c(n): ways n lines can meet¶

In one sentence: \(c(n)\) is the number of different ways \(n\) lines in the plane, no two parallel, can meet — two ways being the same when they produce the same list of how many lines pass through each intersection point.

This page is about the types that can actually be drawn: every known term with its evidence, and the types that are still undecided.

Terms¶

\(c(n)\) for \(n = 1, \dots, 12\): 1, 1, 2, 3, 5, 9, 16, 28, 53, 98, 180, 336; \(n = 13\): 620 … 659. Combinatorially possible but not realizable: \(0, 0, 0, 0, 0, 0, 1, 1, 3, 8, 22, 48\) types for \(n \le 12\).

Status through \(n = 12\): computer-verified; \(c(13)\) is a bound. The evidence is of two kinds and every type has one of them:

• Realizable types: exact certificates. For each of the 1 + 1 + 2 + 3 + 5 + 9 + 16 + 28 + 53 + 98 + 180 + 336 types a set of \(n\) lines with coordinates in \(\mathbb Q\) (all but six), \(\mathbb Q(\sqrt 2)\) or \(\mathbb Q(\sqrt 5)\) is stored in data/certificates/. \(\{5, 3^{10}\}\) for \(n = 10\) provably needs \(\sqrt 5\); for \(\{3^{16}\}\) (\(n = 11\)) and three types for \(n = 12\) we only know realizations over \(\mathbb Q(\sqrt 5)\); the \(\mathbb Q(\sqrt 2)\) type for \(n = 12\) also has rational realizations. The verifier recomputes the pencil type of each certificate with exact arithmetic; it runs in continuous integration on every change of the repository.
• Non-realizable types: a proof. Each of the \(b(n) - c(n)\) combinatorially possible but non-realizable types is excluded by Melchior's inequality (Fano \(\{3^7\}\), dual Hesse \(\{3^{12}\}\), and most others), by the Csima–Sawyer bound \(t_2 \ge 6n/13\) (\(\{3^{11}\}\) at \(n = 9\), \(\{4, 3^{15}\}\) and \(\{3^{17}\}\) at \(n = 11\)), or because no linear space of the type is orientable (Möbius–Kantor \(\{3^8\}\) at \(n = 8\); \(\{4^2, 3^9\}\) and \(\{3^{13}\}\) at \(n = 10\); nine types at \(n = 11\); fifteen types at \(n = 12\)). For \(n = 12\) five more are excluded individually: \(\{4^9\}\) by a theorem, \(\{3^{20}\}\) and \(\{4^5, 3^9\}\) by computations showing that none of their linear spaces is orientable, and \(\{6, 4, 3^{12}\}\), \(\{6, 3^{14}\}\) by Gröbner certificates although they are orientable (see below). The orientability verdicts come from an exhaustive search (algorithms); the files data/results/classification_n{n}.json list the reason for every type.

n = 12: what is known¶

Of the 384 combinatorially possible types, 336 have verified certificates (332 over \(\mathbb Q\), one over \(\mathbb Q(\sqrt 2)\), three over \(\mathbb Q(\sqrt 5)\); for the \(\sqrt 2\) type a rational realization exists as well, for the three \(\sqrt 5\) types none was found and it is not known whether one exists), 28 are excluded by Melchior's inequality, \(\{4^9\}\) by a theorem (it is the type of the complex Hesse arrangement: theorems), \(\{3^{20}\}\) because none of its five linear spaces is orientable (Proposition C3a), \(\{4^5, 3^9\}\) by a case split of the orientability search (Proposition C3b), 15 more types because no linear space of theirs is orientable, and \(\{6, 4, 3^{12}\}\), \(\{6, 3^{14}\}\) by exact algebraic certificates although they are orientable (Theorems C4, C5). No type remains undecided. The table lists the types that pass the inequalities but were hard to decide, with their status; \(\{3^{18}\}\), the last type to be realized (Example C3c), is at the top.

A dash in the "orientable?" column means that the type was excluded by a theorem without an orientability search.

n = 13¶

620 types have verified certificates (data/certificates/n13.json; three were constructed with the band theorem, 18 by the zone walk, one by the targeted deformation and one from a section of the H4 mirror arrangement, see the algorithms). The last one is \(\{4^9\}\): impossible for 12 lines (it would be the dual Hesse configuration), it is realized by 13 of the 31 planes perpendicular to the symmetry axes of the icosahedron. The combinatorial test, the inequalities and the band theorem (Theorem B4) exclude all other types except 39, so \(620 \le c(13) \le 659\) (bound). Of the excluded types, 28 are excluded by the band theorem alone (41 before more of the combinatorial searches for \(n = 13\) finished).

Fields of definition¶

Every realizable type for \(n \le 14\) in our data has a certificate over \(\mathbb Q\) or over \(\mathbb Q(\sqrt d)\) with \(d \in \{2, 3, 5\}\). The type \(\{5, 3^{10}\}\) (\(n = 10\)) provably needs an irrational number: the regular pentagon is the only way to arrange it, up to projective transformation (Proposition C1). We do not know whether some type for small \(n\) needs a field of degree \(\ge 3\).