2020/09/16 by Lars Blomberg, L. G. Blomberg, Scott R. Shannon +5
Arts and Humanities · Mathematics · Physics and Astronomy · Psychology · #05A16 #05C10 #05C30 #52B05 #Art, Politics, and Modernism #Color Science and Applications #Color perception and design #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05A16 #msc:05C10 #msc:05C30 #msc:52B05
paper · pdf · doi:10.48550/arxiv.2009.07918
25 pages, 26 references, 20 figures, 12 tables. Added a table and a conjecture
openalex publication_date 2020/09/16 · arxiv created 2020/12/22 · arxiv updated 2020/12/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Kn 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 Kn+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.