PUTTING IT ALL TOGETHER--A BASE OF SPATIAL CONCEPTS

During the course of the term, and in the development of your projects, certain themes reappear. These concepts that keep coming up are often useful in providing a broad conceptual framework for more specific work. Think about how these spatial concepts have been a part of your project; for, it is in the context of project development that otherwise abstract themes take on meaning. This framework is more than a simple reiteration of lecture material. The broad ideas cut across the pattern of presentation in order to gain the advantage of looking at the material from a different perspective. Broad themes, common to a wide variety of concerns, can be implemented using any of various tools, including (but not limited to) computerized tools.

Below is a summary of how lecture material fit into one set of spatial concepts. This set of concepts is quite broad, but certainly not all inclusive (nor mutually exclusive), nor is it unique in its capabilities to capture the basic spatial elements of a problem. You may wish to craft a different set of broad spatial concepts into which you work will fall (however, many of these basic concepts are evident in any study that involves spatial components of a problem).

• Position

1. Earth-sun relations.
2. Latitude and longitude

• Distance

1. Distance and scale.
2. Distance and map projections: Sinusoidal, Mollweide, and Goode's homolographic.
3. Manhattan Distance

• Direction/Orientation

1. Jordan Curve Theorem. Basic theorem to separate inside from outside of curves.
2. Four seasons and general astronomical configuration.

• Parallelism

1. Manhattan space
2. Non-Euclidean spaces

• Scale

1. Scale transformation: from large to small scale using integral dimensions
2. Scale transformation: from small to large scale using fractional dimensions…an application to marina construction using fractals to achieve boundary compression.

• Transformation

1. D'Arcy Thompson's fish transformations: projection of biological species shape
2. One Point Compactification Theorem: shows that there can never be a perfect map--that it is impossible to unroll the sphere onto the plane. This demonstration employed stereographic projection of the sphere to the plane--all but one point (the center of projection) maps to the plane.
3. Varieties of map projections. Issues involving suitability of choice in projection.
4. Maps on developable surfaces: cylinder, torus, Mobius strip, Klein bottle. Conic projections. Cylindrical projections.

• Basic dimensional units

1. Point. Measures of scatters of points using centroids migrating over time, centroids based on a grouped mean (and when that grouping is on spatial bases, the resulting centroid is also referred to as the "spatial mean").
2. Line. Latitude and longitude.
3. Area.

• Four Color Theorem. Four colors are necessary and sufficient to color a map in the plane.
• Spatial autocorrelation--clustering of regions of similar or dissimilar character. Measured using the join-count statistic. Expressed using graph theory and the concept of adjacency.
Volume.

• Latitude and longitude.
• Earth-sun relations.
• Map-coloring on sphere…application of One Point Compactification Theorem to coloring.
• Map-coloring on a torus.
• Three dimensional maps.
• Partition

1. Jordan Curve Theorem. Partition of the plane by a curve into mutually exclusive, exhaustive sets.
2. Thematic mapping categories; choice of partition of data (as quantiles, equal area, and so forth) yields vastly different maps.
3. Contours: lines that separate the plane into regions. They are also lines that partition volume.

• Diffusion

1. General concept of diffusion as change in spatial position over time.
2. Hagerstrand's simulation of the diffusion of an innovation.

Given current technology...

1. Students, by the fifth (second?) grade, are competent to make thematic maps using a computer. Doing so reinforces skills in arithmetic and offers a rich source of projects in which to guide further mathematical development of students at a young age (calculating simple statistics; using set theory and principle of inclusion and exclusion; seeing differences between rational and irrational numbers).

It also fosters a strong awareness of the world around them and helps to make them informed, responsible citizens interested in promoting, from various perspectives, the welfare of their planet.

2. Presence of courses in undergraduate/graduate curriculum that deal with some of the more difficult cartographic aspects of mapping. Presented in parallel with the mathematical development of students. Thus, one course after this one might focus entirely on the mathematics of map projections. Another might focus on the historical development of mapping and the importance of archiving maps as well as software. Much of the new mapping capability ignores capturing the efforts of previous research. This effort needs to be made to avoid repeating, as much as possible, the errors of the past.

3. Research interests in a variety of disciplines might probe more deeply the spatial components of that interest. Indeed, the analysis of the spatial components, using whatever associated mathematical or other tools seem appropriate, can lead the research direction. Interests that involve both the spatial and the temporal component (as for example in diffusion) offer an opportunity to link the geographical with the historical and a broadened picture of events.