Order-8 triangular tiling: Difference between revisions

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{{Short description|Concept in geometry}}

{{Uniform hyperbolic tiles db|Reg hyperbolic tiling stat table|U83_2}}

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

== SymmetryUniform colorings ==

The half symmetry [1⁺,8,3] = [(4,3,3)] can be shown with alternating two colors of triangles:

:[[File:Uniform_tiling_433H2_tiling_334-t24.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⁺,4,1⁺,4,1⁺,4)] (222222) is the [[commutator subgroup]] of [(4,4,4)].

A larger subgroup is constructed [(4,4,4^*)], 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

![[Subgroup index|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 notation|Coxeter]]

|[(4,4,4)]<BR>{{CDD|node_c1|split1-44|branch_c3-2|label4}}

|[(1⁺,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⁺,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⁺,4)]<BR>{{CDD|node_c1|split1-44|branch_c3h0|label4}} = {{CDD|label4|branch_c1-3|2a2b-cross|branch_c1-3|label4}}

|[(1⁺,4,1⁺,4,4)]<BR>{{CDD|labelh|node|split1-44|branch_h0c2|label4}}

|[(4⁺,4⁺,4)]<BR>{{CDD|node_h4|split1-44|branch_h2h2|label4}}

|- align=center

!Orbifold

|*444

|colspan=3|[[4242 symmetry|*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⁺,4)]<BR>{{CDD|node_c1|split1-44|branch_h2h2|label4}}

|[(4,4,4⁺)]<BR>{{CDD|node_h2|split1-44|branch_c3h2|label4}}

|[(4⁺,4,4)]<BR>{{CDD|node_h2|split1-44|branch_h2c2|label4}}

|[(4,1⁺,4,1⁺,4)]<BR>{{CDD|node_c1|split1-44|branch_h0h0|label4}}

|[(1⁺,4,4,1⁺,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)]⁺<BR>{{CDD|node_h2|split1-44|branch_h2h2|label4}}

|[(4,4⁺,4)]⁺<BR>{{CDD|labelh|node|split1-44|branch_h2h2|label4}} = {{CDD|label4|branch_h2h2|2xa2xb-cross|branch_h2h2|label4}}

|[(4,4,4⁺)]⁺<BR>{{CDD|node_h2|split1-44|branch_h0h2|label4}} = {{CDD|label4|branch_h2h2|2xa2xb-cross|branch_h2h2|label4}}

|[(4⁺,4,4)]⁺<BR>{{CDD|node_h2|split1-44|branch_h2h0|label4}} = {{CDD|label4|branch_h2h2|2xa2xb-cross|branch_h2h2|label4}}

|colspan=2|[(4,1⁺,4,1⁺,4)]⁺<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)]⁺

|[(4,4,4*)]⁺

|[(4*,4,4)]⁺

|- align=center

!Orbifold

|colspan=3|*22222222

|colspan=3|22222222

|}

== Related polyhedra and tilings ==

[[File:H3_338_UHS_plane_at_infinity.png|thumb|The [[Order-8 tetrahedral honeycomb|{3,3,8}]] honeycomb has {3,8} vertex figures.]]

{{Triangular regular tiling}}

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}}

{{Order-8 regular tilings}}

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

{{Order 4-3-3 tiling table}}

{{Order 4-4-4 tiling table}}

==See also==

{{Commonscat|Order-8 triangular tiling}}

*[[Order-8 tetrahedral honeycomb]]

*[[Tilings of regular polygons]]

*[[List of uniform planar tilings]]

Line 25 ⟶ 150:

{{reflist}}

{{refbegin}}

* [[John Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-StrassStrauss, ''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}}

Line 36 ⟶ 161:

* [http://www.plunk.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch]

{{Tessellation}}

[[Category:tessellation]]

[[Category:Hyperbolic tilings]]

[[Category:Isogonal tilings]]

[[Category:Isohedral tilings]]

[[Category:Order-8 tilings]]

[[Category:Regular tilings]]

[[Category:Triangular tilings]]