Alternated hexagonal tiling honeycomb

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Alternated hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Semiregular honeycomb
Schläfli symbols h{6,3,3}
s{3,6,3}
2s{6,3,6}
2s{6,3[3]}
s{3[3,3]}
Coxeter diagrams CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel branch 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node.png
CDel branch hh.pngCDel split2.pngCDel node h.pngCDel 6.pngCDel node.pngCDel node h0.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node.png
CDel branch hh.pngCDel splitcross.pngCDel branch hh.pngCDel branch hh.pngCDel split2.pngCDel node h.pngCDel 6.pngCDel node h0.pngCDel node h0.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node h0.png
Cells {3,3} Uniform polyhedron-33-t0.png
{3[3]} Uniform tiling 333-t0.png
Faces triangle {3}
Vertex figure Uniform polyhedron-33-t01.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
truncated tetrahedron
Coxeter groups , [3,3[3]]
1/2 , [6,3,3]
1/2 , [3,6,3]
1/2 , [6,3,6]
1/2 , [6,3[3]]
1/2 , [3[3,3]]
Properties Vertex-transitive, edge-transitive, quasiregular

In three-dimensional hyperbolic geometry, the alternated hexagonal tiling honeycomb, h{6,3,3}, CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png or CDel branch 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png, is a semiregular tessellation with tetrahedron and triangular tiling cells arranged in an octahedron vertex figure. It is named after its construction, as an alteration of a hexagonal tiling honeycomb.

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.

Symmetry constructions[]

Subgroup relations

It has five alternated constructions from reflectional Coxeter groups all with four mirrors and only the first being regular: CDel node c1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png [6,3,3], CDel node c1.pngCDel 3.pngCDel node c1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png [3,6,3], CDel node.pngCDel 6.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 6.pngCDel node.png [6,3,6], CDel branch c1.pngCDel split2.pngCDel node c1.pngCDel 6.pngCDel node.png [6,3[3]] and [3[3,3]] CDel branch c1.pngCDel splitcross.pngCDel branch c1.png, having 1, 4, 6, 12 and 24 times larger fundamental domains respectively. In Coxeter notation subgroup markups, they are related as: [6,(3,3)*] (remove 3 mirrors, index 24 subgroup); [3,6,3*] or [3*,6,3] (remove 2 mirrors, index 6 subgroup); [1+,6,3,6,1+] (remove two orthogonal mirrors, index 4 subgroup); all of these are isomorphic to [3[3,3]]. The ringed Coxeter diagrams are CDel node h.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png, CDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png, CDel node.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node.png, CDel branch hh.pngCDel split2.pngCDel node h.pngCDel 6.pngCDel node.png and CDel branch hh.pngCDel splitcross.pngCDel branch hh.png, representing different types (colors) of hexagonal tilings in the Wythoff construction.

Related honeycombs[]

The alternated hexagonal tiling honeycomb has 3 related forms: the cantic hexagonal tiling honeycomb, CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png; the runcic hexagonal tiling honeycomb, CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png; and the runcicantic hexagonal tiling honeycomb, CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png.

Cantic hexagonal tiling honeycomb[]

Cantic hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols h2{6,3,3}
Coxeter diagrams CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.png
Cells r{3,3} Uniform polyhedron-33-t1.png
t{3,3} Uniform polyhedron-33-t01.png
h2{6,3} Uniform tiling 333-t01.png
Faces triangle {3}
hexagon {6}
Vertex figure Cantic hexagonal tiling honeycomb verf.png
wedge
Coxeter groups , [3,3[3]]
Properties Vertex-transitive

The cantic hexagonal tiling honeycomb, h2{6,3,3}, CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png or CDel branch 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.png, is composed of octahedron, truncated tetrahedron, and trihexagonal tiling facets, with a wedge vertex figure.

Runcic hexagonal tiling honeycomb[]

Runcic hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols h3{6,3,3}
Coxeter diagrams CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel branch 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.png
Cells {3,3} Uniform polyhedron-33-t0.png
{}x{3} Triangular prism.png
rr{3,3} Uniform polyhedron-33-t02.png
{3[3]} Uniform tiling 333-t0.png
Faces triangle {3}
square {4}
hexagon {6}
Vertex figure Runcic hexagonal tiling honeycomb verf.png
triangular cupola
Coxeter groups , [3,3[3]]
Properties Vertex-transitive

The runcic hexagonal tiling honeycomb, h3{6,3,3}, CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png or CDel branch 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.png, has tetrahedron, triangular prism, cuboctahedron, and triangular tiling facets, with a triangular cupola vertex figure.

Runcicantic hexagonal tiling honeycomb[]

Runcicantic hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols h2,3{6,3,3}
Coxeter diagrams CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Cells t{3,3} Uniform polyhedron-33-t01.png
{}x{3} Triangular prism.png
tr{3,3} Uniform polyhedron-33-t012.png
h2{6,3} Uniform tiling 333-t01.png
Faces triangle {3}
square {4}
hexagon {6}
Vertex figure Runcicantic hexagonal tiling honeycomb verf.png
rectangular pyramid
Coxeter groups , [3,3[3]]
Properties Vertex-transitive

The runcicantic hexagonal tiling honeycomb, h2,3{6,3,3}, CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png or CDel branch 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.png, has truncated tetrahedron, triangular prism, truncated octahedron, and trihexagonal tiling facets, with a rectangular pyramid vertex figure.

See also[]

  • Convex uniform honeycombs in hyperbolic space
  • Regular tessellations of hyperbolic 3-space
  • Paracompact uniform honeycombs
  • Semiregular honeycomb
  • Hexagonal tiling honeycomb

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)
  • The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
  • Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 (Chapters 16–17: Geometries on Three-manifolds I,II)
  • N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, The size of a hyperbolic Coxeter simplex, Transformation Groups (1999), Volume 4, Issue 4, pp 329–353 [1] [2]
  • N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, Commensurability classes of hyperbolic Coxeter groups, (2002) H3: p130. [3]
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