Mapping the Fourth Dimension

Our 3-dimensional world is the usual subject of 2-dimensional maps; their purpose is to accurately depict geographic extent on a readable/publishable surface, with or without thematic content.  The purpose of adding the 4th dimension of time is somewhat intuitive- maps of the same area through time can give a viewer an understanding of change in some feature.  The problem in depicting the 4th, temporal, dimension on a 2-dimensional surface is remedied by the use of various spatiotemporal visualization techniques, the most effective of which, it could be argued, is the map animation.  The simplest form of this variety of cartographic visualization is the display of two or more maps, adjacent to one another, showing the same geographic area at different times.  The change in the area depicted is evident in the difference(s) between the various maps.  While this method is effective, actual animation on a map is an even more salient method of displaying areal change through time.  

       
The above graphics represent two years- 1790 and 2000- and contain proportional symbols indicating the most populous cities in the U.S. at those times.  The two maps are actually the beginning and ending frames of an animation, which shows the same map for each 10 year period between 1790 and 2000, and thus gives the viewer an idea of where the major centers of population in the U.S. were for each decade.  The ease of doing so with a simple animation, cycling through each decade, is fairly large in comparison to creating and displaying something singular including separate maps, one for each decade.        

Bivariate Maps of the Choropleth Persuasion

The depiction of multiple variables on the same map, displayed to highlight a relationship between the two, is no simple feat of cartography.  The choropleth map, in particular, is best suited for indicating the geographic prevalence of a single, or single incidence of multiple, phenomena.  Mapping with varying color shades to indicate the prevalence of two separate variables- the areas where there is a high prevalence of one and not the other, and areas where there is a high prevalence of both- is an ambitious task by any estimation.  

The use of the bivariate choropleth map may be employed for this depiction of two separate variables, and thus indicates the relationship between the two.  Above is a representation of both rates of obesity and physical inactivity, by U.S. county, with the use of two different hues with separate quantitative color progressions.  The legend in the lower left illustrates these with a 9-square color palette, which varies by hue according to the X- and Y-axes, and by lightness progressing across both.  As the colors progress towards the upper right corner of the cube, the colors are meant to indicate a high prevalence of both variables.  Thus, the viewer is able to use the 9 different colors there to visualize the spatial relationship between the two.  The current version of ArcMap in use, though, doesn't support the automated creation of this variety of legend, and it must be assembled by hand.  Though creating a 9-color legend, which varies in hue and lightness along two axes, is no easy feat, it is unarguably a valuable practical lesson in applied color theory.