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Buffon's problem with a pivot needle

Uwe Bäsel
2015 Elemente der Mathematik  
In this paper, we solve Buffon's needle problem for a needle consisting of two line segments connected in a pivot point.  ...  This is the result of the classical Buffon needle problem.  ...  Introduction The classical Buffon needle problem asks for the probability that a needle of length ℓ thrown at random onto a plane lattice R d of parallel lines at a distance d ≥ ℓ apart will hit one of  ... 
doi:10.4171/em/276 fatcat:3pzi3txd5fgfvkxtgt5pmljxw4

Buffon's problem with a long needle

Persi Diaconis
1976 Journal of Applied Probability  
A needle of length l dropped at random on a grid of parallel lines of distance d apart can have multiple intersections if l > d.  ...  Acknowledgment The author is grateful to Professor Herbert Solomon for suggesting the problem and to Professor Bradley Efron for a flash of insight. Mr.  ...  Using this last bound in the second integral in (2.4) leads to Buffon's problem w i~ha long needle But -I ) , ; ) -2 k 1 B ( ! ( k-I ) , !  ... 
doi:10.2307/3212484 fatcat:sptr5uitpfebzpst5v5xhbka3e

Buffon's problem with a long needle

Persi Diaconis
1976 Journal of Applied Probability  
A needle of length l dropped at random on a grid of parallel lines of distance d apart can have multiple intersections if l > d.  ...  Acknowledgment The author is grateful to Professor Herbert Solomon for suggesting the problem and to Professor Bradley Efron for a flash of insight. Mr.  ...  Using this last bound in the second integral in (2.4) leads to Buffon's problem w i~ha long needle But -I ) , ; ) -2 k 1 B ( ! ( k-I ) , !  ... 
doi:10.1017/s002190020010419x fatcat:sa4vukjj2fg65ff25udi24aiwm

Buffon's problem with a star of needles and a lattice of parallelograms [article]

Uwe Bäsel
2012 arXiv   pre-print
A star of n (n greater than or equal to 2) line segments (needles) of equal length with common endpoint and constant angular spacing is randomly placed onto a lattice which is the union of two families  ...  In [8] , Buffon published the solution of his famous needle problem. It is the calculation of the probability of the event that S 2, ℓ intersects R a .  ...  Duma and Stoka [9] solved the problem for ellipses and R a, b, π/2 . Ren and Zhang [11] and Aleman et al.  ... 
arXiv:1209.5241v1 fatcat:aomtnyxdfbho5o5g4goywlp6cy

On a funicular solution of Buffon's "problem of the needle" in its most general form

J. J. Sylvester
1890 Acta Mathematica  
Buffon's problem of the needle, it will be seen, has now expanded into a problem of n needles rigidly connected, which may be treated as a corollary to that of n entirely separate general contours, the  ...  It is indeed a romantic incident in mathematical history that BuF~o~'s problem of the needle should have led up (as is undoubtedly the case) to CaoF'ro~'s new and striking theorems in the integral calculus  ... 
doi:10.1007/bf02413320 fatcat:h6qaf2uymbeuno33cecpki2rtm

The chance that a convex body is lattice-point free: A relative of Buffon's needle problem

Imre Bárány
2007 Random structures & algorithms (Print)  
The following question, which is a distant relative of Buffon's needle problem, emerged while investigating [BM] the randomized integer convex hull, I L (K) = conv(K ∩ L) of a convex body K ⊂ R d .  ... 
doi:10.1002/rsa.20138 fatcat:rxycptw6lvgzhdlupzcuqdraoe

Bayesian point estimation of Π by Buffon's needle and the inverse problem of measuring related length

Hiroshi Sugiyama
1986 Journal of Mathematical Analysis and Applications  
The fundamental Buffon's needle problem is the following. where p is the probability of a needle of length I randomly thrown on the floor, with equally spaced parallel straight lines with width a, crossing  ...  On the other hand, it has elapsed more than 200 years since the discovery of the following relationship by the French naturalist Buffon in 1773, and the Buffon's needle problems became more and more important  ...  Therefore, the present author wishes to extend our Bayesian approaches for estimating the value of 71, based upon the original Buffon's single grid system, to the modified methods based upon double and  ... 
doi:10.1016/0022-247x(86)90144-7 fatcat:yaz6igv6dfgc5mo7iki3upjxji

A Random-Line-Graph Approach to Overlapping Line Segments [article]

Lucas Böttcher
2020 arXiv   pre-print
We study graphs that are formed by independently-positioned needles (i.e., line segments) in the unit square.  ...  Similar to a variant of Buffon's needle problem that considers curved needles (i.e., "Buffon's noodle problem"), future studies may explore variations in the shape distribution of lines that form RLGs.  ...  A and obtain p(a) = 2a 2 π . (1) Note that the derivation of Eq. (1) is structurally similar to the derivation of the intersection probability in Buffon's needle problem where needles of length a are dropped  ... 
arXiv:1911.10679v2 fatcat:r3bcb5bjefhxzn2pwp6cuv45fm

