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. Reference:  Mark Monmonier and Harm deBlij, How to Lie with Maps, University of Chicago Press. Adjacency patterns:  another viewpoint.  The clustering of regions. How are regions clustered in space?  Are similar ones next to each other or are dissimilar ones next to each other.  Consider for example some of the on-board population that comes with ArcView.  Open up the Michigan Block Group shape file.  Zoom in on southeastern Michigan. Go to Theme|Query;  click on the little "update values" box.  Then, double click on "black" then click on > then double click on "white". Choose "New Set".  Once the selected set comes up, go to to Theme|Convert to Shapefile.  Then, color the new shapefile with a green interior to produce this map of block groups in which quantity of black exceeds quantity of white. Go to Theme|Query;  click on the little "update values" box.  Then, double click on "white" then click on > then double click on "black".  Choose "New Set".  Once the selected set comes up, go to to Theme|Convert to Shapefile.  Then, color the new shapefile with a purple interior to produce this map of block groups in which quantity of white exceeds quantity of black. A more detailed look at the clustering process might consider separating out those block groups in which black exceeds white and has a neighboring block group in which white exceeds black--denote this situation as BW.  There are then four logical alternatives:  BW, WB, BB, and WW.  On the maps, the Ws are always purple and the Bs are always green.  A deeper shade of purple represents a block group that is itself one in which white exceeds black and is one that has at least one neighboring block group in which black exceeds white.  To pick out the appropriate block groups: BW2:  make the B layer active.  In ArcView 3.2: Go to Theme|Select by Theme.  In the top pull-down, select "intersect"; in the next pull down, select the W layer.  Choose "new set".  When the selected polygons come up, convert them to a shape file and color it a deeper green.  This process will pick out the edge of the B layer that is adjacent to the W layer. WB2:  make the W layer active.  Go to Theme|Select by Theme.  In the top pull-down, select "intersect"; in the next pull down, select the B layer.  Choose "new set".  When the selected polygons come up, convert them to a shape file and color it a deeper purple.  This process will pick out the edge of the W layer that is adjacent to the B layer. Look at the whole map.  Clustering of like groups is evident in the Detroit metro area--similar groups are clustered.  In Washtenaw county some dissimilar groups are clustered, some are not.  Clustering of either similar or dissimilar groups is highest in Detroit--this would fit with field evidence.  This process can be iterated indefinitely (limited by the size of the base file) and creates a sort of "contouring". In terms of simple Medelian conditions, each of the four conditions,  BB, BW, WB, WW would be expected, with no constraints, to occur 25% of the time.  When all the block group outline boundaries are removed from the different colors, it is easy to look at a map of the whole state. Policy makers and municipal authorities may find maps such as these useful. Academic research may modify how to interpret what the adjacencies mean and what sort of quantitative significance to assign to them; it may also consider definitional matters, such as how polygons are adjacent--at corners only, at edges only, or at corners and edges.  Chess analogies are often-used jargon to describe such pattern.  The subject of "spatial statistics" delves more deeply into different measures for clustering and for levels of significance once clustering is found.