An 
configuration is a symmetric configuration consisting
of 
abstract points and 
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. The word
"line" here denotes an abstract element of the incidence
structure, not necessarily a straight line.
A geometric 
configuration realizes the prescribed incidences by points
and straight lines in the real
projective plane, with no additional incidences between the selected points
and lines. A topological realization instead uses pseudolines.
These realization conditions are stronger than abstract existence.
Abstract 
configurations exist for every 
, whereas real geometric and topological 
configurations exist exactly for 
(Bokowski and Pilaud 2014). The unique abstract 
configuration is the Fano
plane, and the unique abstract 
configuration is the Möbius-Kantor
configuration; neither has a real geometric
realization.
The numbers of abstract isomorphism types for 
, 2, 3, ... begin 0, 0, 0, 0, 0, 0,
1, 1, 3, 10, 31, 229, 2036, ... (OEIS A001403).
These are abstract counts, not counts of real geometric
realizations. For example, all three 93
configurations are geometrically realizable, while only nine of the ten 103
configurations are (Bokowski and Pilaud 2014). The following named examples illustrate
the distinction between abstract existence and real geometric
realization (Grünbaum 2009, Bokowski and Pilaud 2014, Berman et al. 2025).
| name | type | real geometric realization |
| Fano plane |  | no |
| Möbius-Kantor configuration |  | no |
| Pappus configuration | 93 | yes |
| Desargues configuration | 103 | yes |
| Nauru configuration |  | yes |
| Cremona-Richmond configuration |  | yes |
| Gray configuration |  | yes |
-Configurations." Comput. Geom. 47,
175-186, 2014.