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{{Uniform hyperbolic tiles db|Uniform hyperbolic tiling stat table|U64_12}}
{{Uniform hyperbolic tiles db|Uniform hyperbolic tiling stat table|U64_12}}
In [[geometry]], the '''truncated order-6 square tiling''' is a uniform tiling of the [[Hyperbolic geometry|hyperbolic plane]]. It has [[Schläfli symbol]] of t<sub>0,1</sub>{4,6}.
In [[geometry]], the '''truncated order-6 square tiling''' is a uniform tiling of the [[Hyperbolic geometry|hyperbolic plane]]. It has [[Schläfli symbol]] of t{4,6}.


== Related polyhedra and tiling ==
== Uniform colorings ==


{| class=wikitable width=240
|- valign=top
|[[File:Uniform_tiling_443-t012.png|240px]]<BR>The half symmetry [1<sup>+</sup>,6,4] = [(4,4,3)] can be shown with alternating two colors of octagons, with as [[Coxeter diagram]] {{CDD|branch_11|split2-44|node_1}}.
|}

== Symmetry ==
[[File:Truncated_order-6_square_tiling_with_mirrors.png|thumb|left|Truncated order-6 square tiling with *443 symmetry mirror lines]]
The dual tiling represents the fundamental domains of the *443 orbifold symmetry. There are two reflective subgroup kaleidoscopic constructed from [(4,4,3)] by removing one or two of three mirrors. In these images fundamental domains are alternately colored black and cyan, and mirrors exist on the boundaries between colors.

A larger subgroup is constructed [(4,4,3*)], index 6, as (3*22) with gyration points removed, becomes (*222222).

The symmetry can be doubled as [[642 symmetry]] by adding a mirror bisecting the fundamental domain.
{{-}}
{| class="wikitable collapsible collapsed"
!colspan=12| Small index subgroups of [(4,4,3)] (*443)
|- align=center
![[Subgroup index|Index]]
!1
!colspan=2|2
!6
|- align=center
!Diagram
|[[File:443_symmetry_000.png|150px]]
|[[File:443_symmetry_0a0.png|150px]]
|[[File:443_symmetry_a0a.png|150px]]
|[[File:443_symmetry_z0z.png|150px]]
|- align=center
![[Coxeter notation|Coxeter]]<BR>([[Orbifold notation|orbifold]])
|[(4,4,3)] = {{CDD|node_c1|split1-44|branch_c2}}<BR>(*443)
|[(4,1<sup>+</sup>,4,3)] = {{CDD|labelh|node|split1-44|branch_c2}} = {{CDD|branch_c2|2a2b-cross|branch_c2}}<BR>([[3232 symmetry|*3232]])
|[(4,4,3<sup>+</sup>)] = {{CDD|node_c1|split1-44|branch_h2h2}}<BR>(3*22)
|[(4,4,3*)] = {{CDD|node_c1|split1-44|branch|labels}}<BR>([[222222 symmetry|*222222]])
|-
!colspan=5|Direct subgroups
|- align=center
!Index
!2
!colspan=2|4
!12
|- align=center
!Diagram
|[[File:443_symmetry_aaa.png|150px]]
|colspan=2|[[File:443_symmetry_abc.png|150px]]
|[[File:443_symmetry_zaz.png|150px]]
|- align=center
!Coxeter<BR>(orbifold)
|[(4,4,3)]<sup>+</sup> = {{CDD|node_h2|split1-44|branch_h2h2}}<BR>(443)
|colspan=2|[(4,4,3<sup>+</sup>)]<sup>+</sup> = {{CDD|labelh|node|split1-44|branch_h2h2}} = {{CDD|branch_h2h2|2xa2xb-cross|branch_h2h2}}<BR>(3232)
|[(4,4,3*)]<sup>+</sup> = {{CDD|node_h2|split1-44|branch|labels}}<BR>(222222)
|}

== Related polyhedra and tilings ==
From a [[Wythoff construction]] there are eight hyperbolic [[Uniform tilings in hyperbolic plane|uniform tilings]] that can be based from the regular order-4 hexagonal tiling.

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.
{{Order 6-4 tiling table}}
{{Order 6-4 tiling table}}


It can also be generated from the (4 4 3) hyperbolic tilings:
⚫
{{Template:Truncated figure4 table}}
{{Order 4-4-3 tiling table}}


⚫
{{Truncated figure4 table}}
⚫
==References==
{{Omnitruncated34 table}}
⚫
* [[John Horton Conway|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)
⚫
* {{Cite book|title=The Beauty of Geometry: Twelve Essays|year=1999|publisher=Dover Publications|lccn=99035678|isbn=0-486-40919-8|chapter=Chapter 10: Regular honeycombs in hyperbolic space}}


==See also==
==See also==
{{Commonscat|Uniform tiling 6-8-8}}
*[[Square tiling]]
*[[Square tiling]]
*[[Tilings of regular polygons]]
*[[Tilings of regular polygons]]
*[[List of uniform planar tilings]]
*[[List of uniform planar tilings]]
*[[List of regular polytopes]]
*[[List of regular polytopes]]

⚫
==References==
⚫
* [[John Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, {{isbn|978-1-56881-220-5}} (Chapter 19, The Hyperbolic Archimedean Tessellations)
⚫
* {{Cite book|title=The Beauty of Geometry: Twelve Essays|year=1999|publisher=Dover Publications|lccn=99035678|isbn=0-486-40919-8|chapter=Chapter 10: Regular honeycombs in hyperbolic space}}


