Map makers have long 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, in order 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 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 new 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 US presidential election, lung cancer cases
in the State of New York, and the geographical distribution of stories appearing in
the news.