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Pick's Theorem

Let A be the area of a simply closed lattice polygon. Let B denote the number of lattice points on the polygon edges and I the number of points in the interior of the polygon. Then

A=I+1/2B-1.

The formula has been generalized to three- and higher dimensions using Ehrhart polynomials.

See also

Blichfeldt's Theorem, Ehrhart Polynomial, Lattice Point, Minkowski Convex Body Theorem

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References

Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, p. 17, 2003.Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 209, 1969.DeTemple, D. "Pick's Formula: A Retrospective." Math. Notes Washington State Univ. 32, Nov. 1989.Diaz, R. and Robins, S. "Pick's Formula via the Weierstrass P-Function." Amer. Math. Monthly 102, 431-437, 1995.Ewald, G. Combinatorial Convexity and Algebraic Geometry. New York: Springer-Verlag, 1996.Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, p. 215, 1984.Grünbaum, B. and Shephard, G. C. "Pick's Theorem." Amer. Math. Monthly 100, 150-161, 1993.Haigh, G. "A 'Natural' Approach to Pick's Theorem." Math. Gaz. 64, 173-, 1980.Hammer, J. Unsolved Problems Concerning Lattice Points. London: Pitman, 1977.Kelley, D. A. "Areas of Simple Polygons." Pentagon 20, 3-11, 1960.Khan, M. R. "A Counting Formula for Primitive Tetrahedra in Z^3." Amer. Math. Monthly 106, 525-533, 1999.Morelli, R. "Pick's Theorem and the Todd Class of a Toric Variety." Adv. Math. 100, 183-231, 1993.Niven, I. and Zuckerman, H. S. "Lattice Points and Polygonal Area." Amer. Math. Monthly 74, 1195, 1967.Olds, C. D.; Lax, A.; and Davidoff, G. The Geometry of Numbers. Washington, DC: Math. Assoc. Amer., 2000.Pick, G. "Geometrisches zur Zahlentheorie." Sitzenber. Lotos (Prague) 19, 311-319, 1899.Steinhaus, H. "O polu figur płlaskich." Przeglad Mat.-Fiz., 1924.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 96-98, 1999.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 183-184, 1991.

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Pick's Theorem

Cite this as:

Weisstein, Eric W. "Pick's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PicksTheorem.html

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