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Lattice Polygon

PointLatticeParallelograms

A polygon whose vertices are points of a point lattice. Regular lattice n-gons exists only for n=3, 4, and 6 (Schoenberg 1937, Klamkin and Chrestenson 1963, Maehara 1993). A lattice n-gon in the plane can be equiangular to a regular polygon only for n=4 and 8 (Scott 1987, Maehara 1993).

Maehara (1993) presented a necessary and sufficient condition for a polygon to be angle-equivalent to a lattice polygon in R^n. In addition, Maehara (1993) proved that cos^2(sum_(theta in S)theta) is a rational number for any collection S of interior angles of a lattice polygon.

See also

Bar Graph Polygon, Canonical Polygon, Convex Polygon, Convex Polyomino, Ferrers Graph Polygon, Golygon, Point Lattice, Polyomino, Self-Avoiding Polygon, Stack Polyomino, Staircase Polygon, Three-Choice Polygon

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References

Beeson, M. J. "Triangles with Vertices on Lattice Points." Amer. Math. Monthly 99, 243-252, 1992.Jensen, I. "Size and Area of Square Lattice Polygons." 28 Mar 2000. http://arxiv.org/abs/cond-mat/0003442.Klamkin, M. and Chrestenson, H. E. "Polygon Imbedded in a Lattice." Amer. Math. Monthly 70, 51-61, 1963.Maehara, H. "Angles in Lattice Polygons." Ryukyu Math. J. 6, 9-19, 1993.Schoenberg, I. J. "Regular Simplices and Quadratic Forms." J. London Math. Soc. 12, 48-55, 1937.Scott, P. R. "Equiangular Lattice Polygons and Semiregular Lattice Polyhedra." College Math. J. 18, 300-306, 1987.

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Lattice Polygon

Cite this as:

Weisstein, Eric W. "Lattice Polygon." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LatticePolygon.html

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