Buffon's Needle Problem Research Articles

On the number of tiles visited by a line segment on a rectangular grid

AbstractConsider a line segment placed on a two‐dimensional grid of rectangular tiles. This paper addresses the relationship between the length of the segment and the number of tiles it visits (i.e., has intersection with). The square grid is also considered explicitly, as some of the specific problems studied are more tractable in that particular case. The segment position and orientation can be modeled as either deterministic or random. In the deterministic setting, the maximum possible number of visited tiles is characterized for a given length, and conversely, the infimum segment length needed to visit a desired number of tiles is analyzed. In the random setting, the average number of visited tiles and the probability of visiting the maximum number of tiles on a square grid are studied as a function of segment length. These questions are related to Buffon's needle problem and its extension by Laplace.

The Buffon's needle problem for random planar disk-like Cantor sets

We consider a model of randomness for self-similar Cantor sets of finite and positive 1-Hausdorff measure. We find the sharp rate of decay of the probability that a Buffon needle lands δ-close to a Cantor set of this particular randomness. Two quite different models of randomness for Cantor sets, by Peres and Solomyak, and by Shiwen Zhang, appear to have the same order of decay for the Buffon needle probability: clog⁡1δ. In this note, we prove the same rate of decay for a third model of randomness, which asserts a vague feeling that any “reasonable” random Cantor set of positive and finite length will have Favard length of order clog⁡1δ for its δ-neighbourhood. The estimate from below was obtained long ago by Mattila.

The Buffon needle problem for Lévy distributed spacings and renewal theory

What is the probability that a needle dropped at random on a set of points scattered on a line segment does not fall on any of them? We compute the exact scaling expression of this hole probability when the spacings between the points are independent identically distributed random variables with a power-law distribution of index less than unity, implying that the average spacing diverges. The theoretical framework for such a setting is renewal theory, to which the present study brings a new contribution. The question posed here is also related to the study of some correlation functions of simple models of statistical physics.

Upper and lower bounds on the rate of decay of the Favard curve length for the four-corner Cantor set

The Favard length of a subset of the plane is defined as the average of its orthogonal projections. This quantity is related to the probabilistic Buffon needle problem; that is, the Favard length of a set is proportional to the probability that a needle or a line that is dropped at random onto the set will intersect the set. If instead of dropping lines onto a set, we drop fixed curves, then the associated Buffon curve probability is proportional to the so-called Favard curve length. As we show in our companion paper, a Besicovitch generalized projection theorem still holds in the setting where lines are replaced by curves. Consequently, the Favard curve length of any purely unrectifiable set is zero. Since the four-corner Cantor set is a compact, purely unrectifiable $1$-set with bounded, non-zero Hausdorff measure, then its Favard curve length equals zero. In this article, we estimate upper and lower bounds for the rate of decay of the Favard curve length of the four-corner Cantor set. Our techniques build on the ideas that have been previously used for the classical Favard length.

The object detection by autonomous apparatus as a solution of the Buffon needle problem

The object detection by autonomous apparatus as a solution of the Buffon needle problem

Laplace type problem with non-uniform distribution

In some previous papers the authors consider some Laplace type for different lattice, in particular in (7) the authors consider a Laplace type for a trapezoidal lattice with rectangle body test. In this paper we consider a lattice with fundamental cell composed by a trapetium but for the first time we consider as body test a random rectangle not uniformly distributed. We compute the probability that a random rectangles of constant sides intersects the a side of lattice when the position of rectangle is a random variable with exponential and distibution. I. INTRODUCTION In 1773, at a meeting of the Academic des Sciences de Paris, Buffon posed a that later on should become knows as the famous Buffon needle problem : in a room, the floor of which is merely divided by parallel lines, at a distance a apart, a needle of length is allowed to fall at random: which is the probability that the needle intersects one of the lines? The solution, determined by Buffon by means of empirical methods, was . The and its solution were published in 1777, in the Comptes rends de l'Academie des sciences de Paris . In 1812, Laplace extended the by considering a room paved with equal tiles, shaped as rectangles of sides and , with The solution was , and it is obvious that the probability of Buffon can be obtained from that of Laplace by letting .

Buffon’s problem with a pivot needle

In this paper, we solve Buffon's needle problem for a needle consisting of two line segments connected in a pivot point.

