The Fractal Nature of Maps and Mapping
  A fractal can be simply understood as a set or pattern in which there are far more small things than large ones, e.g., far more small geographic features than large ones on the earth surface, or far more large-scale maps than small-scale maps for a geographic region. This paper attempts to argue and provide evidence for the fractal nature of maps and mapping. It is the underlying fractal structure of geographic features, either natural or human-made, that make reality mappable, large-scale maps generalizable, and cities imageable. The fractal nature is also what underlies the beauty of maps. After introducing some key fractal concepts such as recursion, self-similarity, scaling ratio, and scaling exponent, this paper demonstrates that fractal thought is rooted in long-standing map-making practices such as series maps subdivision, visual hierarchy, and T\"opfer's radical law. Drawing on previous studies on head/tail breaks, mapping can be considered a head/tail breaks process; that is to divide things around an average, according to their geometric, topological and/or semantic properties, into the head (for those above the average) and the tail (for those below the average), and recursively continue the dividing process for the head for map generalization, statistical mapping, and cognitive mapping. Given the fractal nature of maps and mapping, cartography should be considered a perfect combination of science and art, and scaling must be formulated as a law of cartography or that of geography in general.   Keywords: Scaling of geographic features, map generalization, statistical mapping, cognitive mapping, head/tail breaks 
Bin Jiang
math.HO nlin.AO