Graphical enumeration and stained glass windows, 1: rectangular grids
Lars G Blomberg, S. R. Shannon, Neil J.A. Sloane · 2022
This is a survey of enumeration problems arising from the study of planar 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 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. Part 1 discusses planar graphs that are based on rectangular grids.