: The basis for spatial thinking is the structure of space and the operations that we can perform on and in that structure. We can think about spatial structure and spatial operations from a number of perspectives, each of which is built from a root metaphor. For example, geography and cartography give us the map as a way of describing, representing, and understanding spatial relations. Mathematics gives us the analytic power of geometry and topology. Each root metaphor can be expressed in a form that is inherently spatial. For example, the concepts of maps and mappings exist in cartography and mathematics, and they can lead to a remarkable range of representational forms: cartographic maps, tree diagrams, graphs of phase spaces, cross tabulations, flow charts, networks, nonplanar graphs, etc. To illustrate the basis for spatial thinking, we use a combination of the map, geometry, topology, and graphics metaphors. The idea of spatial structure can be understood in terms of sets of primitives and the concepts that can be derived from them. The idea of spatial operations can be understood in terms of the transformations that are possible within the space and the interpretations that can be generated from the spatial structures. Spatial thinking can be decomposed into competencies that allow us to understand four ideas:
(1) we can start with a set of primitives,
(2) to which we can add some languages of space,
(3) from which we can derive spatial concepts, and
(4) on the basis of which we can perform operations.
: 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.