== External links ==
== External links ==
Line 24: Line 84:
* [http://geometrygames.org/KaleidoTile/index.html KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings]
* [http://geometrygames.org/KaleidoTile/index.html KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings]
* [http://www.plunk.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch]
* [http://www.plunk.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch]

⚫
[[Category:Tessellation]]
{{Tessellation}}

[[Category:Hyperbolic tilings]]
[[Category:Isogonal tilings]]
[[Category:Order-6 tilings]]
⚫
[[Category:Square tilings]]
[[Category:Truncated tilings]]
[[Category:Uniform tilings]]

Latest revision as of 21:58, 12 December 2023

Truncated order-6 square tiling
Truncated order-6 square tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 8.8.6
Schläfli symbol t{4,6}
Wythoff symbol 2 6 | 4
Coxeter diagram
Symmetry group [6,4], (*642)
[(3,3,4)], (*334)
Dual Order-4 hexakis hexagonal tiling
Properties Vertex-transitive

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

Uniform colorings

[edit]

The half symmetry [1+,6,4] = [(4,4,3)] can be shown with alternating two colors of octagons, with as Coxeter diagram .

Symmetry

[edit]
Truncated order-6 square tiling with *443 symmetry mirror lines

The dual tiling represents the fundamental domains of the *443 orbifold symmetry. There are two reflective subgroup kaleidoscopic constructed from [(4,4,3)] by removing one or two of three mirrors. In these images fundamental domains are alternately colored black and cyan, and mirrors exist on the boundaries between colors.

A larger subgroup is constructed [(4,4,3*)], index 6, as (3*22) with gyration points removed, becomes (*222222).

The symmetry can be doubled as 642 symmetry by adding a mirror bisecting the fundamental domain.

[edit]

From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular order-4 hexagonal tiling.

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.

Uniform tetrahexagonal tilings
Symmetry: [6,4], (*642)
(with [6,6] (*662), [(4,3,3)] (*443) , [∞,3,∞] (*3222) index 2 subsymmetries)
(And [(∞,3,∞,3)] (*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
V64 V4.12.12 V(4.6)2 V6.8.8 V46 V4.4.4.6 V4.8.12
Alternations
[1+,6,4]
(*443) 		[6+,4]
(6*2) 		[6,1+,4]
(*3222) 		[6,4+]
(4*3) 		[6,4,1+]
(*662) 		[(6,4,2+)]
(2*32) 		[6,4]+
(642)

=
=
=
=
=
=
h{6,4} s{6,4} hr{6,4} s{4,6} h{4,6} hrr{6,4} sr{6,4}

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

Uniform (4,4,3) tilings
Symmetry: [(4,4,3)] (*443) [(4,4,3)]+
(443) 		[(4,4,3+)]
(3*22) 		[(4,1+,4,3)]
(*3232)
h{6,4}
t0(4,4,3) 		h2{6,4}
t0,1(4,4,3) 		{4,6}1/2
t1(4,4,3) 		h2{6,4}
t1,2(4,4,3) 		h{6,4}
t2(4,4,3) 		r{6,4}1/2
t0,2(4,4,3) 		t{4,6}1/2
t0,1,2(4,4,3) 		s{4,6}1/2
s(4,4,3) 		hr{4,6}1/2
hr(4,3,4) 		h{4,6}1/2
h(4,3,4) 		q{4,6}
h1(4,3,4)
Uniform duals
V(3.4)4 V3.8.4.8 V(4.4)3 V3.8.4.8 V(3.4)4 V4.6.4.6 V6.8.8 V3.3.3.4.3.4 V(4.4.3)2 V66 V4.3.4.6.6
*n42 symmetry mutation of truncated tilings: n.8.8
Symmetry
*n42
[n,4] 		Spherical Euclidean Compact hyperbolic Paracompact
*242
[2,4] 		*342
[3,4] 		*442
[4,4] 		*542
[5,4] 		*642
[6,4] 		*742
[7,4] 		*842
[8,4]... 		*∞42
[∞,4]
Truncated
figures
Config. 2.8.8 3.8.8 4.8.8 5.8.8 6.8.8 7.8.8 8.8.8 ∞.8.8
n-kis
figures
Config. V2.8.8 V3.8.8 V4.8.8 V5.8.8 V6.8.8 V7.8.8 V8.8.8 V∞.8.8
*n32 symmetry mutation of omnitruncated tilings: 6.8.2n
Sym.
*n43
[(n,4,3)] 		Spherical Compact hyperbolic Paraco.
*243
[4,3] 		*343
[(3,4,3)] 		*443
[(4,4,3)] 		*543
[(5,4,3)] 		*643
[(6,4,3)] 		*743
[(7,4,3)] 		*843
[(8,4,3)] 		*∞43
[(∞,4,3)]
Figures
Config. 4.8.6 6.8.6 8.8.6 10.8.6 12.8.6 14.8.6 16.8.6 ∞.8.6
Duals
Config. V4.8.6 V6.8.6 V8.8.6 V10.8.6 V12.8.6 V14.8.6 V16.8.6 V6.8.∞

See also

[edit]
Wikimedia Commons has media related to Uniform tiling 6-8-8.

References

[edit]
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
  • "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
[edit]
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