The Buffon needle problem

An animation

The Buffon needle problem says that if we drop a needle of length d on a floor with lines parallel that are l inches apart, then the probability, p, that the needle crosses a line is

\(p=\frac{2}{\pi}\frac{d}{l} \)

If the needle is d = 1/2 inches long and the lines are l = 1 inch apart, then

\(p=\frac{1}{\pi} \hspace{0.25in} \text{ or } \hspace{0.25in} \pi = \frac{1}{p}\)

In other words, we have a way to estimate π by way of a simulation. The animation below does just that. On the left side are the needles being “dropped,” and on the right is the associated inverse probability that a needle crosses the line. This is done 100 times. The convergence is on the slow side, so the estimation with 100 trials (a carefully chosen 100 trials) has an estimate of 3.225806. The R code is below.

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R Code

## Packages

library(dplyr)
library(tidyr)
library(ggplot2)
library(scales)
library(animation)
library(patchwork)

## Colors

MyBlue <- "#437fca"
MyRed <- "#be4242"
MyPurple <- "#5B005B"
MyLightP <- "#dfdbdf" 
MyLightP2 <-  "#f8f4f8"  
MyLightP3 <- "#fcfafc" 

## Defined Variables

CaptionData <- ""

## Create Data

Pins <- 100

Buffon <- data.frame("Drop" = 1:Pins,
                     "xPoint" = runif(Pins, min = 0, max = 1),
                     "yPoint" = runif(Pins, min = 0, max = 1),
                     "theta"  = runif(Pins, 0, pi)
                     )
Buffon <- Buffon %>%
 mutate( xStart = ifelse(theta < pi / 2, xPoint - cos(theta) / 4,
                         xPoint - cos(pi - theta) / 4),
         yStart = ifelse(theta < pi/2, yPoint - sin(theta) / 4,
                         yPoint + sin(pi - theta) / 4),
         xEnd = ifelse(theta < pi/2, xPoint + cos(theta) / 4,
                       xPoint + cos(pi - theta) / 4),
         yEnd = ifelse(theta < pi/2, yPoint + sin(theta) / 4,
                       yPoint - sin(pi - theta) / 4),
         Cross = ifelse(yStart >= 1 | yStart <= 0 | yEnd >= 1 | yEnd <= 0, 1, 0),
         CrossCount = cumsum(Cross),
         InProb = Drop/CrossCount
        )

# Set Graph Frame
dev.new(width = 1400,height = 700,unit = "px")

## Create Annimation

saveGIF(
{
ani.options(interval = 0.20, nmax = 50)

for(i in 2:Pins){

# Pin Drop Plot

  p1 <- ggplot(Buffon[1:i,], aes(xPoint, yPoint)) +
        geom_point(shape = 16, size = 3, color = MyBlue) +
        geom_hline(yintercept = c(0, 1), color = MyRed, linewidth = 1.5) +
        geom_segment(aes(x = xStart, y = yStart, xend = xEnd, yend = yEnd),
                         linewidth = 1.25)+
        scale_x_continuous(limits = c(-0.25, 1.25), expand = c(0.01, 0.01)) +
        scale_y_continuous(limits = c(-0.25, 1.25), expand = c(0.01, 0.01)) +
        coord_fixed()+
        theme(axis.text = element_text(size = 14),
	          axis.title = element_text(size = 16),
	          plot.title = element_text(size = 20),
             plot.background = element_rect(fill = MyLightP3),
             panel.background = element_rect(fill = MyLightP), 
	         legend.background = element_rect(fill = MyLightP2),
	         plot.caption = element_text(hjust = c(1, 0), size = c(14, 14),
                                        color = c(MyPurple, "black"))) +
        labs(title="1/2 in Needles Dropped on 1 in Lines",
             y = NULL,	x = NULL,
             caption = c("Briefed by Data || Thomas J Pfaff", CaptionData))

# Inverse Probability Plot

  p2 <- ggplot(Buffon[1:i,], aes(x = Drop, y = InProb)) +
        geom_point() + geom_line()+
        geom_hline(yintercept = pi, color = MyPurple, linewidth = 1.5) +
        xlim(1, Pins) +
        scale_y_continuous(labels = function(x){format(x, nsmall = 1)}) +
        theme(axis.text = element_text(size = 14),
	          axis.title = element_text(size = 16),
	          plot.title = element_text(size = 20),
             plot.background = element_rect(fill = MyLightP3),
             panel.background = element_rect(fill = MyLightP), 
	         legend.background = element_rect(fill = MyLightP2),
	         plot.caption = element_text(hjust = c(1, 0), size = c(14, 14),
                                        color = c(MyPurple, "black") ) )+
        labs(title = "Inverse Probability the Needles Cross",
             y = "Inverse Probability", x = NULL,
			caption = c("Briefed by Data || Thomas J Pfaff", CaptionData))
 
print(p1 + p2 + 
      plot_annotation(title = "The Buffon Needle Problem",
                      theme = theme(plot.title = element_text(size = 24))
      ))
ani.pause()
  }

for (k in 1:5){
print(p1 + p2 +
      plot_annotation(title = "The Buffon Needle Problem",
                      theme = theme(plot.title = element_text(size = 30))
      ))
ani.pause()
  }
}, movie.name = "Buffon.gif", ani.width = 1456, ani.height = 936)