Buffon’s Needle Algorithm to Estimate π

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1. Introduction

Buffon’s needle experiment [1] was originally used to provide $\pi $ . Throwing a needle (see Figure 1) onto a flat plane with equally-spaced parallel lines, the probability that the needle touches the parallel line provides an estimate for $\pi $ is

$P=\frac{{\displaystyle {\int}_{0}^{\pi}}\frac{l}{2}\mathrm{sin}\theta d\theta}{\pi d/2}$ (1)

and so

$\pi \mathrm{=}\frac{2l}{Pa}$ (2)

where $P$ is the probability, $l$ the length of the needle and $a$ the spacing with $l\mathrm{<}a$ .

Here, we should note that the Buffon’s needle problem becomes an integration problem (see Figure 2) so the probability is just the ratio of areas.

Many variants of the original Buffon’s needle experiment [2] [3] [4] [5] [6] [7] have been developed. Kendall and Moran [2] and Diaconis [3] examined several aspects of the problem for a long needle ( $l\mathrm{>}a$ ) and Siniksaran [7] used Mathematics to review various statistical aspects of the experiment.

With the advent of computers, Buffon’s needle has been pedagogically used as an example in Monte Carlo method introductory classes and many websites pro- vide Buffon’s needle algorithm implementations. However, in these programs, the value of $\pi $ is used for the random sampling of the needle angular direction. Thus $\pi $ is used to obtain the value of $\pi $ itself.

This note presents a sampling algorithm for the random direction of the needle that avoids using $\pi $ within the algorithm. We then compare the Buffon’s needle and the Hit-and-Miss integration algorithms using the common Monte Carlo algorithm comparison method, the time consumption (laboriousness) [8] .

Figure 1. Buffon’s needle algorithm with spacing $a$ and needle length $l$ .

Figure 2. Buffon’s needle problem can be converted to an integration problem.

2. Buffon’s Needle Algorithm

For random direction sampling of the needle, we use a square enclosing a tightly fitted circle (see Figure 3).

Since the directions of the random points inside the circle are uniform, we obtain uniform needle random directions.

3. Comparison with the Hit-and-Miss Algorithm

The Hit-and-Miss integration algorithm can also provide the value of $\pi $ . Using one quadrant of the circle circumscribed by a square (see Figure 4), we can obtain $\pi \mathrm{=4}h$ , where, $h$ is the probability of hitting the shaded area, when we generate a pair of random numbers to sample a random point inside of the square.

We use the common Monte Carlo algorithm comparison, time consumption (laboriousness): [8] $t\underset{\_}{D}\xi $ where $t$ is the CPU time expended in calculating a single estimate and $\underset{\_}{D}\xi $ estimate variance. Figure 5 shows the log-log plot of the number of Monte Carlo steps v.s. the errors of the two algorithms, and Table 1 compare the two algorithms. The Hit-and-Miss algorithm is superior to the Buffon’s needle algorithm for calculation of #Math_28#, which seems reasonable given the Buffon’s needle algorithm requires more numerical operations for a Monte Carlo step. Of course, we can improve Buffon’s algorithm performance using a tighter non-constant proposed probability distribution for the acceptance-rejec- tion sampling method. Using quasi-Monte Carlo random numbers [9] for the angle sampling will also improve the convergence.

4. Conclusion

Buffon’s needle algorithm has been pedagogically used as an introduction to

Figure 3. Needle random direction sampling. After generating a ran- dom point inside the circle, we obtain the random direction vector for the needle.

Figure 4. Hit-and-Miss diagram. Using a pair of random num- bers, we obtain the probability of hitting inside of the marked quadrant.

Figure 5. Monte Carlo steps and estimation errors. The linear regression slopes for Buffon’s needle and Hit-and-Miss algorithms (solid and dotted lines res- pectively) are $-0.502$ and $-0.505$ with correlation coefficients $-0.9994$ and $-0.9999$ respectively.

Table 1. Time consumption of Buffon’s needle and Hit-and-Miss algorithm. Variances were obtained from 100 independent runs and the number of random walks per run was ${10}^{7}$ .

Monte Carlo methods. However, $\pi $ is used for needle angle sampling inside the original algorithm. This note presents a method for the needle angle sampling without using $\pi $ and make the Buffon’s needle algorithm a Monte Carlo method to estimate $\pi $ . We compared the Buffon’s needle and Hit-and-Miss integration algorithms and found that the Buffon’s needle algorithm is not superior to the Hit-and-Miss integration algorithm.

Acknowledgements

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A2003025). Also, later this work was supported by the GIST Research Institute (GRI) in 2017. In addition, this research was a collaborative research project, supercomputing infrastructure service and application, supported by the Korea institute of science and technology information.

References

[1] Buffon, G. (1777) Essai da’rithme’tique morale. Histoire Naturelle, Générale, et Particuliére, Supplément, 4, 685.

[2] Morton, R.A. (1966) The Expected Number and Angle of Intersections between Random Curves in a Plane. Journal of Applied Probability, 3, 559-562.

https://doi.org/10.1017/S0021900200114342

[3] Diaconis, P. (1976) Buffon’s Problem with a Long Needle. Journal of Applied Probability, 13, 614-618.

[4] Periman, M.D. and Wichura, M.J. (1975) Sharpening Buffon’s Needle. Amer. Stat, 29, 157.

[5] Solomon, H. (1987) Geometric Probability. SIAM, Philadelphia.

[6] Wood, G.R. and Robertson, J.M. (1998) Information in Buffon Experiments. Journal of Statistical Planning and Inferences, 66, 415-421.

[7] Siniksaran, E. (2008) Throwing Buffon’s Needle with Mathematica. The Mathematical Journal, 11, 71.

[8] Sobol, I.M. (1994) A Primer for the Monte Carlo Method. CRC Press, Washington DC.

[9] Lemieux, C. (2009) Monte Carlo and Quasi-Monte Carlo Sampling. Springer-Verlag, New York.