Divide and Conquer: Proving Pick's Theorem for Lattice Polygons

Abstract

If you like to play Tetris, then you might like this project. You will learn something interesting about the mathematics of complex shapes as you try to prove Pick's Theorem.

The strange shape below is an example of a lattice polygon, which is a polygon whose vertices lie on points in the plane that have integer coordinates.

As you can see, it is a complex shape, but there is an easy way to calculate its area, by simply counting lattice points! If you count the number of lattice points on the boundary of the polygon (b), and the number of lattice points inside the polygon (i), then the area (A) of the polygon is given by Pick's Theorem:

Equation 1: A = i + b/2 - 1.

A good way to explore lattice polygons is with a geoboard. A physical geoboard is a piece of wood with pegs (or nails) arranged in a regular grid. The wood represents a section of the plane, and the pegs or nails are the lattice points. You stretch rubber bands over the lattice points to create polygons. You can make or buy a geoboard for this project (it would make a nice addition to your display), but it is not absolutely necessary. You can also just draw a lattice and polygons on a piece of paper, or search online for a geoboard simulator.

Using whatever technique you choose, try experimenting with creating different polygon shapes, and see if Pick's theorem is correct. Start out with simple shapes and gradually move to more complex ones. Can you come up with a mathematical proof for Pick's Theorem?

Objective

To study lattice polygons and prove that Pick's Theorem is correct.

Credits

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Last edit date: 2020-11-20

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