View source for Order-8 triangular tiling

In [[Geometry|geometry]], the '''order-8 triangular tiling''' is a regular tiling of the [[Hyperbolic geometry|hyperbolic plane]]. It is represented by [[Schläfli symbol]] of ''{3,8}'', having eight regular [[Triangle|triangle]]s around each vertex.

== Uniform colorings ==
The half symmetry [1<sup>+</sup>,8,3] = [(4,3,3)] can be shown with alternating two colors of triangles:
:[[File:H2_tiling_334-4.png|240px]]

== Symmetry==
[[File:Octagonal_tiling_with_444_mirror_lines.png|thumb|left|Octagonal tiling with *444 mirror lines, {{CDD|node_c1|split1-44|branch_c3-2|label4}}.]]
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<sup>+</sup>,4,1<sup>+</sup>,4,1<sup>+</sup>,4)] (222222) is the [[Commutator subgroup|commutator subgroup]] of [(4,4,4)].

A larger subgroup is constructed [(4,4,4<sup>*</sup>)], 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.
{{-}}
{| class=wikitable
|+ Small index subgroups of [(4,4,4)] (*444)
|- align=center
!Index
!1
!colspan=3|2
!colspan=2|4
|- align=center
!Diagram
|[[File:444_symmetry_mirrors.png|120px]]
|[[File:444_symmetry_a00.png|120px]]
|[[File:444_symmetry_0a0.png|120px]]
|[[File:444_symmetry_00a.png|120px]]
|[[File:444_symmetry_ab0.png|120px]]
|[[File:444_symmetry_xxx.png|120px]]
|- align=center
!Coxeter
|[(4,4,4)]<BR>{{CDD|node_c1|split1-44|branch_c3-2|label4}}
|[(1<sup>+</sup>,4,4,4)]<BR>{{CDD|labelh|node|split1-44|branch_c3-2|label4}} = {{CDD|label4|branch_c3-2|2a2b-cross|branch_c3-2|label4}}
|[(4,1<sup>+</sup>,4,4)]<BR>{{CDD|node_c1|split1-44|branch_h0c2|label4}} = {{CDD|label4|branch_c1-2|2a2b-cross|branch_c1-2|label4}}
|[(4,4,1<sup>+</sup>,4)]<BR>{{CDD|node_c1|split1-44|branch_c3h0|label4}} = {{CDD|label4|branch_c1-3|2a2b-cross|branch_c1-3|label4}}
|[(1<sup>+</sup>,4,1<sup>+</sup>,4,4)]<BR>{{CDD|labelh|node|split1-44|branch_h0c2|label4}}
|[(4<sup>+</sup>,4<sup>+</sup>,4)]<BR>{{CDD|node_h4|split1-44|branch_h2h2|label4}}
|- align=center
!Orbifold
|*444
|colspan=3|*4242
|2*222
|222×
|- align=center
!Diagram
|
|[[File:444_symmetry_0bb.png|120px]]
|[[File:444_symmetry_b0b.png|120px]]
|[[File:444_symmetry_bb0.png|120px]]
|[[File:444_symmetry_0b0.png|120px]]
|[[File:444_symmetry_a0b.png|120px]]
|- align=center
!Coxeter
|
|[(4,4<sup>+</sup>,4)]<BR>{{CDD|node_c1|split1-44|branch_h2h2|label4}}
|[(4,4,4<sup>+</sup>)]<BR>{{CDD|node_h2|split1-44|branch_c3h2|label4}}
|[(4<sup>+</sup>,4,4)]<BR>{{CDD|node_h2|split1-44|branch_h2c2|label4}}
|[(4,1<sup>+</sup>,4,1<sup>+</sup>,4)]<BR>{{CDD|node_c1|split1-44|branch_h0h0|label4}}
|[(1<sup>+</sup>,4,4,1<sup>+</sup>,4)]<BR>{{CDD|labelh|node|split1-44|branch_c3h2|label4}} = {{CDD|label4|branch_c3h2|2a2b-cross|branch_c3h2|label4}}

