  A survey of enumeration problems arising from the study of graphs formed when
the edges of a polygon are marked with evenly spaced points and every pair of
points is joined by a line. A few of these problems have been solved, a
classical example being the the graph K_n formed when all pairs of vertices of
a regular n-gon are joined by chords, which was analyzed by Poonen and
Rubinstein in 1998. Most of these problems are unsolved, however, and this
two-part article provides data from a number of such problems as well as
colored illustrations, which are often reminiscent of stained glass windows.
The polygons considered include rectangles, hollow rectangles (or frames),
triangles, pentagons, pentagrams, crosses, etc., as well as figures formed by
drawing semicircles joining equally-spaced points on a line. %The paper ends
with a brief discussion of the problem of how to %design aesthetically pleasing
colorings for these graphs. This first part discusses rectangular grids. The 1
X n grids, or equally the graphs K_{n+1,n+1}, were studied by Legendre and
Griffiths, and here we investigate the number of cells with a given number of
edges and the number of nodes with a given degree. We have only partial results
for the m X n rectangles, including upper bounds on the numbers of nodes and
cells.
