Sesión DivulgaciónDropping Needles, Casting Candies, and Rolling Dice: Unconventional Pathways to Demystifying \(\pi\)
Miguel Aguiar Carvalho dos Santos
Instituto Federal do Paraná - Campus Jaguariaíva, Brasil - Esta dirección de correo electrónico está siendo protegida contra los robots de spam. Necesita tener JavaScript habilitado para poder verlo.
Unconventional numbers such as \(\pi\) and e can be particularly challenging for young mathematics students, being an opportunity for mathematical communication instead of an obstacle. Those numbers, said transcendental, did not present integer roots challenging our orthodox understandings of infinity as an exceptional catalyst for mathematical outreach. Originating from ancient Greek geometry, the concept of \(\pi\) was initially understood as the elementary, tangible ratio between a circle’s circumference and its diameter. When in the 19th century researchers proved \(\pi\) is transcendental, the concept of it being a simple geometric measurement changed, becoming a profound symbol of the limits of rational calculation. This study investigates three unconventional methodologies for estimating \(\pi\): the Geometric Area Ratio, the Buffon’s Needle Problem (using probability and shapes), and the Coprimality of Random integers (looking at numbers that share no common factors). We compare these methods through physical experimentation and Monte Carlo simulations (computational randomized testing), evaluating their convergence rates (how fast they get to the right answer), standard deviations (how much the results vary), and relative efficiencies. To implement these experimental approaches. The Geometric Area Ratio uses the comparison of the areas within a bounded geometry. In this method, \(n\) items (gummy candies) are randomly distributed over a square of side r encompassing a quarter of a circle of radius the same length as the square side. The probability \(P\) of a drop falling within the target region, given the target area Atarget (quarter of the circle with radius \(r\)) and the area of the square with side \(r\), Asquare, is given by:
\[P = \frac{A_{\text{target}}}{A_{\text{square}}}\]
Thus, \(\pi\) is estimated from the ratio of successful hits (\(N_hits\)) to the total number of drops (\(N_{\text{total drops}}\)): \[\pi \approx 4\frac{N_{\text{hits}}}{N_{\text{total drops}}}\] Buffon’s needle method is a classic geometric probability experiment. Considering a needle of length \(1\) and a plane marked with parallel lines separated by distance \(d\) (\(l \le d\)), the probability of a dropping needle crossing a line is: \[P = \frac{2l}{\pi d}\]
Finding \(\pi\) involves performing \(n\) drops and calculating: \[\pi \approx \frac{2ln}{d \cdot \text{crossings}}\] Coprimality of Random Integers is based on the probability that two randomly chosen large integers are coprime (meaning they share no common divisors other than 1) is \(\frac{6}{\pi^2}\). So \(\pi\) is found by \[\pi \approx \sqrt{\frac{6}{\text{probability of coprimality}}}\] The probability of coprimality is given by selecting n pairs of integers, each one generated by rolling three 10-sided dice, finding a random 3-digit natural number, and determining how many are coprime (using a standard method called the Euclidean algorithm).
Finding \(\pi\) now involves probability, convergence problems, and computational methods. The performance of the estimation methods is modeled using a standard probability formula, the Binomial distribution \(X \sim B(n,p)\), where n represents the total number of trials and p the success probability. The precision of each approximation was evaluated by applying error propagation and the Delta Method to track how small mistakes carry over, using the standard Monte Carlo Method. The standard deviation (the spread of our results) for all estimators shrinks predictably is dictated by a fundamental rule of statistics called the Central Limit Theorem and the standard error of the binomial proportion. In conclusion, while all three evaluated methodologies share the same asymptotic convergence rate governed by the Central Limit Theorem, the Coprimality of Random Integers method demonstrates higher numerical efficiency and stability in computational simulations. Despite this, the inherent didactics of using items like gummy candies provide an intuitive, hands-on way to understand abstract probability that strictly computational methods cannot replicate. Beyond these technical performance metrics, engaging students and the general public with these unconventional estimation techniques serves as a powerful instrument for mathematical outreach. By connecting these abstract concepts, like transcendental numbers, with playful and tangible experiments—such as casting candies, dropping needles, or rolling dice—we demystify higher mathematics, transforming intimidating theoretical limits into accessible, captivating experiences that foster widespread scientific curiosity.
Trabajo en conjunto con: Kaike Nogueira Krzyzanowski (Instituto Federal do Paraná - Campus Jaguariaíva, Brasil), Ray Augusto de Miranda (Instituto Federal do Paraná - Campus Jaguariaíva, Brasil), Tiago Henrique Gonsalves da Silva (Instituto Federal do Paraná - Campus Jaguariaíva, Brasil) y Leandro Mariano (Instituto Federal do Paraná - Campus Jaguariaíva, Brasil).