symmetry construction as the order-4 hexagonal tiling" target="_blank">tiling.


      See also


      Uniform honeycombs in hyperbolic space
      List of regular polytopes


      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

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