Congruent triangles in arrangements of lines. (English) Zbl 1396.52022

Zamfirescu, Carol T.

Ars Math. Contemp. 14, No. 2, 359-373 (2018).

In the paper under review, the author studies the maximal number of congruent triangles determined by finite arrangements of \(\ell\) lines in \(\mathbb{R}^{2}\). In order to formulate the main results of the paper, we need to recall some notions. An arrangement \(\mathcal{A}\) of lines is always a finite family of \(\ell\) lines \(L_{1},\dots,L_{\ell}\), which is not a pencil. Denote by \(\mathfrak{U}_{\ell}\) the set of all arrangements of \(\ell \geq 3\) lines. For an arrangement \(\mathcal{A} \in \mathfrak{U}_{\ell}\) we can associate a graph \(\Gamma_{\mathcal{A}}\): the vertices of \(\Gamma_{\mathcal{A}}\) correspond to the intersection points of lines and the edges of \(\Gamma_{\mathcal{A}}\) correspond to the line-segments between these vertices. A triangle in \(\mathcal{A} \in \mathfrak{U}_{\ell}\) is the convex hull of the set of intersection points of three non-concurrent pairwise non-parallel lines in \(\mathcal{A}\). We denote by \(F^{\mathcal{A}}\) the set of all triangles in \(\mathcal{A}\). If two triangles \(\triangle_{1}, \triangle_{2} \in F^{\mathcal{A}}\) are congruent, then we write \(\triangle_{1} \sim \triangle_{2}\). Let \(F_{1}^{\mathcal{A}}, \dots, F_{p}^{\mathcal{A}}\) be the equivalence classes with respect to \(\sim\) such that \(\# F_{1}^{\mathcal{A}} \geq \dots \geq \# F_{p}^{\mathcal{A}}\). We call a triangle \(\triangle \in F^{\mathcal{A}}\) facial if it is a face of \(\Gamma_{\mathcal{A}}\), i.e., \(L \cap \text{int} \triangle = \emptyset\) for all \(L \in \mathcal{A}\). Denote by \(G^{\mathcal{A}} \subset F^{\mathcal{A}}\) the set of all facial triangles, and we denote by \(G_{1}^{\mathcal{A}}, \dots ,G_{q}^{\mathcal{A}}\) the equivalence classes with respect to \(\sim\) such that \(\# G_{1}^{\mathcal{A}} \geq \dots \geq \# G_{q}^{\mathcal{A}}\). Now we define \[ f(\ell) = \max_{\mathcal{A} \in \mathfrak{U}_{\ell}} \# F_{1}^{\mathcal{A}}, \quad \quad g(\ell) = \max_{\mathcal{A} \in \mathfrak{U}_{\ell}} \# G_{1}^{\mathcal{A}}. \] Additionally, we define \(F(\ell)\) (\(G(\ell)\), respectively) as the set of all integers \(u\) such that there exists an arrangement on \(\ell\) lines having exactly \(u\) congruent triangles (congruent facial triangles, respectively). We write \([s..t]\) for the set of all integers \(u\) such that \(s \leq u\leq t\), put \(H\) for \(F\) or \(G\), and \(h\) is equal to \(f\) if \(H=G\) or \(g\) if \(H = G\). Whenever \(H(\ell) = [0..h(\ell)]\), we say that \(H(\ell)\) is complete.
Theorem 1. One has \(f(5) = g(5) = 5\) while \(F(5)\) and \(G(5)\) are complete.
Thereom 2. One has \(f(6) = 8, \, 6 \leq g(6) \leq 7\), \(F(6)\) is complete, and \([0..6] \subset G(6)\).
Theorem 3. One has \(f(7) = 14, \, 9 \leq g(7) \leq 11, [0..10]\cup\{14\} \subset F(7)\), and \([0..9] \subset G(7)\).
Theorem 4. One has \(16 \leq f(8) \leq 22, \, 12 \leq g(8) \leq 15\), \([0..16] \setminus \{13\} \subset F(8)\) and \([0..12] \subset G(8)\).
In the last section, the author formulates some natural conjectures, for instance the author predicts that \(g(6)=6\).

Reviewer: Piotr Pokora (Kraków)

MSC:

52C10	Erdős problems and related topics of discrete geometry
52C30	Planar arrangements of lines and pseudolines (aspects of discrete geometry)

Keywords:

arrangement of lines; congruent triangles

Software:

OEIS

References:

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