Adjacency patterns:  one viewpoint.  The dot density map.
It is interesting to consider how to use a GIS to do some analysis of mapped information--using a bit of creative effort.
  • Dot density maps:  layer of randomization, layer of observation--scale change; absolute representation (1 dot represents 1000 people) and relative representation (1 dot represents 0.1% of the population of the state).  Use ArcView.
  • The concept of clustering is tied to scale.
    • Equal Area Projections and dot density maps--one way to look for clustering in geographic space.
      • Select a distribution that can usefully be represented as dot scatter--such as population.
      • Then, choose polygonal nets of at least two different scales--such as state and county boundaries.
      • Let the map with the smallest spatial units (counties) be used as the randomizing layer--the dot scatter is spread around randomly within each unit.
      • View the scatter through polygons (states) that are larger than are those of the randomizing layer (counties).  Map 1 shows the results of removing the county boundaries; Map 2 shows the national picture with state boundaries; and, Map 3 views the dot scatter through the national border lens.  The clustering of dots at the state level means something; at the county level it is merely random.
      • Select an equal area projection (such as an Albers Equal Area Conic for the U.S.).
    • Because the underlying projection is an equal area projection, a unit square (or other polygon) may be placed anywhere on the map and comparisons made between one location and another.  Indeed, urban or rural measures might be held up to a value associated with similar polygon tossed out randomly.
    Interesting reading on related topics:  Mark Monmonier and Harm deBlij, How to Lie with Maps, University of Chicago Press.