Triangular Tiling - Uniform Colorings

Uniform Colorings

There are 9 distinct uniform colorings of a triangular tiling. (Naming the colors by indices on the 6 triangles around a vertex: 111111, 111112, 111212, 111213, 111222, 112122, 121212, 121213, 121314)

Four of the colorings are generated by Wythoff constructions. Seven of the nine distinct colorings can be made as reductions of the four coloring: 121314. The remaining two, 111222 and 112122, have no Wythoff constructions.

Coloring
indices
111111 121212 121314 121213
Coloring
Symmetry *632
(p6m)
*333
(p3m1)
] =
333
(p3)
]+
3*3
(p31m)
Wythoff symbol 6 | 3 2 3 | 3 3 | 3 3 3
Coxeter-Dynkin =
Coloring
indices
111222 112122 111112 111212 111213
Coloring
Symmetry 2*22
(cmm)
2222
(p2)
+
*333
(p3m1)
]
*333
(p3m1)
]
333
(p3)
]+

Read more about this topic:  Triangular Tiling

Famous quotes containing the word uniform:

    The sugar maple is remarkable for its clean ankle. The groves of these trees looked like vast forest sheds, their branches stopping short at a uniform height, four or five feet from the ground, like eaves, as if they had been trimmed by art, so that you could look under and through the whole grove with its leafy canopy, as under a tent whose curtain is raised.
    Henry David Thoreau (1817–1862)