Truncated order-4 hexagonal tiling

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Truncated order-4 hexagonal tiling
Truncated order-4 hexagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 4.12.12
Schläfli symbol t{6,4}
tr{6,6}
Wythoff symbol 2 4 | 6
2 6 6 |
Coxeter diagram CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 6.pngCDel node 1.png
Symmetry group [6,4], (*642)
[6,6], (*662)
Dual Order-6 tetrakis square tiling
Properties Vertex-transitive

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.

Constructions

There are two uniform constructions of this tiling, first from [6,4] kaleidoscope, and a lower symmetry by removing the last mirror, [6,4,1+], gives [6,6], (*662).

Two uniform constructions of 4.6.4.6
Name Tetrahexagonal Truncated hexahexagonal
Image Uniform tiling 64-t01.png Uniform tiling 66-t012.png
Symmetry [6,4]
(*642)
CDel node c1.pngCDel 6.pngCDel node c2.pngCDel 4.pngCDel node c3.png
[6,6] = [6,4,1+]
(*662)
CDel node c1.pngCDel split1-66.pngCDel nodeab c2.png = CDel node c1.pngCDel 6.pngCDel node c2.pngCDel 4.pngCDel node h0.png
Symbol t{6,4} tr{6,6}
Coxeter diagram CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 6.pngCDel node 1.png

Dual tiling

Order-6 tetrakis square tiling.png Hyperbolic domains 662.png
The dual tiling, order-6 tetrakis square tiling has face configuration V4.12.12, and represents the fundamental domains of the [6,6] symmetry group.

Related polyhedra and tiling

Symmetry

File:Truncated order-4 hexagonal tiling with mirrors.png
Truncated order-4 hexagonal tiling with *662 mirror lines

The dual of the tiling represents the fundamental domains of (*662) orbifold symmetry. From [6,6] (*662) symmetry, there are 15 small index subgroup (12 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 subgroup index-8 group, [1+,6,1+,6,1+] (3333) is the commutator subgroup of [6,6].

Larger subgroup constructed as [6,6*], removing the gyration points of (6*3), index 12 becomes (*333333).

The symmetry can be doubled to 642 symmetry by adding a mirror to bisect the fundamental domain.

References

  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
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See also

External links