|- align=center
!Orbifold
|
|colspan=3|4*22
|colspan=2|2*222

|- align=center
!colspan=7|Direct subgroups
|- align=center
!Index
!2
!colspan=3|4
!colspan=2|8
|- align=center
!Diagram
|[[File:444_symmetry_aaa.png|120px]]
|[[File:444_symmetry_abb.png|120px]]
|[[File:444_symmetry_bab.png|120px]]
|[[File:444_symmetry_bba.png|120px]]
|colspan=2|[[File:444_symmetry_abc.png|120px]]
|- align=center
!Coxeter
|[(4,4,4)]<sup>+</sup><BR>{{CDD|node_h2|split1-44|branch_h2h2|label4}}
|[(4,4<sup>+</sup>,4)]<sup>+</sup><BR>{{CDD|labelh|node|split1-44|branch_h2h2|label4}} = {{CDD|label4|branch_h2h2|2xa2xb-cross|branch_h2h2|label4}}
|[(4,4,4<sup>+</sup>)]<sup>+</sup><BR>{{CDD|node_h2|split1-44|branch_h0h2|label4}} = {{CDD|label4|branch_h2h2|2xa2xb-cross|branch_h2h2|label4}}
|[(4<sup>+</sup>,4,4)]<sup>+</sup><BR>{{CDD|node_h2|split1-44|branch_h2h0|label4}} = {{CDD|label4|branch_h2h2|2xa2xb-cross|branch_h2h2|label4}}
|colspan=2|[(4,1<sup>+</sup>,4,1<sup>+</sup>,4)]<sup>+</sup><BR>{{CDD|labelh|node|split1-44|branch_h0h0|label4}} = {{CDD|node_h4|split1-44|branch_h4h4|label4}}

|- align=center
!Orbifold
|444
|colspan=3|4242
|colspan=2|222222

|- align=center
!colspan=7|Radical subgroups
|- align=center
!Index
!colspan=3|8
!colspan=3|16
|- align=center
!Diagram
|[[File:444 symmetry 0zz.png|120px]]
|[[File:444 symmetry z0z.png|120px]]
|[[File:444 symmetry zz0.png|120px]]
|[[File:444 symmetry azz.png|120px]]
|[[File:444 symmetry zaz.png|120px]]
|[[File:444 symmetry zza.png|120px]]
|- align=center
!Coxeter
|[(4,4*,4)]
|[(4,4,4*)]
|[(4*,4,4)]
|[(4,4*,4)]<sup>+</sup>
|[(4,4,4*)]<sup>+</sup>
|[(4*,4,4)]<sup>+</sup>
|- align=center
!Orbifold
|colspan=3|*22222222
|colspan=3|22222222
|}

== Related polyhedra and tilings ==
[[File:H3_338_UHS_plane_at_infinity.png|thumb|The {3,3,8} honeycomb has {3,8} vertex figures.]]


From a [[Wythoff construction]] there are ten hyperbolic [[Uniform tilings in hyperbolic plane|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.
{{Octagonal tiling table}}


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


==See also==
*Order-8 tetrahedral honeycomb
*Tilings of regular polygons
*List of uniform planar tilings
*List of regular polytopes

==References==
{{reflist}}
{{refbegin}}
* [[Biography: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}}
{{refend}}

== External links ==
*{{MathWorld | urlname= HyperbolicTiling | title = Hyperbolic tiling}}
*{{MathWorld | urlname=PoincareHyperbolicDisk | title = Poincaré hyperbolic disk }}
* [http://bork.hampshire.edu/~bernie/hyper/ Hyperbolic and Spherical Tiling Gallery]
* [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]

{{Tessellation}}

[[Category:Hyperbolic tilings]]
[[Category:Isogonal tilings]]
[[Category:Isohedral tilings]]
[[Category:Regular tilings]]
[[Category:Triangular tilings]]

{{Sourceattribution|Order-8 triangular tiling}}