Jump to main content

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

Summary

Areas of Science
Difficulty
 
Time Required
Short (2-5 days)
Prerequisites
None
Material Availability
Readily available
Cost
Very Low (under $20)
Safety
No issues
Credits
Science Buddies
*Note: For this science project you will need to develop your own experimental procedure. Use the information in the summary tab as a starting place. If you would like to discuss your ideas or need help troubleshooting, use the Ask An Expert forum. Our Experts won't do the work for you, but they will make suggestions and offer guidance if you come to them with specific questions.

If you want a Project Idea with full instructions, please pick one without an asterisk (*) at the end of the title.

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.

Drawing of a lattice polygon on a pegboard
Figure 1. Example of a lattice polygon.

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.

Bibliography

This Mudd Math Fun Facts webpage has some information on Pick's Theorem, and a hint for how to go about proving it.

icon scientific method

Ask an Expert

Do you have specific questions about your science project? Our team of volunteer scientists can help. Our Experts won't do the work for you, but they will make suggestions, offer guidance, and help you troubleshoot.

Careers

If you like this project, you might enjoy exploring these related careers:

Career Profile
Mathematicians are part of an ancient tradition of searching for patterns, conjecturing, and figuring out truths based on rigorous deduction. Some mathematicians focus on purely theoretical problems, with no obvious or immediate applications, except to advance our understanding of mathematics, while others focus on applied mathematics, where they try to solve problems in economics, business, science, physics, or engineering. Read more
Career Profile
Do you dream of building big? Civil engineering technicians help build some of the largest structures in the world—from buildings, bridges, and dams to highways, airfields, and wastewater treatment facilities. Many of these construction projects are "public works," meaning they strengthen and benefit a community, state, or the nation. Read more

News Feed on This Topic

 
, ,

Cite This Page

General citation information is provided here. Be sure to check the formatting, including capitalization, for the method you are using and update your citation, as needed.

MLA Style

Science Buddies Staff. "Divide and Conquer: Proving Pick's Theorem for Lattice Polygons." Science Buddies, 20 Nov. 2020, https://www.sciencebuddies.org/science-fair-projects/project-ideas/Math_p009/pure-mathematics/picks-theorem-lattice-polygons. Accessed 3 Oct. 2022.

APA Style

Science Buddies Staff. (2020, November 20). Divide and Conquer: Proving Pick's Theorem for Lattice Polygons. Retrieved from https://www.sciencebuddies.org/science-fair-projects/project-ideas/Math_p009/pure-mathematics/picks-theorem-lattice-polygons


Last edit date: 2020-11-20
Top
Free science fair projects.