Dynamical Fractal: Theory and Case Study

TL;DR

This paper introduces a theoretical framework for analyzing and extrapolating the evolution of dynamic fractals, applied to urban growth in Boston, using fractal dimensions and differential equations to model changes over time.

Contribution

It develops a novel approach to analyze changing fractals and applies it to urbanization, integrating fractal theory with differential equations for future trend prediction.

Findings

Fractal dimension trends can be modeled with differential equations.

Urban growth exhibits fractal complexity that can be quantified.

Logistic models fit the evolution of fractal dimensions and population.

Abstract

Urbanization is a phenomenon of concern for planning and public health: projections are difficult because of policy changes and natural events, and indicators are multiple. There are previous studies of development that used fractals, but none for this specific problem, nor extrapolating the future trend. In the first part of this paper, we construct a theoretical framework for analyzing dynamic (changing) fractals and extrapolating their future trends based on their fractal dimension, a measure of the complexity of the fractal. We believe this approach holds enormous potential for applications in analyzing changing fractals in the real world, such as urban growth, cells, cancers, etc., all of which are invaluable to research. This theoretical framework may shed light on a factor overlooked in past research: the trend of how fractals change. In the second part of this paper, we apply this theoretical framework to study the urbanization of Boston. We compare several maps and measurements of fractal dimensions, produce code that reads the maps and divides the city into subsections, and ultimately graph the fractal dimension over time using both differential and difference equations. Finally, we postulate the logistic equation as a model to fit the evolution of the fractal dimension, as well as the total population derived from census data, which serves as a component for comparing dynamical fractals.