Scale - Spatial Mathematics: Theory and Practice through Mapping - page 112

National Atlas Mapmaker permits users to change scale, it is appropriate that
it employs bar scale to change in response to scale change.
This observation involving bar scale in relation to map scale change often
becomes important in municipal planning applications. An urban planner
may make a fine map on a huge piece of paper showing plans for a new
shopping mall, complete with new tree location, parking spaces, onsite water
retention plans, and so forth. But, unless that large original contains a bar
scale, it will be wrong when it is reduced in size to present to the municipal
planning commission for consideration. Municipal decision makers need to
have accurate facts, including accurate maps—based on mathematics, in order
to make informed decisions.
5.3 The dot density map: Theory and example
Probably we have all seen maps in the media that portray distributions as
dots in geographic space. Often, these are simply ways to present material
that might otherwise be captured by putting a wall map on a bulletin board
and inserting push-pins at points of interest. In this model, the accuracy of
pin placement determines whether the map is “right” or “wrong.” Too often,
though, this sort of bulletin board model is confused with a powerful, but
perhaps underappreciated, style of map that shows clustering and relies on
change of scale to do so. The latter is called a dot density map; we illustrate
it, in theory and practice, below.
5.3.1 Construction of a dot density map
Figure 5.3 shows the beginning of a dot density map designed to represent
the clustering of population in southeast Michigan. In it, the dots do not
reflect single homeowner locations. A single dot represents some number of
people living within a given county. The county boundaries nest inside the
state boundaries; the county boundaries at the edge match perfectly with the
state boundaries. The dot scatter is assigned to random locations within a
base of polygons (in this case, counties). The bulletin board model described
above is “accurate”; the dot density map is “precise.” Mistakes in reading dot
density maps come about by trying to assign “accuracy” where there may be
little or none, such as in assuming that dot coordinates on the map represent
accurate real-world locations at the latitude-longitude level rather than simply
at the county level.
The concept of clustering is tied to scale and to scale change. The dots in a dot
density map are simply a way to partition data. If the population of a county
is 1,000,000 and a single dot represents a thousand people, then 1000 dots
scattered randomly in that county might be used to represent that county's
population. Viewed on its own, the dot scatter within a county boundary map
such as this has little meaning ( Figure 5.3 ) . The county layer is being used