  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.
