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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.
[[File:H3_338_UHS_plane_at_infinity.png|thumb|The [[Order-8 tetrahedral honeycomb|{3,3,8}]] honeycomb has {3,8} vertex figures.]]
==
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==
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<sup>+</sup>,4,1<sup>+</sup>,4,1<sup>+</sup>,4)] (222222) is the [[commutator subgroup]] of [(4,4,4)].
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