Some Open Problems Regarding the Number of Lines and Slopes in Arrangements that Determine Shapes
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866909344817217536 |
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| author | Haridis, Alexandros |
| author_facet | Haridis, Alexandros |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2011_10700 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Some Open Problems Regarding the Number of Lines and Slopes in Arrangements that Determine Shapes Haridis, Alexandros General Mathematics 51-02, 51A05, 51E26, 00A67 G.2 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. |
| title | Some Open Problems Regarding the Number of Lines and Slopes in Arrangements that Determine Shapes |
| topic | General Mathematics 51-02, 51A05, 51E26, 00A67 G.2 |
| url | https://arxiv.org/abs/2011.10700 |