Question

What are some modern day applications of Pick's theorem? 

Answer 1

We know that picks theorem is useful method for determining the area of any polygon whose vertices are points on the lattice, a regularly spaced array of points.

1)A cute, quick little application of Pick's theorem is this:

You cannot draw an equilateral triangle neatly on graph paper, by placing vertices at grid points.

                           Ceci n'est pas un triangle équilatéral.

The reason is simple: Pick's Theorem says that the area of a polygon drawn on graph paper (with vertices at grid points) is the number of interior grid points plus half the number of border grid points minus 1. In particular, the area of such a polygon is a rational number. But the area of an equilateral triangle with base aa is 3?4a234a2, and no matter how you orient the triangle, the square of the base a2a2 is an integer (by Pythagoras). Since 3–?3 is irrational, The equilateral triangle cannot be a polygon drawn on graph paper.

2) A lattice point on the cartesian plane is a point [(x,y) \in \mathbb R^2] where both coordinates are integers. Let P be a polygon on the cartesian plane such that every vertex is a lattice point (we call it a lattice polygon). Pick’s theorem tells us that the area of P can be computed solely by counting lattice points:

The area of P is given by [i + \frac b 2 -1] , where i = number of lattice points in P and b = number of lattice points on the boundary of P.