Order-8 triangular tiling: Difference between revisions

Revision as of 00:48, 10 November 2014

Revision as of 00:55, 10 November 2014

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== Symmetry==

[[File:Octagonal_tiling_with_444_mirror_lines.png|thumb|left|Octagonal tiling with *444 mirror lines

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).

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

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|+ Small index subgroups of [(4,4,4)] (*444)