The affine Hall plane of order 
, where 
is a prime power, is a
finite geometry constructed from the projective
plane over 
.
Choose a line at infinity and a copy of the projective
line over 
within it. Keep the affine points and the ordinary affine
lines whose directions lie outside this subline. The other
new lines are the affine parts of copies of the projective
plane over 
containing the chosen subline. Its projective completion is a non-Desarguesian finite
projective plane (Lia et al. 2026).
Although Pappus's hexagon theorem does not hold for every admissible choice of six points in a Hall plane, many Pappus
configurations exist. Lia et al. (2026) proved that for 
, any triple of distinct points
on one affine line extends to a Pappus
configuration using three points on any other affine
line. When the lines meet, all six points must avoid their
intersection.
