uniform coloring
{{one source |date=May 2024}}
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| colspan=3|The hexagonal tiling has 3 uniform colorings. |
Image:Square tiling uniform colorings.png has 9 uniform colorings:
1111, 1112(a), 1112(b),
1122, 1123(a), 1123(b),
1212, 1213, 1234.]]
In geometry, a uniform coloring is a property of a uniform figure (uniform tiling or uniform polyhedron) that is colored to be vertex-transitive. Different symmetries can be expressed on the same geometric figure with the faces following different uniform color patterns.
A uniform coloring can be specified by listing the different colors with indices around a vertex figure.
n-uniform figures
In addition, an n-uniform coloring is a property of a uniform figure which has n types vertex figure, that are collectively vertex transitive.
Archimedean coloring
A related term is Archimedean color requires one vertex figure coloring repeated in a periodic arrangement. A more general term are k-Archimedean colorings which count k distinctly colored vertex figures.
For example, this Archimedean coloring (left) of a triangular tiling has two colors, but requires 4 unique colors by symmetry positions and become a 2-uniform coloring (right):
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References
- {{cite book | author=Grünbaum, Branko | author-link=Branko Grünbaum | author2=Shephard, G. C. | author2-link=G.C. Shephard | title=Tilings and Patterns | publisher=W. H. Freeman and Company | year=1987 | isbn=0-7167-1193-1 | url-access=registration | url=https://archive.org/details/isbn_0716711931 }} Uniform and Archimedean colorings, pp. 102–107
External links
- {{MathWorld | urlname=PolyhedronColoring | title=Polyhedron coloring }}
- [http://www2u.biglobe.ne.jp/~hsaka/mandara/ue2 Uniform Tessellations on the Euclid plane]
- [http://www.orchidpalms.com/polyhedra/tessellations/tessel.htm Tessellations of the Plane]
- [https://web.archive.org/web/20080602030052/http://www.tess-elation.co.uk/index.htm David Bailey's World of Tessellations]
- [https://web.archive.org/web/20060909053826/http://www.uwgb.edu/dutchs/SYMMETRY/uniftil.htm k-uniform tilings]
- [http://probabilitysports.com/tilings.html n-uniform tilings]
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