Comparison Between Manhattan and Euclidean Metrics in Discrete Spaces: A Computational Analysis of the Growth of Metric Balls

This paper investigates the growth behavior of metric balls in two-dimensional discrete spaces, comparing the Manhattan (L1) and Euclidean (L2) metrics. Through a computational simulation implemented in Python, the number of integer points contained within metric balls of increasing radius centered at the origin was counted over an N × N grid, with N = 301, for radii r ranging from 1 to 60. The results show that both metrics exhibit quadratic growth, but with distinct coefficients. Furthermore, the ratio between the number of points in the Euclidean ball and the Manhattan ball converges to a value close to π/2 ≈ 1.5708, in agreement with classical theoretical results from the geometry of numbers. The numerical and graphical analysis highlights the structural differences between the two metrics, with direct implications for search algorithms, machine learning, and image processing.