Pappus configurations also occur in non-Pappian finite geometries. Lia et al. (2026) proved that in the affine Hall plane of
order ,
with
a prime power and , every triple of distinct points
on one line can be completed by a triple on any other distinct
line. If the lines intersect, all six points must avoid
the intersection. This is an existence statement for a suitable second triple, not
Pappus's hexagon theorem for every admissible
six-tuple.
Berman, L. W.; Gévay, G.; and Pisanski, T. "Polycyclic Geometric Realizations of the Gray Configuration." 20 Feb 2025. https://arxiv.org/abs/2502.14484.Coxeter,
H. S. M. "Self-Dual Configurations and Regular Graphs." Bull.
Amer. Math. Soc.56, 413-455, 1950.Grünbaum, B. Configurations
of Points and Lines. Providence, RI: Amer. Math. Soc., pp. 1-2 and 9,
2009.Lia, S.; Sheekey, J.; and Van de Voorde, G. "Pappus Configurations
in the Hall Plane." 21 Oct 2026. https://arxiv.org/abs/2610.29080.
,
with 
a prime power and 
, every triple of distinct points
on one line can be completed by a triple on any other distinct
line. If the lines intersect, all six points must avoid
the intersection. This is an existence statement for a suitable second triple, not
Pappus's hexagon theorem for every admissible
six-tuple.
