Tessellation

A tessellation is a subdivision of a space into nonoverlapping regions that fill the space completely. Tessellations are considered regular if the regions are of the same size and shape (e.g., a remotely sensed image composed of square pixels) and are considered irregular otherwise (e.g., the provinces of Sri Lanka). Tessellations are said to nest if they are organized hierarchically, with the individual regions of a tessellation at one level being subdivided at a lower level; for example, each of the states forming a tessellation of the continental United States may be subdivided into counties. When a tessellation is used to display data collected for its regions by means of shading symbols, it is called a choropleth map.

The most frequently encountered tessellations in human geography are various types of administrative units. However, some phenomena, such as the trade areas of supermarkets in a city, sometimes are modeled as tessellations even though the individual regions are not mutually exclusive. For some tessellations the individual regions might have one or more associated internal entities (e.g., counties and county seats), whereas for others they might not (e.g., census tracts). Even when the phenomenon under consideration does not take the form of a tessellation, it still may be possible to generate a tessellation from the original objects. One example, in which the regions are known as Voronoi (or Thiessen or proximal) polygons, often is used in locational decision making involving service facilities. Here a region is created for each facility by assigning to it all locations in space that are closer to that facility than to any other facility so that the regions collectively form a tessellation.

Because irregular tessellations of administrative units are created by humans, they involve a degree of[Page 482] arbitrariness and are modifiable. For example, gerrymandering is said to occur when boundaries of electoral districts are drawn in a manner that is most advantageous for a particular political party. Often more than one tessellation may exist for a given space. For example, a metropolitan area may be divided into school districts, health districts, police precincts, and so forth. Furthermore, any one of these tessellations may change over time. Reconciling different data collected from more than one tessellation usually is problematic and requires some form of areal interpolation. When choropleth maps are constructed for a variable collected at an individual level using different tessellations, different representations can occur because the tessellations may have different relationships to the underlying spatial pattern of the variable. Similarly, the relationships between variables may also change with different tessellations. The same thing may also be observed when a variable is mapped using tessellations at different levels. These two situations are referred to as the zoning problem and the aggregation problem, respectively; collectively, they are referred to as the modifiable areal unit problem (MAUP).