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Hexagon Tiling


HexagonalGrid

A hexagon tiling is a tiling of the plane by identical hexagons.

The regular hexagon forms a regular tessellation, also called a hexagonal grid, illustrated above.

HexagonTiling

There are at least three tilings of irregular hexagons, illustrated above.

HexagonalTile

They are given by the following types:

 A+B+C=360 degrees a=d; A+B+D=360 degrees a=d,c=e; A=C=E=120 degrees a=b,c=d,e=f
(1)

(Gardner 1988). Note that the periodic hexagonal tessellation is a degenerate case of all three tilings with

 A=B=C=D=E=F
(2)

and

 a=b=c=d=e=f.
(3)

Amazingly, the number of plane partitions PL(a,b,c) contained in an a×b×c box also gives the number of hexagon tilings by rhombi for a hexagon of side lengths a, b, c, a, b, c (David and Tomei 1989, Fulmek and Krattenthaler 2000). The asymptotic distribution of rhombi in a random hexagon tiling by rhombi was given by Cohn et al. (1998). A variety of enumerations for various explicit positions of rhombi are given by Fulmek and Krattenthaler (1998, 2000).


See also

Hexagon, Hexagonal Grid, Plane Partition, Tessellation, Tiling

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References

Cohn, H.; Larsen, M.; and Propp, J. "The Shape of a Typical Boxed Plane Partition." New York J. Math. 4, 137-166, 1998.David, G. and Tomei, C. "The Problem of the Calissons." Amer. Math. Monthly 96, 429-431, 1989.Gardner, M. "Tilings with Convex Polygons." Ch. 13 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 162-176, 1988.Fulmek, M. and Krattenthaler, C. "The Number of Rhombus Tilings of a Symmetric Hexagon which Contains a Fixed Rhombus on the Symmetry Axis, I." Ann. Combin. 2, 19-40, 1998.Fulmek, M. and Krattenthaler, C. "The Number of Rhombus Tilings of a Symmetric Hexagon which Contains a Fixed Rhombus on the Symmetry Axes, II." Europ. J. Combin. 21, 601-640, 2000.

Referenced on Wolfram|Alpha

Hexagon Tiling

Cite this as:

Weisstein, Eric W. "Hexagon Tiling." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HexagonTiling.html

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