Andy Woodruff discusses the value-by-alpha map, an alternative to the cartogram that he, Robert Roth and Zachary Johnson have developed (and have written a paper about in The Cartographic Journal): “[V]alue-by-alpha is essentially a bivariate choropleth technique that ‘equalizes’ a base map so that the visual weight of a map unit corresponds to some data value. Whereas cartograms accomplish this by varying size, VBA modifies the alpha channel (transparency, basically) of map units overlain on a neutral color background. Thus shapes and sizes are not distorted (except necessarily by the map projection, of course), but the lower-impact units with lower alpha values fade into the background and make for a map that is visually equalized by the data.” An example of which (at right) shows the 2008 presidential election results, where brighter colours reflect counties with larger populations.