A sporadic simplicial arrangement of $51$ lines

Abstract

We construct a sporadic simplicial arrangement of $51$ lines in the real projective plane. It is obtained from the icosahedral arrangement $\mathcal{A}(21,2)$ by adjoining the $30$ images of the line $y+\sqrtφ\,z=0$ under the rotation group of the icosahedron, where $φ=(1+\sqrt5)/2$. Its intersection multiplicities are $t_2=120$, $t_3=190$, $t_5=36$, and $t_6=15$, with all other $t_r$ zero. These counts give equality in Melchior's inequality and hence prove simpliciality. The arrangement is irreducible, belongs to neither of the two known infinite families of irreducible simplicial arrangements, and is not supersolvable. It therefore disproves the $37$-line bound for sporadic arrangements conjectured by Cuntz and Mücksch. We give explicit coordinates, a finite incidence certificate, a geometric explanation of the parameter $\sqrtφ$, and a standalone exact-arithmetic verifier.