Efficacy of 3D Monte Carlo Simulations vis-à-vis 2D in the Estimation of Pi: A Multifaceted Approach

Abstract

$Pi(\pi )$ occupies a central place in multiple physical phenomena such as diffraction, radiation, quantum mechanics and in various applications such as crystallography, Fourier analysis etc. Hence a suitable simulation based numerical/statistical method to determine it accurately is of great significance. Keeping this objective into consideration, we describe a simulation-based computational method to implement Monte Carlo (MC) simulations in a non-Euclidean 3D geometry for the first time in this paper. To visualize the MC simulations and generate data, codes in the python programming language were developed efficiently. The data were generated by (i) varying the number of simulations, that is, the number of random data points, and (ii) repeating the process I times (iterations) for a fixed number of simulations and pooling the data generated from (i) and (ii). A dual approach was adopted to analyze the data with the help of statistical methods implemented through python code, which comprised random sampling using the ‘central limit theorem’ (CLT) and ‘clustering technique’. Similar extensive studies were carried out for Euclidean 2D geometry, and the results were analyzed and compared through a detailed error analysis. It has been reported that with an increase in the number of simulations/iterations, the value of pi attains better stabilization and conforms better to the normal distribution. We report that the errors are $\propto {n^{-0.5}}$ as a function of number of simulation n for 3D geometry, whereas the absolute error $\epsilon (n)$ decreases exponentially in the case of 2D geometry. All the results obtained through the dual approach were consistent for both the geometries, and a stabilization in pi value up to $10^{-5}$ in the 2D geometry and up to $10^{-4}$ in the 3D geometry was achieved using random sampling. We also report that the simulation developed in our work has yielded better and more accurate mean values of pi between (0.99979–0.99998)$\pi$ for both the geometries and has ‘least error’ compared to the work reported by others in literature.