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A survey of enumeration problems arising from the study of graphs formed whenthe edges of a polygon are marked with evenly spaced points and every pair ofpoints is joined by a line. A few of these problems have been solved, aclassical example being the the graph K_n formed when all pairs of vertices ofa regular n-gon are joined by chords, which was analyzed by Poonen andRubinstein in 1998. Most of these problems are unsolved, however, and thistwo-part article provides data from a number of such problems as well ascolored 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 bydrawing semicircles joining equally-spaced points on a line. %The paper endswith a brief discussion of the problem of how to %design aesthetically pleasingcolorings for these graphs. This first part discusses rectangular grids. The 1X n grids, or equally the graphs K_{n+1,n+1}, were studied by Legendre andGriffiths, and here we investigate the number of cells with a given number ofedges and the number of nodes with a given degree. We have only partial resultsfor the m X n rectangles, including upper bounds on the numbers of nodes andcells.

Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids