Order-8 triangular tiling

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Order-8 triangular tiling
Order-8 triangular tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex figure 38
Schläfli symbol {3,8}
(3,4,3)
Wythoff symbol 8 | 3 2
4 | 3 3
Coxeter diagram CDel node.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.png
CDel label4.pngCDel branch.pngCDel split2.pngCDel node 1.png
Symmetry group [8,3], (*832)
[(4,3,3)], (*433)
[(4,4,4)], (*444)
Dual Octagonal tiling
Properties Vertex-transitive, edge-transitive, face-transitive

In geometry, the order-8 triangular tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {3,8}, having eight regular triangles around each vertex.

Uniform colorings[edit]

The half symmetry [1+,8,3] = [(4,3,3)] can be shown with alternating two colors of triangles:

H2 tiling 334-4.png

Symmetry[edit]

Octagonal tiling with *444 mirror lines, CDel node c1.pngCDel split1-44.pngCDel branch c3-2.pngCDel label4.png.

From [(4,4,4)] symmetry, there are 15 small index subgroups (7 unique) by mirror removal and alternation operators. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. Adding 3 bisecting mirrors across each fundamental domains creates 832 symmetry. The subgroup index-8 group, [(1+,4,1+,4,1+,4)] (222222) is the commutator subgroup of [(4,4,4)].

A larger subgroup is constructed [(4,4,4*)], index 8, as (2*2222) with gyration points removed, becomes (*22222222).

The symmetry can be doubled to 842 symmetry by adding a bisecting mirror across the fundamental domains. The symmetry can be extended by 6, as 832 symmetry, by 3 bisecting mirrors per domain.

Small index subgroups of [(4,4,4)] (*444)
Index 1 2 4
Diagram 444 symmetry mirrors.png 444 symmetry a00.png 444 symmetry 0a0.png 444 symmetry 00a.png 444 symmetry ab0.png 444 symmetry xxx.png
Coxeter [(4,4,4)]
CDel node c1.pngCDel split1-44.pngCDel branch c3-2.pngCDel label4.png
[(1+,4,4,4)]
CDel labelh.pngCDel node.pngCDel split1-44.pngCDel branch c3-2.pngCDel label4.png = CDel label4.pngCDel branch c3-2.pngCDel 2a2b-cross.pngCDel branch c3-2.pngCDel label4.png
[(4,1+,4,4)]
CDel node c1.pngCDel split1-44.pngCDel branch h0c2.pngCDel label4.png = CDel label4.pngCDel branch c1-2.pngCDel 2a2b-cross.pngCDel branch c1-2.pngCDel label4.png
[(4,4,1+,4)]
CDel node c1.pngCDel split1-44.pngCDel branch c3h0.pngCDel label4.png = CDel label4.pngCDel branch c1-3.pngCDel 2a2b-cross.pngCDel branch c1-3.pngCDel label4.png
[(1+,4,1+,4,4)]
CDel labelh.pngCDel node.pngCDel split1-44.pngCDel branch h0c2.pngCDel label4.png
[(4+,4+,4)]
CDel node h4.pngCDel split1-44.pngCDel branch h2h2.pngCDel label4.png
Orbifold *444 *4242 2*222 222×
Diagram 444 symmetry 0bb.png 444 symmetry b0b.png 444 symmetry bb0.png 444 symmetry 0b0.png 444 symmetry a0b.png
Coxeter [(4,4+,4)]
CDel node c1.pngCDel split1-44.pngCDel branch h2h2.pngCDel label4.png
[(4,4,4+)]
CDel node h2.pngCDel split1-44.pngCDel branch c3h2.pngCDel label4.png
[(4+,4,4)]
CDel node h2.pngCDel split1-44.pngCDel branch h2c2.pngCDel label4.png
[(4,1+,4,1+,4)]
CDel node c1.pngCDel split1-44.pngCDel branch h0h0.pngCDel label4.png
[(1+,4,4,1+,4)]
CDel labelh.pngCDel node.pngCDel split1-44.pngCDel branch c3h2.pngCDel label4.png = CDel label4.pngCDel branch c3h2.pngCDel 2a2b-cross.pngCDel branch c3h2.pngCDel label4.png
Orbifold 4*22 2*222
Direct subgroups
Index 2 4 8
Diagram 444 symmetry aaa.png 444 symmetry abb.png 444 symmetry bab.png 444 symmetry bba.png 444 symmetry abc.png
Coxeter [(4,4,4)]+
CDel node h2.pngCDel split1-44.pngCDel branch h2h2.pngCDel label4.png
[(4,4+,4)]+
CDel labelh.pngCDel node.pngCDel split1-44.pngCDel branch h2h2.pngCDel label4.png = CDel label4.pngCDel branch h2h2.pngCDel 2xa2xb-cross.pngCDel branch h2h2.pngCDel label4.png
[(4,4,4+)]+
CDel node h2.pngCDel split1-44.pngCDel branch h0h2.pngCDel label4.png = CDel label4.pngCDel branch h2h2.pngCDel 2xa2xb-cross.pngCDel branch h2h2.pngCDel label4.png
[(4+,4,4)]+
CDel node h2.pngCDel split1-44.pngCDel branch h2h0.pngCDel label4.png = CDel label4.pngCDel branch h2h2.pngCDel 2xa2xb-cross.pngCDel branch h2h2.pngCDel label4.png
[(4,1+,4,1+,4)]+
CDel labelh.pngCDel node.pngCDel split1-44.pngCDel branch h0h0.pngCDel label4.png = CDel node h4.pngCDel split1-44.pngCDel branch h4h4.pngCDel label4.png
Orbifold 444 4242 222222
Radical subgroups
Index 8 16
Diagram 444 symmetry 0zz.png 444 symmetry z0z.png 444 symmetry zz0.png 444 symmetry azz.png 444 symmetry zaz.png 444 symmetry zza.png
Coxeter [(4,4*,4)] [(4,4,4*)] [(4*,4,4)] [(4,4*,4)]+ [(4,4,4*)]+ [(4*,4,4)]+
Orbifold *22222222 22222222

Related polyhedra and tilings[edit]

The {3,3,8} honeycomb has {3,8} vertex figures.

From a Wythoff construction there are ten hyperbolic uniform tilings that can be based from the regular octagonal and order-8 triangular tilings.

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 10 forms.

It can also be generated from the (4 3 3) hyperbolic tilings:

See also[edit]

References[edit]

External links[edit]