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Some Open Problems Regarding the Number of Lines and Slopes in Arrangements that Determine Shapes

Alexandros Haridis

Abstract

A set $L$ of straight lines and a set $P$ of points in the Euclidean plane define an arrangement $\mathcal{A}$ = ($L$, $P$) of construction lines and registration marks, if and only if: (1) any point in $P$ is a point of intersection of at least two lines in $L$, and (2) any two nonparallel lines in $L$ have a unique point of intersection in $P$. This expository article discusses the following open problems regarding such point-line arrangements. Suppose $k \geq 0$ number of points are given in the plane. How many construction lines $k$ points must determine? How many distinct slopes, or directions, are defined by construction lines that $k$ points determine? How many distinct sets of construction lines partition the plane, such that the lines meet at exactly $k$ points? Empirical evidence is reported for small numbers of $k$, offering partial answers to the three problems. A conjecture is also stated for the first problem, on the number of construction lines, after examining a related problem about finite linear spaces from incidence geometry. This paper contributes to the body of work related to the mathematics of shapes in the area of shape grammar theory.

Some Open Problems Regarding the Number of Lines and Slopes in Arrangements that Determine Shapes

Abstract

A set of straight lines and a set of points in the Euclidean plane define an arrangement = (, ) of construction lines and registration marks, if and only if: (1) any point in is a point of intersection of at least two lines in , and (2) any two nonparallel lines in have a unique point of intersection in . This expository article discusses the following open problems regarding such point-line arrangements. Suppose number of points are given in the plane. How many construction lines points must determine? How many distinct slopes, or directions, are defined by construction lines that points determine? How many distinct sets of construction lines partition the plane, such that the lines meet at exactly points? Empirical evidence is reported for small numbers of , offering partial answers to the three problems. A conjecture is also stated for the first problem, on the number of construction lines, after examining a related problem about finite linear spaces from incidence geometry. This paper contributes to the body of work related to the mathematics of shapes in the area of shape grammar theory.

Paper Structure

This paper contains 16 sections, 1 theorem, 20 figures.

Table of Contents

  1. Overview of arrangements that determine shapes
  2. Line and slope problems
  3. Empirical evidence
  4. k = 0
  5. k = 1
  6. k = 2
  7. k = 3
  8. Discussion
  9. Distinct arrangements of construction lines with $k = 4, 5, 6, 7$ registration marks
  10. Comparing arrangements
  11. Projectively equivalent arrangements
  12. Combinatorially equivalent arrangements
  13. 1. Number of registration marks, construction lines, and distinct slopes
  14. 2. Line type, slope type, and point type
  15. 3. Point-line degree type
  16. ...and 1 more sections

Key Result

Theorem 1

(de Bruijn-Erdős, 1948) Let $P$ be a set of $k \geq 3$ points, and $L$ a set of $n > 1$ lines such that any two distinct points of $P$ are on exactly one line of $L$. Then

Figures (20)

  • Figure 1: Top: Examples of shapes. Bottom: The same shapes superimposed on their construction lines and registration marks.
  • Figure 2: Examples of shapes with the same underlying arrangement of construction lines and registration marks.
  • Figure 3: Some sets of points and lines that do not define arrangements of construction lines and registration marks.
  • Figure 4: Three types of arrangements that exist for any number of points greater than or equal to three.
  • Figure :
  • ...and 15 more figures

Theorems & Definitions (2)

  • Theorem 1
  • Conjecture 1