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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} or t { 6 6 } {\displaystyle t{\begin{Bmatrix}6\\6\end{Bmatrix}}}
Wythoff symbol 2 4 | 6
2 6 6 |
Coxeter diagram
or
Symmetry group , (*642)
, (*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 kaleidoscope, and a lower symmetry by removing the last mirror, , gives , (*662).

Two uniform constructions of 4.6.4.6
Name Tetrahexagonal Truncated hexahexagonal
Image
Symmetry
(*642)
 =
(*662)
=
Symbol t{6,4} tr{6,6}
Coxeter diagram

Dual tiling

The dual tiling, order-6 tetrakis square tiling has face configuration V4.12.12, and represents the fundamental domains of the symmetry group.

Related polyhedra and tiling

*n42 symmetry mutation of truncated tilings: 4.2n.2n
Symmetry
*n42
 Spherical Euclidean Compact hyperbolic Paracomp.
*242
 *342
 *442
 *542
 *642
 *742
 *842
... 		*∞42
Truncated
figures
Config. 4.4.4 4.6.6 4.8.8 4.10.10 4.12.12 4.14.14 4.16.16 4.∞.∞
n-kis
figures
Config. V4.4.4 V4.6.6 V4.8.8 V4.10.10 V4.12.12 V4.14.14 V4.16.16 V4.∞.∞
Uniform tetrahexagonal tilings
Symmetry: , (*642)
(with (*662), (*443) , (*3222) index 2 subsymmetries)
(And (*3232) index 4 subsymmetry)

=

=
=
=
=
=

=

=

=
=
=


=
{6,4} t{6,4} r{6,4} t{4,6} {4,6} rr{6,4} tr{6,4}
Uniform duals
V6 V4.12.12 V(4.6) V6.8.8 V4 V4.4.4.6 V4.8.12
Alternations

(*443)
(6*2)
(*3222)
(4*3)
(*662)
(2*32)
(642)

=
=
=
=
=
=
h{6,4} s{6,4} hr{6,4} s{4,6} h{4,6} hrr{6,4} sr{6,4}
Uniform hexahexagonal tilings
Symmetry: , (*662)
=
= 		=
= 		=
= 		=
= 		=
= 		=
= 		=
=
{6,6}
= h{4,6} 		t{6,6}
= h2{4,6} 		r{6,6}
{6,4} 		t{6,6}
= h2{4,6} 		{6,6}
= h{4,6} 		rr{6,6}
r{6,4} 		tr{6,6}
t{6,4}
Uniform duals
V6 V6.12.12 V6.6.6.6 V6.12.12 V6 V4.6.4.6 V4.12.12
Alternations

(*663)
(6*3)
(*3232)
(6*3)
(*663)
(2*33)
(662)
= = =
h{6,6} s{6,6} hr{6,6} s{6,6} h{6,6} hrr{6,6} sr{6,6}

Symmetry

Truncated order-4 hexagonal tiling with *662 mirror lines

The dual of the tiling represents the fundamental domains of (*662) orbifold symmetry. From (*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, (3333) is the commutator subgroup of .

Larger subgroup constructed as , 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.

Small index subgroups of (*662)
Index 1 2 4
Diagram
Coxeter

=
=
=
=
Orbifold *662 *663 *3232 *3333 33×
Direct subgroups
Diagram
Coxeter



= =
= =
= =
= =
Orbifold 6*3 2*33 3*33
Direct subgroups
Index 2 4 8
Diagram
Coxeter

=
=
= 		=
= = =
Orbifold 662 663 3232 3333
Radical subgroups
Index 12 24
Diagram
Coxeter



Orbifold *333333 333333

References

See also

External links

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