From: Gerry Myerson Subject: Re: Buffon's Noodle Date: Thu, 09 Mar 2000 09:29:22 +1100 Newsgroups: sci.math.research Summary: [missing] In article , "Mark Thornber" wrote: > I'm trying to find good references for the extension of the well-known > Buffon's needle problem to curved line segments (eg rectifiable curves). > This is referred to in several texts as "Buffon's Noodle", but I can't > find one with a decent treatment. A search of Math Reviews turned this up. **************************************** 95g:60021a 60D05 (60J65) Waymire, Ed(1-ORS) Buffon noodles. Amer. Math. Monthly 101 (1994), no. 6, 550--559. 95g:60021b 60D05 (60J65) Waymire, Edward C.(1-ORS) Addendum: "Buffon noodles". Amer. Math. Monthly 101 (1994), no. 8, 791. One tosses a needle of length $L$ onto a plane surface marked by parallel lines of width $D>L$ units apart and asks for the probability of the event $C$ that the needle intersects its closest line. This is the classical Buffon needle problem, whose answer is a function $p\sb 0(x)\coloneq P(C)=(2L)/(\pi D)=\gamma\sb 0x$, where $x$ is the ratio $x=L/D$ and $\gamma\sb 0=2/\pi$. In this paper, in place of a needle, the author considers "noodles", which are allowed to become randomly tangled. The case considered is that of a two-dimensional standard Brownian motion $\{B(t)=(B\sb 1(t),B\sb 2(t))\}$ over a time interval of unit length. Although the total length of the Brownian noodle is $\infty$, one may introduce $L$ as a scale parameter by taking $S(t)=(\sqrt 2/2)LB(t)$ $(0\leq t\leq 1)$. The answer in this case is $$\multline p\sb 1(x)\coloneq P(C)=\\ \frac{8}{\pi\sp 2}\sum\sp \infty\sb {m=1}\frac{1}{(2m-1)\sp 2}\bigg(1-\exp\bigg(-(2m-1)\sp 2\bigg(\frac{\pi\sp 2}{4}\bigg)\bigg(\frac{L}{D}\bigg)\sp 2\bigg)\bigg).\endmultline$$ Next, a real string of $n$ needles, each of length $L/\sqrt n$, strung together in independent random directions, is considered. The total length is then $n(L/\sqrt n)=L\sqrt n$ and the gap between parallel lines is chosen as $D/\sqrt n$. In this case the functional central limit theorem provides an approximation to the distribution of the string of needles by the Brownian noodle. In the limit the approximation involves a noodle $\{S(t)\colon 0\leq t\leq1\}$ of length $\infty$ in a gap of width $\infty$, but in the ratio $x=L/D$. In particular, one obtains that $P(C)\leq (4/n)x\sp 2+\frac23$. The case of large deviations is also considered in detail. Reviewed by L. A. Santalo ************************* Gerry Myerson (gerry@mpce.mq.edu.au)