Truncated order-8 hexagonal tiling

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Truncated order-8 hexagonal tiling
H2 tiling 268-6.png
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 8.12.12
Schläfli symbol t{6,8}
Wythoff symbol 6
Coxeter diagram CDel node.pngCDel 8.pngCDel node 1.pngCDel 6.pngCDel node 1.png
Symmetry group [8,6], (*862)
Dual Order-6 octakis octagonal tiling
Properties Vertex-transitive

In geometry, the truncated order-8 hexagonal tiling is a semiregular tiling of the hyperbolic plane. It has Schläfli symbol of t{6,8}.

Contents

Uniform colorings

This tiling can also be constructed from *664 symmetry, as t{(6,6,4)}.

H2 tiling 466-7.png

From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-6 octagonal tiling.

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 7 forms with full [8,6] symmetry, and 7 with subsymmetry.

Uniform octagonal/hexagonal tilings
Symmetry: [8,6], (*862)
CDel node 1.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node.pngCDel node 1.pngCDel 8.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel node.pngCDel 8.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel node.pngCDel 8.pngCDel node 1.pngCDel 6.pngCDel node 1.pngCDel node.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node 1.pngCDel node 1.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node 1.pngCDel node 1.pngCDel 8.pngCDel node 1.pngCDel 6.pngCDel node 1.png
H2 tiling 268-1.png H2 tiling 268-3.png H2 tiling 268-2.png H2 tiling 268-6.png H2 tiling 268-4.png H2 tiling 268-5.png H2 tiling 268-7.png
{8,6} t{8,6}
r{8,6} 2t{8,6}=t{6,8} 2r{8,6}={6,8} rr{8,6} tr{8,6}
Uniform duals
CDel node f1.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node.pngCDel node f1.pngCDel 8.pngCDel node f1.pngCDel 6.pngCDel node.pngCDel node.pngCDel 8.pngCDel node f1.pngCDel 6.pngCDel node.pngCDel node.pngCDel 8.pngCDel node f1.pngCDel 6.pngCDel node f1.pngCDel node.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node f1.pngCDel node f1.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node f1.pngCDel node f1.pngCDel 8.pngCDel node f1.pngCDel 6.pngCDel node f1.png
H2chess 268b.png H2chess 268f.png H2chess 268a.png H2chess 268e.png H2chess 268c.png H2chess 268d.png H2checkers 268.png
V86V6.16.16V(6.8)2V8.12.12V68V4.6.4.8V4.12.16
Alternations
[1+,8,6]
(*466)
[8+,6]
(8*3)
[8,1+,6]
(*4232)
[8,6+]
(6*4)
[8,6,1+]
(*883)
[(8,6,2+)]
(2*43)
[8,6]+
(862)
CDel node h1.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node.pngCDel node h.pngCDel 8.pngCDel node h.pngCDel 6.pngCDel node.pngCDel node.pngCDel 8.pngCDel node h1.pngCDel 6.pngCDel node.pngCDel node.pngCDel 8.pngCDel node h.pngCDel 6.pngCDel node h.pngCDel node.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node h1.pngCDel node h.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node h.pngCDel node h.pngCDel 8.pngCDel node h.pngCDel 6.pngCDel node h.png
H2 tiling 466-1.png H2 tiling 388-1.png Uniform tiling 86-snub.png
h{8,6}s{8,6}hr{8,6}s{6,8}h{6,8}hrr{8,6} sr{8,6}
Alternation duals
CDel node fh.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node.pngCDel node fh.pngCDel 8.pngCDel node fh.pngCDel 6.pngCDel node.pngCDel node.pngCDel 8.pngCDel node fh.pngCDel 6.pngCDel node.pngCDel node.pngCDel 8.pngCDel node fh.pngCDel 6.pngCDel node fh.pngCDel node.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node fh.pngCDel node fh.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node fh.pngCDel node fh.pngCDel 8.pngCDel node fh.pngCDel 6.pngCDel node fh.png
H2chess 466b.png
V(4.6)6V3.3.8.3.8.3V(3.4.4.4)2V3.4.3.4.3.6V(3.8)8V3.45V3.3.6.3.8

Symmetry

The dual of the tiling represents the fundamental domains of (*664) orbifold symmetry. From [(6,6,4)] (*664) symmetry, there are 15 small index subgroup (11 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. The symmetry can be doubled to 862 symmetry by adding a bisecting mirror across the fundamental domains. The subgroup index-8 group, [(1+,6,1+,6,1+,4)] (332332) is the commutator subgroup of [(6,6,4)].

A large subgroup is constructed [(6,6,4*)], index 8, as (4*33) with gyration points removed, becomes (*38), and another large subgroup is constructed [(6,6*,4)], index 12, as (6*32) with gyration points removed, becomes (*(32)6).

