Scale - Spatial Mathematics: Theory and Practice through Mapping - page 116

In a dot density map satisfying Principle 1, if a more global layer is composed
of polygons that contain the randomizing layer in a nested fashion, then there
is no problem of overlapping regions and possible confusion about the assign-
ment of dot to polygon. Thus,
Principle 2: Optimization principle
A nested hierarchy of layers provides optimized assignment of dots to
more global layers.
Often, unfortunately, one is not able to obtain data arranged in a nested spa-
tial hierarchy, such as the state and county units, tract and block group units,
or another nested hierarchy. It may be desired, for example, to use Census
data to obtain information about zip code polygons, school districts, minor
civil divisions, and so forth. Census boundaries, however, are not generally
commensurate with zip code boundaries and various other spatial units. In
cases such as this, numerous creative approaches may be needed to align data
sets to make appropriate comparisons.
In creating a dot density map, randomizing at the most local level is often,
but not always, best. The reason it may not be best often involves zooming
out so much that the highly localized randomization introduces clutter into
the map—the dots create one big blob on the map. Conceptually, to reduce
dot clutter, one might randomize at a more global level or one might alter
what the dots represent. Generally, it is best to alter what the dot represents,
instead of its position.
Principle 3: Scale principle
When a change in scale produces dot clutter from dot density, random-
ize at the most local scale available and alter dot representation to
retain a dot density map satisfying Principles 1 and 2.
Map projections on which a geometric unit square represents the same
amount of geographic area, independent of position, are called “equal area
projections.” On an equal area projection, relative land mass sizes appear as
they do on the globe: Brazil is larger than Greenland by nearly 21 times, but
on maps that are not in equal area projections, Greenland appears larger, for
example. Dot density maps can be used to make comparisons. Unit (or other)
squares represent the same amount of area on the Earth and comparisons at
different latitudes are valid.
Principle 4: Projection principle
A dot density map must be based on an equal area projection.
If any projection other than an equal area projection is employed in making
a dot density map, then comparisons involving area will be in error. This
principle is of particular importance in constructing global maps and is less
important in constructing local maps.