Dotting the Dot Map, Revisited

Dot maps show the geographic distribution of a phenomenon in an area by placing dots representing a certain quantity of the phenomenon where it is most likely to occur. The fundamental steps in dot mapping are selecting the dot size, determining the dot unit value, and placing the correct number of dots in a manner that correctly reflects the geographic distribution of features. Selecting the dot size is a subjective decision, but the dot unit value has long been determined with the aid of the Mackay nomograph. Close examination of the nomograph finds it is not appropriate for determining the dot unit value when dot placement is based on computer-generated random numbers that result in overlapping dots. A new graphical aid for the determination of dot unit value was created by modeling aggregate area of dots and amount of dot overlap using a truncated form of the unification equation from probability theory. Aggregate dot areas predicted by this equation were tested against actual random dots created for several common dot sizes, and high agreement was found between measured and predicted aggregate areas. Pseudo-random dot placement with a maximum overlap constraint for dot pairs appears to mimic more closely how cartographers have traditionally placed dots. Pseudo-random dot placement can be thought of as similar to rigid random placement of circles in a square with maximum circle overlap limits from mutually exclusive to completely random dots. Thinking of dot placement in this manner allowed a general equation for aggregate dot area to be devised as a weighted combination of the mutually exclusive and completely random dot endpoint equations. Aggregate areas predicted by this general equation were found to match actual assemblages of pseudo-random dots with differing maximum dot pair overlaps closely.