Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids

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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.

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We consider the number of triangles formed by the intersecting diagonals of a regular polygon. Basic geometry provides a slight overcount, which is corrected by applying a result of Poonen and Rubinstein [1]. The number of triangles is 1, 8, 35, 110, 287, 632, 1302, 2400, 4257, 6956 for polygons with 3 through 12 sides.
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In this expository compendium, a new approach for assessing com-plexity of classical geometric constructions is presented. Everyone who has experience of making geometric constructions in practice knows how much attention must be paid to a careful placement of the compass and the straightedge in each step of the construction in order to achieve as accurate results as possible. The accuracy of these placements is now described by a simple statistical model and the accuracy of the entire construction is estimated on this basis. Then it is natural to consider the accuracy of the construction as a measure of its complexity. This measure is expected to give better possibilities for comparing complexities of constructions than the characteristics of Lemoine's geometrography. Our approach is mainly computational. Although the error distribution of placements is defined precisely, the error distributions related to entire con-structions are so complicated that the only way is to use Monte Carlo simula-tion for estimating essential statistics. The constructions are described by a special code in Survo which is a gen-eral environment for statistical computing and related areas. A new GEOM program module of Survo is used for creating constructions both in accurate and randomized forms. Graphs based on these constructions and summaries of simulations are accomplished by standard tools of Survo. As applications, various constructions for a regular pentagon and for ap-proximate circle squaring are presented. Especially in the latter task, this statistical approach gives a new possibility for comparison of various approx-imations. It is shown how the goodness of each construction depends funda-mentally on the initial accuracy.
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In an m × n rectangular grid of points (here m = 4, n = 6) lines through at least 2 points of the grid (136 lines) and points of intersection of these lines inside the grid (1961 points)
It is known that a minimal teaching set of any threshold function on the two-dimensional rectangular grid consists of 3 or 4 points. We derive exact formulae for the numbers of functions corresponding to these values and further refine them in the case of a minimal teaching set of size 3. We also prove that the average cardinality of the minimal teaching sets of threshold functions is asymptotically 7/2. We further present corollaries of these results concerning some special arrangements of lines in the plane.
The problem of deciphering the threshold functions of k-valued logic is considered in the Shannon formulation. Upper and lower estimates for the complexity of this problem are established. The construction of the lower estimates is based on the analysis of the structure of the learning set of a threshold function f(x). The learning set is defined as a set of points of the domain of definition such that the values of the function at these points are sufficient for the single-valued definition of f(x) at all remaining points.
The regular drawing of the complete bipartite graph K n,n produces a striking pat-tern comprising simple and multiple crossings. We compute the number c(n) of cross-ings and give an asymptotic estimate for this sequence.
We calculate here both exact and asymptotic formulas for the number of regions enclosed by the edges of a regular drawing of the complete bipartite graph Kn,n. This extends the work of Legendre, who considered the number of crossings arising from such a graph. We also show that the ratio of regions to crossings tends to a rational constant as n increases without limit.
In the series of "Census" papers, of which this is the fourth, we attempt to lay the groundwork of an enumerative theory of planar maps (12, 13, 14). The maps concerned are rooted in the sense that some edge is fixed as the root , and a positive sense of description and right and left sides are specified for it. This device simplifies the theory by ruling out the possibility of a map being symmetrical.
We describe the gfun package which contains functions for manipulating sequences, linear recurrences or differential equations and generating functions of various types. This document is intended both as an elementary introduction to the subject and as a reference manual for the package. Gfun: Un package Maple pour la manipulation de fonctions g'en'eratrices et holonomes en une variable R'esum'e Nous d'ecrivons le package gfun qui contient des fonctions permettant de manipuler des suites, des r'ecurrences ou des 'equations diff'erentielles lin'eaires ainsi que des fonctions g'en'eratrices de types vari'es. Ce document est concu `a la fois comme une introduction 'el'ementaire au domaine et comme un manuel de r'ef'erence pour le package. To appear in the ACM Transactions on Mathematical Software. Gfun: A Maple Package for the Manipulation of Generating and Holonomic Functions in One Variable Bruno Salvy Paul Zimmermann Algorithms ...
We give a formula for the number of interior intersection points made by the diagonals of a regular $n$-gon. The answer is a polynomial on each residue class modulo 2520. We also compute the number of regions formed by the diagonals, by using Euler's formula $V-E+F=2$.
On the number of two-dimensional threshold functions
M. A. Alekseyev, On the number of two-dimensional threshold functions, SIAM J. Discr. Math., 24:4 (2010), 1617-1631.
Graphical enumeration and stained glass windows, 2: Polygons, frames, crosses, etc
  • L Blomberg
  • S R Shannon
  • N J A Sloane
L. Blomberg, S. R. Shannon, and N. J. A. Sloane, Graphical enumeration and stained glass windows, 2: Polygons, frames, crosses, etc., in preparation, 2020.
A very minimal introduction to TikZ
  • J Crémer
J. Crémer, A very minimal introduction to TikZ, March 11, 2011; minimaltikz.pdf.
On lines going through a given number of points in a rectangular grid of points
  • S Mustonen
S. Mustonen, On lines going through a given number of points in a rectangular grid of points, May 12, 2010;
The On-Line Encyclopedia of Integer Sequences
The OEIS Foundation Inc., The On-Line Encyclopedia of Integer Sequences, 2020;
Large chambers in a lattice polygon
  • M E Pfetsch
  • G M Ziegler
M. E. Pfetsch and G. M. Ziegler, Large chambers in a lattice polygon, December 13, 2004; http: //
The email servers and Superseeker
  • N J A Sloane
N. J. A. Sloane, The email servers and Superseeker, 2010;
The PGF/TikZ Programming Language, Version 2.10, CTAN Org
  • T Tantau
T. Tantau, The PGF/TikZ Programming Language, Version 2.10, CTAN Org., October 25 2010.
On the complexity of deciphering threshold functions in two variables, (Russian)
  • N Yu
  • Zolotykh
N. Yu. Zolotykh, On the complexity of deciphering threshold functions in two variables, (Russian), in Proc. 11th Internat. School Seminar "Synthesis and complexity of control systems," Part I, Center of Applied Research, Moscow State Univ. Faculty of Mechanics and Mathematics, Moscow, Russia, 2001, pp. 74-79.