A 
configuration is an abstract configuration of 9
points and 9 lines, with exactly three lines incident with
each point and exactly three points
incident with each line. Two distinct points are incident
with at most one common line. A geometric realization
represents these incidences by points and straight
lines in the real projective plane, without
adding incidences between the selected points and lines.
There are three abstract 
configurations up to isomorphism,
and all three have real geometric realizations
(Bokowski and Pilaud 2014). One is the Pappus
configuration, arising from Pappus's hexagon
theorem. The other two have no standard proper names. Kocay (2021, p. 3)
labels them 
and 
.
-Configurations." Comput.
Geom. 47, 175-186, 2014. 
." Art Discrete Appl. Math. 4, #P3.15,
2021.