From The Cover: Diffusion-based method for producing density-equalizing maps

Abstract

Map makers have for many years searched for a way to construct cartograms, maps in which the sizes of geographic regions such as countries or provinces appear in proportion to their population or some other analogous property. Such maps are invaluable for the representation of census results, election returns, disease incidence, and many other kinds of human data. Unfortunately, to scale regions and still have them fit together, one is normally forced to distort the regions' shapes, potentially resulting in maps that are difficult to read. Many methods for making cartograms have been proposed, some of them are extremely complex, but all suffer either from this lack of readability or from other pathologies, like overlapping regions or strong dependence on the choice of coordinate axes. Here, we present a technique based on ideas borrowed from elementary physics that suffers none of these drawbacks. Our method is conceptually simple and produces useful, elegant, and easily readable maps. We illustrate the method with applications to the results of the 2000 U.S. presidential election, lung cancer cases in the State of New York, and the geographical distribution of stories appearing in the news.

Fig. 1.

U.S. population cartogram constructed with the method of Gusein-Zade and Tikunov. [Reproduced with permission from ref. (Copyright 1993, American Congress on Surveying and Mapping)]

Fig. 2.

Population cartogram of Britain by county. (Left) The original map. (Right) Cartogram generated with the cellular automaton algorithm of Dorling. [Reproduced with permission from Dorling (6) (Copyright 1996, University of East Anglia)].

Fig. 3.

(a) Population density with Gaussian blur, as described in the text. The width σ of the Gaussian is ≈80 km. (b) Gaussian blur with width ≈8 km. Results of the 2000 U.S. presidential election shown on a standard Albers conic projection (c), on a cartogram based on the population density in a (d), and on a cartogram constructed with the finer population density of b (e). The latter results in greater distortion of some state boundaries, most noticeably for Pennsylvania and Indiana. (f) A cartogram based on states' representation in the electoral college. The density of electors was calculated by spreading each state's electors evenly across the state.

Fig. 4.

Lung cancer cases among males in the state of New York, 1993–1997. Each dot represents 10 cases, randomly placed within the zip-code area of occurrence. (a) The original map. (b) Cartogram with a coarse-grained population density with σ = 0.3°.(c) Cartogram with a much finer-grained population density with σ = 0.04°. (Data are from the New York State Department of Health.)

Fig. 5.

The distribution of news stories by state in the United States. (a) Albers conic projection. (b) Cartogram in which the sizes of states are proportional to the frequency of their appearance in news stories. States are the same shades in a and b.