  Understanding distance metrics in high-dimensional spaces is crucial for
various fields such as data analysis, machine learning, and optimization. The
Manhattan distance, a fundamental metric in multi-dimensional settings,
measures the distance between two points by summing the absolute differences
along each dimension. This study investigates the behavior of Manhattan
distance as the dimensionality of the space increases, addressing the question:
how does the Manhattan distance between two points change as the number of
dimensions n increases?. We analyze the theoretical properties and statistical
behavior of Manhattan distance through mathematical derivations and
computational simulations using Python. By examining random points uniformly
distributed in fixed intervals across dimensions, we explore the asymptotic
behavior of Manhattan distance and validate theoretical expectations
empirically. Our findings reveal that the mean and variance of Manhattan
distance exhibit predictable trends as dimensionality increases, aligning
closely with theoretical predictions. Visualizations of Manhattan distance
distributions across varying dimensionalities offer intuitive insights into its
behavior. This study contributes to the understanding of distance metrics in
high-dimensional spaces, providing insights for applications requiring
efficient navigation and analysis in multi-dimensional domains.
