Los Angeles Math Circle

4/7 -- High School Circle: Pick's Theorem and Ehrhart Theory (Mo Omar)

Pick's Theorem relates the area of a lattice polygon to the number of lattice points on its boundary and the number of lattice points in its interior. We prove this theorem from the ground up, by starting with rectangles, building up to general triangles, and then using triangulations. This is the first of two weeks, the second of which will discuss generalizations of this theory to higher dimensions (Ehrhart Theory).