A remark on buffon's needle problem

Ulrich Abel
1984
A Remark on Buffon's Needle Problem Let Gn be a grid in the Euclidean space Rn with coordinates xx,x29 ...9xn determined by hyperplanes parallel to the hyperplanes with the equation Xi 0 separated by  ...  In his generalization of Buffon's Needle Problem to n dimensions Stoka [2] gives an expression for the probability that a segment co of length L which will be "thrown" in a random fashion into the _Rn  ... 
doi:10.5169/seals-38015 fatcat:kyhui2up25axjiy7cbecnuz7bi

Buffon's Problem with a Star of Needles and a Lattice of Rectangles II

Uwe Bäsel, Vittoria Bonanzinga, Lucia Fiorino
2009 unpublished
In this paper we study Buffon type problems with multiple intersections for lattices of rectangles and a star consisting of four or six needles as test body.  ...  Stars with six needles Theorem 3.1. A star S 6, is thrown at random onto the lattice R a, b .  ...  We denote by R a, b the lattice of rectangles of sides a and b, λ := /a and µ := /b, where is the length of the needles of the star (see figure 1 ).  ... 
fatcat:ba3ir62upvcm5pwj4uvuugvgnu

Buffon's Needle Algorithm to Estimate π

Chi-Ok Hwang, Yeongwon Kim, Cheolgi Im, Sunggeun Lee
2017 Applied Mathematics  
Buffon's needle experiment was originally devised to get the value of π . With  ...  Figure 1 . 1 Buffon's needle algorithm with spacing a and needle length l . Figure 2 . 2 Buffon's needle problem can be converted to an integration problem.  ...  Introduction Buffon's needle experiment [1] was originally used to provide π .  ... 
doi:10.4236/am.2017.83022 fatcat:4z4v2h54nzdrxkzrjs3hqrkrlu

Removing the inherent paradox of the Buffon's needle Monte Carlo simulation using Fixed-Point iteration method

Maximilian J. Wang, Jin Wang
2014 Proceedings of the Winter Simulation Conference 2014  
However, there is a common misconception concerning the Buffon's needle simulation. Erroneously, the simulation of the needle drop cannot be used to evaluate .  ...  In teaching simulation, the Buffon's needle is a popular experiment to use for designing a Monte Carlo simulation to approximate the number .  ...  BUFFON'S NEEDLE PROBLEM In this section, we introduce the Buffon's needle problem. The solution of this problem is derived through basic probability and elementary calculus.  ... 
doi:10.1109/wsc.2014.7020196 dblp:conf/wsc/WangW14a fatcat:dobycnkwqzaw7ek5wz3sx5eg4y

Throwing Buffon's Needle with Mathematica

Enis Siniksaran
2009 The Mathematica Journal  
We wrote the package BuffonNeedle to carry out the most common forms of Buffon's needle experiments.  ...  It has long been known that Buffon's needle experiments can be used to estimate p. Three main factors influence these experiments: grid shape, grid density, and needle length.  ...  ‡ Introduction Buffon's needle problem is one of the oldest problems in the theory of geometric probability. It was first introduced and solved by Buffon [1] in 1777.  ... 
doi:10.3888/tmj.11.1-4 fatcat:evdxsqgrq5dxpe6d56khdkszia

Geometric probability, psychophysics and invariance [article]

Yuval Hart, L Mahadevan
2019 bioRxiv   pre-print
To test this psychophysically, we use a set of simple experiments to distinguish between probability distributions of planar line images connected with Buffon's needle and Bertrand's paradox, two classic  ...  The copyright holder for this preprint . http://dx.doi.org/10.1101/531889 doi: bioRxiv preprint first posted online Jan. 28, 2019; FIG. 1 . 1 Psychophysics of Buffon's needle. a) Buffon's needle problem  ...  To convert Buffon's needle problem to a visual psychophysical task, we use a variant of the problem that presents the simultaneous realizations of multiple lines.  ... 
doi:10.1101/531889 fatcat:idrl5vlffndsxktlafgocdaxke

A Quantized Johnson Lindenstrauss Lemma: The Finding of Buffon's Needle [article]

Laurent Jacques
2015 arXiv   pre-print
In 1733, Georges-Louis Leclerc, Comte de Buffon in France, set the ground of geometric probability theory by defining an enlightening problem: What is the probability that a needle thrown randomly on a  ...  In this work, we show that the solution to this problem, and its generalization to $N$ dimensions, allows us to discover a quantized form of the Johnson-Lindenstrauss (JL) Lemma, i.e., one that combines  ...  reduce that distortion by a factor √ M . 3 Buffon's needle problem Initial formulation and solution Let us rephrase Buffon's needle problem stated in the Introduction in a more formal way.  ... 
arXiv:1309.1507v6 fatcat:p7slt7fcqbhkzfir6yjbwzlty4
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