Pick's Theorem
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Given a simple polygon constructed on a grid of equal-distanced points (i.e., points with integer coordinates) such that all the polygon''s vertices are grid points,The result was first described by Georg Alexander Pick in 1899. 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. However, there is a generalization in higher dimensions via Ehrh...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Given a simple polygon constructed on a grid of equal-distanced points (i.e., points with integer coordinates) such that all the polygon''s vertices are grid points,The result was first described by Georg Alexander Pick in 1899. 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. However, there is a generalization in higher dimensions via Ehrhart polynomials. The formula also generalizes to surfaces of polyhedra.
