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 Uniform polyhedron-33-t2.png
tetrahedron
Uniform tiling 333-t0.png
Triangular tiling
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-uniform, edge-transitive, quasiregular

In 3-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, with tetrahedron and triangular tiling cells, in an octahedron vertex figure. It is named by 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[edit]

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[edit]

It has 3 related form cantic hexagonal tiling honeycomb, CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png, runcic hexagonal tiling honeycomb, CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png, 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[edit]

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 Uniform polyhedron-33-t1.png
octahedron
Uniform polyhedron-33-t12.png
Truncated tetrahedron
Uniform tiling 333-t01.png
trihexagonal tiling
Faces Triangle {3}
Hexagon {6}
Vertex figure Cantic hexagonal tiling honeycomb verf.png
Coxeter groups , [3,3[3]]
Properties Vertex-uniform

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.

Runcic hexagonal tiling honeycomb[edit]

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 Uniform polyhedron-33-t0.png
cube
Triangular prism.png
triangular prism
Uniform polyhedron-33-t02.png
cuboctahedron
Uniform tiling 333-t0.png
Triangular tiling
Faces Triangle {3}
Hexagon {6}
Vertex figure Runcic hexagonal tiling honeycomb verf.png
Coxeter groups , [3,3[3]]
Properties Vertex-uniform

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.

Runcicantic hexagonal tiling honeycomb[edit]

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 Uniform polyhedron-33-t01.png
Truncated cube
Triangular prism.png
triangular prism
Uniform polyhedron-33-t012.png
Truncated octahedron
Uniform tiling 333-t01.png
trihexagonal tiling
Faces Triangle {3}
Square {4}
Hexagon {6}
Vertex figure Runcicantic hexagonal tiling honeycomb verf.png
Coxeter groups , [3,3[3]]
Properties Vertex-uniform

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.

See also[edit]

References[edit]