A simplicial arrangement is a central real hyperplane arrangement whose chambers are simplicial polyhedral cones. For a central arrangement in three-dimensional Euclidean space, this means that the associated lines in the real projective plane divide that plane into triangular regions.
Silva (2026) constructed an arrangement of 51 real projective lines with
 |
where 
counts points incident with exactly 
lines, and all other 
are zero. It has 361 vertices,
1080 edges, and 720 triangular chambers in the real
projective plane. It is irreducible and belongs to neither of the two known infinite
families of irreducible simplicial arrangements, making it a sporadic example. It
refutes the conjectured bound of 37 lines for sporadic arrangements
(Silva 2026), not a bound on all simplicial arrangements.
Silva (2026) supplies explicit coordinates and an exact-arithmetic verifier, and reports that Claude assisted with the computations.
