A preprint reports a clear split among four rank-four crystallographic Coxeter arrangements: A4 and B4 admit a special vertex, while D4 and F4 do not. The study examines irreducible simplicial arrangements of projective planes in real projective three-space through combinatorial and projective-geometric criteria.
The result also extends through a four-stage deletion chain inside B4. The arrangements named A31 (13), A31 (14), A31 (15) and A31 (16), with the last identified with B4, are all described as irreducible, simplicial and special-vertex. Here, simpliciality is checked through the paper’s stated relations among faces and chambers.
Turning geometry into counts
The paper’s main tools reduce a geometric question to several equivalent counting checks. For an essential arrangement, simpliciality is equivalent to saying that the number of two-dimensional cells is twice the number of chambers. It is also equivalent to an alternating combination of the numbers of zero-, one- and two-dimensional cells equalling zero. These identities allow the same structure to be checked from different parts of the induced cell decomposition.
A second route uses restrictions: the arrangement is examined plane by plane, and each plane carries a reduced line arrangement whose chambers can be counted. Adding those chamber counts gives the total number of two-dimensional faces, so the arrangement is simplicial exactly when the sum equals twice the total chamber count. The paper also states an equivalent characteristic-polynomial criterion.
The analysis gives a third version in terms of incidences, meaning which vertices lie on which intersection lines. It weights vertices according to how many planes meet there and then weights the corresponding vertex-line incidences according to line multiplicity. The resulting identity is another if-and-only-if test for simpliciality, so the conclusion can be reached through incidence data as well as face counts.
For the two negative cases, D4 and F4, the paper uses an exact finite enumeration of candidate planes in the dual normal-vector configurations. It forms planes from triples of noncollinear points, represents the relevant vectors with primitive integers, checks determinants and ranks exactly, removes duplicates, and tests whether a candidate covers the required points. The procedure avoids floating-point tolerance when deciding exact incidence.
Why D4 and F4 fall short
For D4, the relevant configuration contains eleven points distinct from q∞, but the largest covered set has size ten. The point represented by [e3 + e4] can remain uncovered. That finite result is used to classify A(D4), also denoted A31 (12), as irreducible and simplicial but not special-vertex.
The F4 check is larger. Candidate planes containing four points reached a maximum coverage of ten, while candidates containing nine points reached nineteen. Full coverage would require twenty-three points, so the paper classifies A(F4), or A31 (24), as irreducible and simplicial but not special-vertex.
Together, the exact checks support the paper’s stated rank-four classification: the special-vertex property is present for A4 and B4 and absent for D4 and F4. The scope is specific. It covers these named crystallographic Coxeter arrangements and the displayed B4 deletion chain, not every irreducible simplicial arrangement in real projective three-space.
A second set of measures points in different directions
The preprint also compares seven named arrangements: the four full Coxeter arrangements A4, D4, B4 and F4, plus the three intermediate B4 deletion-chain rows. It records two numerical defects. The rank-flat difference is the number of rank-three projective flats minus the number of rank-two projective flats. The Purdy defect adds the number of planes and two to that difference; a nonnegative value is equivalent to the corresponding lower bound on the number of rank-three flats.
Every arrangement in that comparison has a negative rank-flat difference and a nonnegative Purdy defect. At the endpoints reported in the analysis, F4 has a rank-flat difference of minus two, B4 has minus eighteen, and B4’s Purdy defect is zero. B4 is therefore the equality case among the named arrangements for the Purdy inequality, but the comparison does not establish that inequality for all irreducible simplicial arrangements.
Within the B4 deletion chain, the reported formulas show a regular linear pattern from A31 (13) through A31 (16): the rank-flat difference changes by two as the index increases, while the Purdy defect changes by one. The paper presents these as formulas for that named range, not as a universal law for other arrangements.
The Grünbaum–Shephard defect measures the number of ordinary intersection lines against the number of non-ordinary ones, with the conjectured condition requiring a value greater than zero. All seven listed arrangements have positive values: 5 for A4, 2 for D4, 2 for A31 (13), 4 for A31 (14), 8 for A31 (15), 14 for B4 and 22 for F4. None has an intersection line contained in more than four planes.
A result bounded by its examples
The special-vertex result covers the four named crystallographic rank-four Coxeter arrangements and the displayed B4 deletion chain. It is not a classification of every irreducible simplicial arrangement in real projective three-space. The defect findings are similarly descriptive for the seven arrangements in the comparison.
The preprint therefore does not establish special-vertex status outside those named cases, prove the Purdy inequality universally, or prove the Grünbaum–Shephard conjecture universally. Its exact enumeration supports the conclusions for D4 and F4, while the defect calculations identify patterns that remain limited to the arrangements examined.
The document is a preprint identified as arXiv:2608.28254v1. Its supplied front matter shows August 31, 2026, while the arXiv line gives August 28, 2026. Funding is listed from Poland’s National Science Centre through Sonata Bis Grant 2023/50/E/ST1/00025.
Paper data and sources
Original title: Simplicial arrangements in real projective three-space revisited
Authors: Marek Janasz, Piotr Pokora
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
Original paper · Full text