  A type of fractal dimension definition is based on the generalized entropy
function. Both entropy and fractal dimension can be employed to characterize
complex spatial systems such as cities and regions. Despite the inherent
connect between entropy and fractal dimension, they have different application
scopes and directions in urban studies. This paper focuses on exploring how to
convert entropy measurement into fractal dimension for the spatial analysis of
scale-free urban phenomena using ideas from scaling. Urban systems proved to be
random prefractal and multifractals systems. The entropy of fractal cities
bears two typical properties. One is the scale dependence. Entropy values of
urban systems always depend on the scales of spatial measurement. The other is
entropy conservation. Different fractal parts bear the same entropy value. Thus
entropy cannot reflect the spatial heterogeneity of fractal cities in theory.
If we convert the generalized entropy into multifractal spectrums, the problems
of scale dependence and entropy homogeneity can be solved to a degree for urban
spatial analysis. The essence of scale dependence is the scaling in cities, and
the spatial heterogeneity of cities can be characterized by multifractal
scaling. This study may be helpful for the students to describe and understand
spatial complexity of cities.
