which we are operating (by zooming in or out) or change the distance metric (e.g., using a Manhattan or city-block metric versus a Euclidean or as-the-crow-flies metric) or change the dimensionality of the space (collapsing from three to two dimensions).

Through processes of simplification, generalization, and classification, we can identify patterns in distributions of objects (see Chapter 3.8). We could describe patterns as random versus systematic, recognizing that these descriptions suggest something about the processes that may have given rise to the patterns, thus linking space and time. Systematic patterns can be clustered or uniform; uniform patterns in two-space can be built on either a rectangular or a triangular lattice. Shapes and patterns can display symmetry or be asymmetrical. We can look for outliers to patterns, breaks or discontinuities, and distortions in portions of the pattern.

We could identify higher-order structures in the spatial structure such as systems, networks, or hierarchies based on concepts of sequence, linkage, dominance, and subordination. We can overlay sets of objects in the same space, looking for associations and correlations, or disaggregate complex spatial patterns into separate layers. We look for correlations (positive or negative) between layers. We can identify—and try to interpret—outliers or exceptions that do not conform to a pattern. We can interpolate between or extrapolate from objects. We can bring to bear interpretive axioms: for example, nearby objects are likely to be similar, but closer objects are likely to be more similar. From this we can consider nearest neighbors, distance decay effects, spatial autocorrelation, and so forth. (All of these operations can be performed on a GIS working with geospatial data; see Chapters 7 and 8.)

At this point, the basis for the power of spatial thinking is clear: it lies in the range of operations that we can bring to bear on the description and explanation of spatial structures and the range of representations that we can use to capture those spatial structures. We can appreciate that power in another way, as well. This discussion of three sets of ideas—the language of space, spatial concepts, and operations—is based on only one member of the set of four primitives—spatial location. Each of the other three primitives—identity, magnitude, and temporal specificity and duration—can be approached spatially. Thus, identity gives rise to taxonomies and a range of spatial representations can be used to express the structure of classifications (trees, Venn diagrams, etc.). We can capture branching relations and ordination (super- and subordinates) and think about families, hierarchies, etc. The property of time gives rise to ideas such as growth, change, and development, all of which can be spatialized and represented. Magnitude can be considered an ordered series and therefore easily spatialized.

2.6 THE PROCESS OF SPATIAL THINKING

Spatial thinking is not unitary in character and operation (as demonstrated in Section 2.2.2). It can appear in many flavors and varieties—some appropriate for one task, some for another. For example, mental rotation is involved in describing the world as it appears from another’s point of view, while distinguishing figure from ground is involved in finding a face in a crowd. Individuals may excel at some aspects of spatial thinking and not at others. Facility in using the components increases with experience, most obviously expressed in expertise in a knowledge domain, such as finding tumors in X-rays, inferring the presence of oil-bearing strata in a geological cross section, or imagining three-dimensional shapes from two-dimensional architectural drawings (see Chapter 3).

What follows is a framework for organizing the components of spatial thinking (Tversky, 2005). Any complex spatial reasoning task, such as comprehending a weather map or planning a route, will use several components in concert. To characterize the nature and varieties of spatial thinking, we have to make a distinction about thinking in general.



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