A Brief History of the Most Remarkable Numbers $$\pi $$ π , $$g$$ g and $$\delta $$ δ in Mathematical Sciences with Applications

This paper deals with a brief history of the most remarkable numbers $$\pi , g \text {and} \delta $$ in mathematical sciences with their many examples of applications. Several series, products, continued fractions and integral representations of $$\pi $$ are discussed with examples. The celebrated Newton method of approximation of $$\pi $$ to many decimal places is included. The appearance of $$\pi $$ in many problems, formulas, elliptic integrals and in probability and statistics is presented with examples of applications including the Tchebycheff problem of prime numbers, the Buffon needle problem and the Euler quadratic polynomial. The golden number $$g$$ and its applications to algebra and geometry are briefly discussed. The Feigenbaum universal constant, $$\delta $$ is discovered in 1978 and it is found to occur in many period doubling bifurcation phenomena in the celebrated logistic map and the Lorenz differential equation system with chaotic (or aperiodic) solutions. Included is a numerically computed Lorenz attractor which resembles a butterfly or figure eight. The Lorenz attractor is a strange attractor because it has a non-integer (or fractal) dimension. The major focus of this article is to provide basic pedagogical information through historical approach to mathematics teaching and learning of the fundamental knowledge and skills required for students and teachers at all levels so that they can understand the concepts of mathematics, and mathematics education in science and technology and pursue further research.

Simulations of Linearity of Buffon’s Needle Problem

The expected number of crossings of Buffon's needle problem is linear if the distance d between the lines of the ruled paper is less than or equal to the length l of the needle. With the help of Dynamic Geometry Super Sketchpad software, two vivid simulations of the linearity of Buffon's problem are given, that is, the expected number E(l ) satisfies E(x+y) = E(x) + E(y) and E(rx) = rE(x).

A generalization of Buffon's needle problem

A generalization of Buffon's needle problem

How needless are Buffon's needles?

The study of statistical physics requires introductory preparation regarding probability theory. Understanding its fundamental concepts (randomness, distributions, fluctuations), and some experience in application of the basic concepts of statistics can be obtained in several ways. We found that the basic training in probability and statistics needed for physics and engineering study can be achieved by focusing on Buffon's needle problem. We believe this approach could help university specialists make study more efficient when probability and statistics play an important role. Buffon's experiment, with its convincing simplicity and flexibility, as well as its attractiveness, is in our opinion a useful tool in physics education at university level.

On generalized buffon needle problem for lattices

Two new concepts, the generalized support function and restricted chord function, both referring to a convex set, were introduced in [1]. General formulae to yield the kinematic measure of a segment of fixed length in a convex set were established based on these concepts. In this article, using the partial intersection method, we consider the generalized Buffon problem for three kinds of lattices. We determine the probability of intersection of a body test needle of length l, l < α.

The Buffon Needle Problem Revisited in a Pedagogical Perspective

The Buffon Needle Problem Revisited in a Pedagogical Perspective

Four Shots for a Convex Quadrilateral

probability of a re-entrant quadrilateral.) This problem has an interesting history [4] which is intimately tied to the development of the notion of probability mea sure. In fact, Sylvester's problem, together with Buffon's needle problem, is one of the prime paradigms in geometric probability theory. In 1917 Blaschke [1], [2, ??24? 25] proved that the probability of K is maximal (namely 0.70448 ... ) when the shape from which the points are chosen is an ellipse and minimal (namely |) when it is a triangle. Blaschke's results refer to convex shapes only; for nonconvex shapes the situation becomes combinatorially more challenging, see [5]. In geometric probability theory the random objects (points, lines, etc.) are usu ally assumed to be uniformly distributed with respect to a natural geometric measure (angle, area, etc.), so the relevant probabilities are directly proportional to geometric quantities like area. Let us now imagine a marksman firing four shots at the origin of the (x, ;y)-plane. Under these circumstances the four bullet holes will not be uniformly distributed on a circle or a square. It is much more natural to model their distribution by a bivari ate Gaussian whose parameters depend on the markman's skill, the quality of his rifle and maybe other factors. So we pose Sylvester's question under these new assump tions: What is the probability that the four bullet holes form the vertices of a convex quadrilateral? In this note we shall prove the following result:

