Order-8 triangular tiling: Difference between revisions

Revision as of 02:32, 3 November 2014

Revision as of 03:16, 4 November 2014

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

:[[File:H2_tiling_334-4.png|240px]]

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. The symmetry can be doubled to [[842 symmetry]] by adding a bisecting mirror across the fundamental domains. 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).

{| 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 ==