Small index subgroups of [(6,6,4)] (*664)
Fundamental
domains
H2checkers 466.png H2chess 466e.png
H2chess 466b.png
H2chess 466f.png
H2chess 466c.png
H2chess 466d.png
H2chess 466a.png
H2chess 466b.png
H2chess 466c.png
H2chess 466a.png
Subgroup index 124
Coxeter [(6,6,4)]
CDel node c1.pngCDel split1-66.pngCDel branch c3-2.pngCDel label4.png
[(1+,6,6,4)]
CDel node c1.pngCDel split1-66.pngCDel branch h0c2.pngCDel label4.png
[(6,6,1+,4)]
CDel node c1.pngCDel split1-66.pngCDel branch c3h0.pngCDel label4.png
[(6,1+,6,4)]
CDel labelh.pngCDel node.pngCDel split1-66.pngCDel branch c3-2.pngCDel label4.png
[(1+,6,6,1+,4)]
CDel node c1.pngCDel split1-66.pngCDel branch h0h0.pngCDel label4.png
[(6+,6+,4)]
CDel node h4.pngCDel split1-66.pngCDel branch h2h2.pngCDel label4.png
Orbifold *664 *6362 *4343 2*3333332×
Coxeter[(6,6+,4)]
CDel node h2.pngCDel split1-66.pngCDel branch c3h2.pngCDel label4.png
[(6+,6,4)]
CDel node h2.pngCDel split1-66.pngCDel branch h2c2.pngCDel label4.png
[(6,6,4+)]
CDel node c1.pngCDel split1-66.pngCDel branch h2h2.pngCDel label4.png
[(6,1+,6,1+,4)]
CDel labelh.pngCDel node.pngCDel split1-66.pngCDel branch c3h0.pngCDel label4.png
[(1+,6,1+,6,4)]
CDel labelh.pngCDel node.pngCDel split1-66.pngCDel branch h0c2.pngCDel label4.png
Orbifold6*324*333*3232
Direct subgroups
Subgroup index248
Coxeter[(6,6,4)]+
CDel node h2.pngCDel split1-66.pngCDel branch h2h2.pngCDel label4.png
[(1+,6,6+,4)]
CDel node h2.pngCDel split1-66.pngCDel branch h0h2.pngCDel label4.png
[(6+,6,1+,4)]
CDel node h2.pngCDel split1-66.pngCDel branch h2h0.pngCDel label4.png
[(6,1+,6,4+)]
CDel labelh.pngCDel node.pngCDel split1-66.pngCDel branch h2h2.pngCDel label4.png
[(6+,6+,4+)] = [(1+,6,1+,6,1+,4)]
CDel node h4.pngCDel split1-66.pngCDel branch h4h4.pngCDel label4.png = CDel labelh.pngCDel node.pngCDel split1-66.pngCDel branch h0h0.pngCDel label4.png
Orbifold66463624343332332

See also

Related Research Articles

Truncated trioctagonal tiling

In geometry, the truncated trioctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one hexagon, and one hexadecagon (16-sides) on each vertex. It has Schläfli symbol of tr{8,3}.

Truncated tetrahexagonal tiling

In geometry, the truncated tetrahexagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one dodecagon on each vertex. It has Schläfli symbol of tr{6,4}.

Rhombitetrahexagonal tiling

In geometry, the rhombitetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{6,4}. It can be seen as constructed as a rectified tetrahexagonal tiling, r{6,4}, as well as an expanded order-4 hexagonal tiling or expanded order-6 square tiling.

Truncated order-4 hexagonal tiling

In geometry, the truncated order-4 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{6,4}. A secondary construction tr{6,6} is called a truncated hexahexagonal tiling with two colors of dodecagons.

Truncated order-6 square tiling

In geometry, the truncated order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{4,6}.

Truncated tetrapentagonal tiling

In geometry, the truncated tetrapentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1,2{4,5} or tr{4,5}.

Order-8 triangular tiling

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.

Truncated order-6 hexagonal tiling

In geometry, the truncated order-6 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{6,6}. It can also be identically constructed as a cantic order-6 square tiling, h2{4,6}

Truncated tetraheptagonal tiling

In geometry, the truncated tetraheptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of tr{4,7}.

Truncated order-4 octagonal tiling

In geometry, the truncated order-4 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{8,4}. A secondary construction t0,1,2{8,8} is called a truncated octaoctagonal tiling with two colors of hexakaidecagons.

Truncated tetraoctagonal tiling Semiregular tiling in geometry

In geometry, the truncated tetraoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one hexakaidecagon on each vertex. It has Schläfli symbol of tr{8,4}.

Truncated order-8 octagonal tiling

In geometry, the truncated order-8 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{8,8}.

Order-3 apeirogonal tiling

In geometry, the order-3 apeirogonal tiling is a regular tiling of the hyperbolic plane. It is represented by the Schläfli symbol {∞,3}, having three regular apeirogons around each vertex. Each apeirogon is inscribed in a horocycle.

Truncated triapeirogonal tiling

In geometry, the truncated triapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of tr{∞,3}.

Truncated order-4 apeirogonal tiling

In geometry, the truncated order-4 apeirogonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{∞,4}.

Truncated infinite-order square tiling

In geometry, the truncated infinite-order square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{4,∞}.

Truncated tetraapeirogonal tiling

In geometry, the truncated tetraapeirogonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one apeirogon on each vertex. It has Schläfli symbol of tr{∞,4}.

Truncated hexaoctagonal tiling

In geometry, the truncated hexaoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one dodecagon, and one hexakaidecagon on each vertex. It has Schläfli symbol of tr{8,6}.

Truncated order-6 octagonal tiling

In geometry, the truncated order-6 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{8,6}.

Hexaoctagonal tiling

In geometry, the hexaoctagonal tiling is a uniform tiling of the hyperbolic plane.

References