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

Given a convex body K ⊂Rd, what is the probability that a randomly chosen congruent copy, K*, of K is lattice-point free, that is, K*∩Zd = ∅︁? Here Zd is the usual lattice of integer points in Rd. Luckily, the underlying probability is well defined since integer translations of K can be factored out. The question came up in connection with integer programming. We explain what the answer is for convex bodies of large enough volume. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007

W3MCSim: an online and reconfigurable Monte Carlo simulator for interactive probabilistic/statistical modeling

We have implemented a Monte Carlo simulator (W3MCSim) as an Internet software tool, primarily for interactive use by students, educators, life and other physical scientists, as well as other practitioners of probabilistic and statistical modeling. Interested users download, install and run W3MCSim by visiting the application website. This application incorporates three freely available Microsoft web technologies, namely the Internet Explorer web browser, the Component Object Model software framework and the JScript web page script interpreter. We define the software architecture here, as a web application model, and show how incorporation of these technologies provides an efficient solution to W3MCSim software deployment. We demonstrate the usability and versatility of this simulator with three distinct tutorial examples: simulating the sum of six-sided dice, estimating intersection frequency in "Buffon's needle problem", and testing an animal experiment design model a priori. We also show how the program components can be reconfigured into other programs.

Stereological tools in biomedical research.

Stereological studies are more and more frequent in literature, particularly in the development/evolution, pathology, and neurosciences areas. The stereology challenge is to understand the structural inner three-dimensional arrangement based on the analysis of the structure slices only showing two-dimensional information. Cavalieri and Scherle's methods to estimate volume, and Buffon's needle problem, are commented in the stereological context. A group of actions is needed to appropriately quantify morphological structures (unbiased and reproducibly), e.g. sampling, isotropic and uniform randomly sections (Delesse's principle), and updated stereological tools (disector, fractionator, nucleator, etc). Through the correct stereology use, a quantitative study with little effort could be performed: efficiency in stereology means a minimum slices sample counting (little work), low cost (slices preparation), but good accuracy. In the present text, a short review of the main stereological tools is done as a background basis to non-expert scientists.

The Buffon's needle problem and the design of a geophysical survey

A logical procedure for designing a geophysical survey when sampling an area with a regular grid can be summarized as follows: model the expected anomaly, estimate the expected noise level, estimate the area of the anomaly above the noise level, choose the spacing, in both the x‐ and y‐directions, of the measurement grid. This last step can be approached according to two main strategies: either when applying the sampling theorem to the shortest dimension of the anomaly or when using a coarser grid, leaving a more complete definition of the anomaly to a later fitting. When following this second option, it can be constructive to estimate the probability of intercepting a given anomaly with a specific segment of profile and a given profile spacing. This latter procedure is analysed by considering a rectangle approximating the plane projection of the anomaly shape and taking into account various ratios between the grid spacing and the rectangle sides. The formulae for estimating the probability of intersecting a given anomaly with a given segment of a given profile spacing are calculated. To demonstrate the accuracy of the results, a Monte‐Carlo simulation on a synthetic magnetic map was performed, obtaining, for different ratios between the sides, segment length and profile interval, an agreement better than 0.1% with the analytical formulae.

Buffon's needle problem with a twist

Some simple variations of Buffon's well-known needle problem in probability are discussed, and an interesting observation connecting the corresponding results is then made.

Buffon's Needle and the Probability of Intercepting Short‐Distance Trips by Multiple Screen‐Line Surveys

For transport planning purposes, information on origin‐destination movements may be obtained by stopping traffic at the roadside and interviewing drivers. The roadside interview stations are usually located so that they lie on one of a number of lines (called screen lines) that cross from one side of the survey area to another. In this way, all movements with the origin on one side of a screen line and the destination on the other are intercepted. For regional surveys, a grid of screen‐lines may be used. The movements intercepted are not, however, representative of all those in the region. That is because shorter‐distance movements are underrepresented; the coarser the grid, the worse the effect. The extent of this underrepresentation is estimated in this paper by calculating the probability of intercepting a trip of given direct length, under the assumption that the screen lines constitute a uniform rectangular grid, and that trips of a given length are distributed over the region at random. The result is an extension to the Buffon needle problem. Ways in which such a result, obtained for an idealized situation, may be extended to apply to more realistic situations, are discussed: in particular it is shown how the trip‐length frequency distribution of all trips may be estimated from that of intercepted trips, using a simple formula that is applicable to a much more general pattern of screen lines.