Pick's theorem

This result is known as Pick's theorem.

The theory of Ehrhart polynomials can be seen as a higher-dimensional generalization of Pick's theorem in the Euclidean plane.

Pick's Theorem by Ed Pegg, Jr., the Wolfram Demonstrations Project.

In 1957 Reeve used this tetrahedron as a counterexample to show that there is no simple equivalent of Pick's theorem in R or higher-dimensional spaces.

Study of properties of digital sets; see, for example, Pick's theorem, digital convexity, digital straightness, or digital planarity.

The Reeve tetrahedron shows that there is no analogue of Pick's theorem in three dimensions that expresses the volume of a polytope by counting its interior and boundary points.

Exploration Lab topics have included relationships between Pick's theorem and the Euler characteristic, linear fractional transformations, Linear Diophantine equations, integer partitions and compositions, finite differences, and Chebyshev polynomials.

If the polygon can be drawn on an equally spaced grid such that all its vertices are grid points, Pick's theorem gives a simple formula for the polygon's area based on the numbers of interior and boundary grid points.

See Pick's theorem for a technique for finding the area of any arbitrary lattice polygon (one drawn on a grid with vertically and horizontally adjacent lattice points at equal distances, and with vertices on lattice points).

This fact may be deduced e.g. with the help of Pick's theorem which expresses the area of a plane triangle whose vertices have integer coordinates in terms of the number v of lattice points (strictly) inside the triangle and the number v of lattice points on the boundary of the triangle.