Buffon\'s needle: fun and fundamentals

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from Buffon\'s needle to equilibrium partitioning of polymers in confining pores

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Buffon\'s needle: fun and fundamentals

  1. 1. Buffon’s needle:fun and fundamentals Yanwei Wang
  2. 2. Georges-Louis Leclerc, Comte de Buffon (7 September 1707 – 16 April 1788)
  3. 3. Histoire Naturelle, Générale et Particulière (1749-88, 36 volumes)
  4. 4. Essai d’arithmétique morale(or “Essay of moral arithmetic”)
  5. 5. The game of franc-carreau (The clean tile problem)
  6. 6. The game of franc-carreau (The clean tile problem)
  7. 7. The game of franc-carreau (The clean tile problem)
  8. 8. The clean tile problem represents the firstattempt towards computing probabilitiesby using geometry instead of analysis. A.M. Mathai : An Introduction to Geometrical Probability: Distributional Aspects with Applications, Gordon and Breach, Newark, (1999).
  9. 9. The needle problem
  10. 10. • Ever helpful, Buffon points out that "On peut jouer ce jeu sur un damier avec une aiguille à coudre ou une épingle sans tête." (You can play this game on a checkerboard with a sewing-needle or a pin without a head.)
  11. 11. The probability that a needle (L<D) cut a line is 2L P = πDThe solution required a geometrical (rather than combinatorial)approach and was obtained by using integral calculus, for thefirst time in the development of probability. A.M. Mathai : An Introduction to Geometrical Probability: Distributional Aspects with Applications, Gordon and Breach, Newark, (1999).
  12. 12. The probability that a needle (L<D) cut a line is 2L P = πDBuffons needle problem established the theoreticalbasis for design-based methods to estimate thetotal length and total surface area of non-classicallyshaped objects.The field is known as Stereology.
  13. 13. Proc Biol Sci. 2000 April 22; 267(1445): 765–770.
  14. 14. The probability that a needle (L<D) cut a line is 2L P = πDIn 1812, Laplace suggested using Buffon’s needleexperiments to estimate π
  15. 15. The Monte Carlo CasinoVon Neumann chose the name "Monte Carlo".
  16. 16. The probability that a needle (L<D) cut a line is 2L P = πDIn 1812, Laplace suggested using Buffon’s needleexperiments to estimate π wrong!
  17. 17. An italian mathematician, Mario Lazzarini performed theBuffon’s needle experiment in 1901. His needle was 2.5cm long, and his parallel lines were separated by 3.0 cmapart. He dropped the needle 3408 times and observed1808 hits.1808 2 2.5 =3408 π 3.0 ˆ 355 π= ˆ = 3.1415929... 113
  18. 18. An italian mathematician, Mario Lazzarini performed theBuffon’s needle experiment in 1901. His needle was 2.5cm long, and his parallel lines were separated by 3.0 cmapart. He dropped the needle 3408 times and observed1808 hits.1808 2 2.5 =3408 π 3.0 ˆ 355 π= ˆ = 3.1415929... 113 The Zu Chongzhi Pi rate, obtained around 480 using Liu Huis algorithm applied to a 12288-gon Zu Chongzhi (429–500)
  19. 19. What is the average number of needle-line crossings? This version was introduced by Émile Barbier (1839-1889) in 1870.
  20. 20. If L<D, the number of crossing in one throw caneither be 1 or 0 with probabilities P and 1-P.The throws are Bernoulli trials. 2L P = πDWhen a needle is dropped at random T times, theexpected number of cuts is PT.
  21. 21. The Buffon Noodle Problem
  22. 22. Suppose the noodle is piecewise linear, i.e. consistsof N straight pieces. Let Xi be the number of timesthe i-th piece crosses one of the parallel lines. Theserandom variables are not independent, but theexpectations are still additive.E(X1+X2+···+XN)=E(X1)+E(X2)+···+E(XN) Ramaley, J. F. (1969). "Buffons Noodle Problem". The American Mathematical Monthly 76 (8, October 1969): 916–918
  23. 23. When a noodle is dropped at random T times,the expected number of cuts is PT. 2L P = πDwhere L is the contour length of the noodle.
  24. 24. Consider a circle with diameter, D, the same asthe grid spacing. The total length of the circle(circumference) is π D. The expected numberof cuts per throw is then 2 πD =2 π D
  25. 25. The Buffon-Laplace problem
  26. 26. Summary• Mr. Buffon and his three classical problems• Mr. Lazzarini’s lucky estimate of π• The Buffon noodle problem• Inspirations to our research

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