Buffon’s Needle

By Allan Roberts

Figure 1. A simulation of Buffon's Needle Experiment.

Figure 1. A simulation of Buffon’s Needle Experiment. An estimate for Pi has been computed from the proportion of needles crossing the horizontal lines (n = 100,000; not all lie within the margins of the figure).

Today is π Day. Which is sort of actually a real holiday (Lamb 2012). (But not, presumably, a rational holiday.) And what better way to celebrate π Day than with a probabilistic estimation of π? The R script provided below will simulate a version of Buffon’s Needle experiment, and use the proportion of needles crossing the horizontal lines to estimate π. (For details on Buffon’s Needle see the Wikipedia article cited below.)

π, what is it good for? Well, it gets used in the formulas for the area of a circle, and the volume of a sphere. It’s also a constant in the formula for the probability density function for the normal distribution. Thus every time you make an assumption of normality, you are making a tacit reference to π.

R Script

BuffonsNeedle <- function(n=100, sd=2){
#Implemented in R by Allan Roberts, 2013.

par( usr = c(-2,12,-2,12), mar=c(4,4,4,4) );
arrows(rep(-4,16),-2:12, rep(14,16),-2:12, length=0);

X <- rnorm(n,5,sd);
Y <- rnorm(n,5,sd);
Angle <- runif(n,0,2*pi); #Note: the script itself uses pi. Homework: write a pi-less version.
X2 <- X + cos(Angle);
Y2 <- Y + sin(Angle);
CrossesLine <- ( floor(Y) != floor(Y2) );



p <- sum(CrossesLine)/n;

BuffonsNeedle( );


Buffon’s needle. Wikipedia. http://en.wikipedia.org/wiki/Buffon%27s_needle

Lamb, Evelyn. July 21, 2012. Pi Approximation Day Celebrated July 22, How Much Pi Do You Need? Huffington Post.

Pi Day. Wikipedia.


One thought on “Buffon’s Needle

  1. Pingback: Randomization: Estimating the Area of a Circle | The Madreporite

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s