  A standard scientific study comprises two processes: one is to describe a
thing, and the other is to understand how the thing works. In order to
understand the principle of urban growth, a number of shape indexes are
proposed to describe the size and shape of cities. However, the comparability
of a shape index is often influenced by the resolution of remote sensing images
or digital maps because the calculated values depend on spatial measurement
scales. This paper is devoted to exploring the scaling in classical shape
indexes with mathematical methods. Two typical regular fractals, Koch's island
and Vicsek's figure, are employed to illustrate the property of scale
dependence of many shape indexes. A set of formula are derived from the
geometric measure relation to associate fractal dimension for boundary lines
with shape indexes. Based on the ideas of fractals, two main problems are
solved in this work. First, several different correlated shape indexes such as
circularity ratios and compactness ratios are normalized and unified by
substituting Feret's diameter for the major axis of urban figure. Second, three
levels of scaling in shape indexes are revealed, and shape indexes are
classified into three groups. The significance of this study lies in two
aspects. One is helpful for realizing the deep structure of urban form, and the
other is useful for clarifying the spheres of application of different types of
shape indexes.
