Current investigation continues to consider both theory and far-flung applications of theory.

• Theory:  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 it continues.
• Connection:  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, as well.
• Real-world:  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.