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  A highly tailored map projection that distorts area or distance either to promote legibility or to reveal patterns not readily apparent on a more traditional base map. Coined around 1860 to describe comparatively abstract small-scale maps of statistical data, the word cartogram acquired the connotation of a purposeful non-conventional map projection in the 1960s, after Waldo Tobler applied the mathematics of equal-area map projection to maps on which the size of areal units represents a transforming variable such as population or wealth (Snyder, 1993, pp. 262-4).

Among the earliest cartograms were nineteenth-century railroad maps deliberately distorted to make room for the names of closely spaced stations. By blatantly stretching political boundaries and rivers, these maps conveniently portrayed the sponsoring company\'s routes as less contorted and more direct than competing railways (Modelski, 1984, p. xviii). Purposeful stretching is also apparent in the maps of urban mass transit systems, which imitate the London Underground map devised by Harry Beck in 1933. By reducing map scale in the suburbs, where routes diverge and stations are more widely separated, Beck found room for greater detail in the inner city, where stations are more closely spaced and converging routes have complex connections. In a similar vein, the enlargement of very small areal units on a \'visibility base map\' can promote the accurate decoding of all symbols on a choropleth map (Monmonier, 1993, pp. 178-80).

The most widely used cartogram is the \'value-by-area cartogram\', on which the size of each areal unit represents its population or relative importance. As a base map for choropleth symbolization, the value-by-area cartogram avoids the misinterpretation that can occur when the dark or light tones of large but relatively unimportant areal units with extraordinarily high or low rates overshadow the symbols of smaller but markedly more important areal units. A classic example is the electoral cartogram, which provides a more reliable cartographic portrait of national elections than conventional maps, easily dominated by rural trends. By adjusting for geographic variation in population density, the \'demographic base map\' often affords a clearer, more meaningful view of political, economic and mortality data.

Early value-by-area cartograms were so geographically crude that Erwin Raisz (1934), a prominent advocate, chose not to call them maps. As implied by the title of his seminal article, \'The rectangular statistical cartogram\', Raisz drew all areal units — the 48 conterminous United States — as rectangles that abutted along vertical and horizontal boundaries. With few visual cues, the map-reader eager to find a specific state or identify a particular rectangle had to rely on abbreviated names, relative position (interior or periphery) and neighbours.

As Tobler (1963) later pointed out, the value-by-area cartogram is a graphic-geometric problem with many mathematical-computational solutions not constrained by horizontal and vertical lines. Although some solutions are more visually pleasing than others, a multiplicity of areal units and wide variation in density can easily thwart a visually pleasing result (see examples in Tobler, 1973). Because of difficulties in finding aesthetically satisfying computational solutions, map authors eager for a cartogram base map typically have adopted an iterative, largely manual trial-and-error strategy of preserving the readily identifiable caricatures of key areal units wherever possible (Dent, 1972; Eastman et al., 1981).

The cartograms of Raisz and Tobler are contiguous area cartograms in which areal units sharing a common boundary are not allowed to separate. A markedly different approach, the noncontiguous area cartogram, preserves shape by freezing the areal unit with the largest density and allowing the perimeters of other areas to contract inward by reducing scale (Olson, 1976; Jackel, 1997). Easily programmed for a computer, the noncontiguous area cartogram can yield an aesthetically awkward solution in which numerous tiny, scarcely identifiable icons surround a few large, easily recognized shapes. Although moving distant area symbols together can provide a more efficient use of the space, abbreviations might be needed to identify areas lacking a distinct shape or detached from a better-known neighbour. Even so, area cartograms with widely separated icons or badly distorted shapes might be useful if the map author supplements the display with a more familiar reference map.

A similar special-purpose projection is the distance cartogram, which portrays geographic distortions resulting from transportation or telecommunication rates (Monmonier, 1993, pp. 198-200). A tailored variation of the azimuthal equidistant projection, the distance cartogram describes relative accessibility to a focal place by moving interacting places closer together or farther apart in accord with telecommunication rates, travel time, frequency of service, migration rates or some other measure of accessibility or interaction. Distance cartograms are especially useful in pairs, to compare different modes of interaction or to illustrate the \'before and after\' effects of new regulations. (MM)

References Dent, B.D. 1972: A note on the importance of shape in cartogram communication. Journal of Geography 71: 393-401. Eastman, J.R., Nelson, W. and Shields, G. 1981: Production considerations in isodensity mapping. Cartographica 18 (1): 24-30. Jackel, C.B. 1997: Using ArcView to create contiguous and noncontiguous area cartograms. Cartography and Geographic Information Systems 24: 101-9. Modelski, A.M. 1984: Railroad maps of North America: the first hundred years.Washington, D.C.: Library of Congress. Monmonier, M. 1993: Mapping it out: expository cartography for the humanities and social sciences. Chicago: University of Chicago Press. Olson, J.M. 1976: Noncontiguous area cartograms. The Professional Geographer 28: 371-80. Raisz, E. 1934. The rectangular statistical cartogram. Geographical Review 24: 282-96. Snyder, J.P. 1993: Flattening the earth: two thousand years of map projection. Chicago: University of Chicago Press. Tobler,W.R. 1963: Geographic area and map projections. Geographical Review 53: 59-78. Tobler, W.R. 1973: A continuous transformation useful for districting. Annals, New York Academy of Sciences 219: 215-20.

Suggested Reading Dent, B.D. 1996: The cartogram: value-by-area mapping. In Cartography: thematic map design, 4th edn. Dubuque, Iowa: William C. Brown, 202-16. Dorling, D. 1994: Cartograms for visualizing human geography. In H.M. Hearnshaw, and D. J. Unwin, eds. Visualization in geographical information systems. Chichester: John Wiley and Sons, 85-102.



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