Team Number: 16
Team Number: 16
School Name: Summer Teacher Institute
Area of Science: Mathematics
Project Title: A Reexamination of Buffon's Needle
Buffon's needle is a classic experiment that approximates the number PI using probability theory and the Monte Carlo method of randomly occurring events. To do the experiment one needs a needle or some other straight object (pencil, stick, or twirling baton). Finally one needs a set of parallel lines drawn on a flat surface separated from each other by a distance equal to the length of the needle. Now the experiment may begin. Drop the needle on the flat surface and take note whether the needle crosses any of the parallel lines or not. Keep a count of the total number of drops and the number of times the needle comes to rest over a line. Do this 1,000 times or 10,000 times or 100,000 times. Now divide the total number of drops by the number of times the needle crossed a line and then multiply the quotient by 2. The value will approximate the number PI.
You say you don't have time to drop a needle 100,000 times or even 1,000 times? Our program allows a computer to simulate needle drops and does it at computer speed.
We have researched the history of the Buffon's Needle Experiment, studied the mathematical theory explaining why the value approximates PI, Located Java apps on the internet that graphically demonstrate the experiment, and written C++ code that simulates the experiment that allows the user to input the number of needle drops.
Georges Buffon (1707-1788) was a French aristocrat formally trained in law, but his passion was natural science. He was also interested in mathematics. Buffon is credited with developing European interest in natural history during the eighteenth century. http://www.ucmp.berkeley.edu/history/buffon2.html
The math theory that explains Buffon's needle shows that the probability of the needle crossing the line is equal the ratio of the area of a rectangle (PI X 1/2 needle length) to area under the curve of the inscribed function D=(1/2)SIN(theta) where D is the distance from the center of the needle to the nearest line and theta is the angle of the needle to the parallel line. http://www.mste.uiuc.edu/reese/buffon/buffon.html.
Buffon's needle experiment is a classic problem. Therefore there a numerous examples on the internet. A particularly good one can be found at http:/www.angelfire.com/wa/hurben/buff.html .
Our C++ code uses random numbers to determine the location and angle of the needle. We use algebra and trigonometry to calculate whether the needle crosses the line. Our program updates the the PI approximation after each drop. The user may choose the number of trials. He may also elect to see the output from each trial or only the final result.
Team Members
Sponsoring Teacher
Project Mentor(s)