Synthesis: Investigations in Progress
investigation continues to consider both theory and far-flung applications
a future volume in the Spatial
Synthesis series will be based on the work that comes from these ventures.
the Diophantine equation K= x2 +
xy + y2 represents, also, an elliptical
paraboloid in three-dimensional Cartesian space with major axis lying along
the line y=x. When this surface is viewed as a generating surface,
different K-values might be seen as level curves (ellipses) dropped down
into the plane in much the way that topographic contours can be seen as
level curves of a mountain. The difference is that here we know the
equation for the "mountain" and so this viewpoint is feasible because the
situation is exact. Consequently, the entire geometry captured in
two-dimensions might be seen as a special case of this more general observation.
Exploration of this approach has been underway for a number of years and
projection and transformation are powerful tools, used frequently in late
20th century pure mathematics as well as in works by D'Arcy Thompson, Waldo
Tobler, and others. Still others, who did not specifically adopt
this sort of view, might have work that readily fits it. One example
might be found in the work of Zipf, in viewing his set of "parallel lines"
in the plane as being dropped down as level curves from an hyperboloid
of two sheets, xy = K. Work progresses in this direction,
the pointillest world of Seurat when captured on a cathode ray tube is
formed using a square brush. What art might be generated using an
hexagonal brush and concepts from central place geometrical hierarchies.
Experiments using software to maneuver images have been underway for several
years and these also continue.
Solstice: An Electronic Journal of Geography and Mathematics,
Institute of Mathematical Geography, Ann Arbor, Michigan.
Volume XVI, Number 1.