Conceptual Material:  More of the MiniMax Principle from the abstract to the "real."
     Spatial analysis of many forms seeks to minimize one property (or set of properties) while maximizing another (hence, MiniMax).  One sees this principle in both conceptual and practical approaches. 

  • A Fractal Construction
    • Maximize contact between humans and natual landscapes in a minimum of terrestrial space with minimal environmental hazard.  Viewed in the context of condos along the Great Lakes's perimeter, one might ask how can individuals be offered the opportunity to secure desirable shore sites without doing extensive overall damage to vast stretches of shore line?  Fractal geometry offers one solution by compressing shoreline through a sequence of cuts involving scale change.
      • Mechanics of  this fractal construction (more detail, in a different context, later):  replace each edge by a shape involving a 95 degree cut (canal). 

      • Characteristics of the construction as applied to condos.
        • Each condo pod has access to road on one side and shoreline on the other.
        • Both road and water networks have successively narrower routes based on likely traffic--a hierarchy of widths
        • The two network trees are separate from each other.  Appropriately selected V-shaped cuts transform a shoreline length of x into one of length 32x, a 32-fold compression without adding any new area.  The chosen self-similarity transformation for generating slips and canals simultaneously produces appropriately scaled roadways from trunk lines to driveways and 466 condominium pods, as well.
        • There are no bridges required--left-hand figure below
          • Any one may have a tall-masted boat
          • Infrastructure (pipes and so forth) is not exposed
        • One may choose relative position on either tree and find a pod to suit those choices--right-hand figure below.  Access to each pod is characterized according to number of turns; the orientation of pod to slip is clockwise across the entire figure (which accounts for lack of bilateral symmetry in shading within the broader bilaterally symmetric geometric pattern).
          • Fewer water than land turns are required to reach 196 black pods--well-suited, perhaps, to the avid sailor who will spend a continuous stretch of time at the shore.
          • Equal number of water and land turns are required to reach 146 gray pods--well suited to the sailor who often comes in and out with a boat trailer.
          • Fewer land than water turns are required to reach 124 white pods--well-suited to the people who spend far more time in the car than on the water (perhaps the elderly or infirm).
Click to see larger image. Click to see larger image.
    • Real-world application

    • Worked with a developer to create such an arrangement on a difficult parcel; dredging costs proved prohibitive.  Sheltered aspect also desirable for this site along Lake St. Clair with a shoreline subjected to continuing wave action.
    Click to see larger image.

    Based on material from: 
    S. Arlinghaus and J. Nystuen, "Geometry of Boundary Exchanges," Geographical Review, Vol. 80, No. 1, 1990, The American Geographical Society, pp. 21-31.

  • Hagerstrand's Simulation of Diffusion--see attached material.  The GIS may also be used to illustrate similar ideas:  spatial infill and extent.
    • Put down a dot for each adopter.  Thus, one shape file has 22 dots. 
    • Create a new field in the attribute table and insert associated random numbers.
    • Partition these dots into the six different categories of the mean information field.
    • Use "Analysis" | "Neighborhood Statistic" to build neighborhoods based on position of cell in relation to center of MIF.
Click to see larger image.