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 𝒜(21,2) by adjoining the $30$ images of the line y+√φ z=0 under the rotation group of the icosahedron, where φ=(1+√5)/2. Its intersection multiplicities are t2=120, t3=190, t5=36, and t6=15, with all other tr 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 √φ, and a standalone exact-arithmetic verifier.
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BibTeX
@misc{silva2026sporadic,
title = {{A sporadic simplicial arrangement of $51$ lines}},
author = {Tomás S. R. Silva},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28874},
eprint = {2610.28874},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28874},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A sporadic simplicial arrangement of $51$ lines
AU - Tomás S. R. Silva
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28874
UR - https://arxiv.org/abs/2610.28874
AB - We construct a sporadic simplicial arrangement of $51$ lines in the real projective plane. It is obtained from the icosahedral arrangement 𝒜(21,2) by adjoining the $30$ images of the line y+√φ z=0 under the rotation group of the icosahedron, where φ=(1+√5)/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 √φ, and a standalone exact-arithmetic verifier.
ER -