Configurations and Squares by Steven H. Cullinane


4x4 and 3x3
For a definition of "abstract configuration," see Dolgachev— 


The 3×3 Square
For the Hesse configuration, see (for instance) the passage from Coxeter quoted in Quaternions in an Affine Galois Plane
http://finitegeometry.org/sc/gen/configs_files/Coxeter-MoebiusKantor.jpg
The (83, 83) Möbius-Kantor configuration here described by Coxeter is of course part of the larger (94, 123) Hesse configuration. Simply add the center point of the 3×3 Galois affine plane and the four lines (1 horizontal, 1 vertical, 2 diagonal) through the center point.
The Hesse Diamond Star configuration

PART II-- September 7,  2011

The Most Important Configuration

A search for some background on Gian-Carlo Rota's remarks
in Indiscrete Thoughts * on a geometric configuration
leads to the following passages in Hilbert and Cohn-Vossen's
classic Geometry and the Imagination
http://finitegeometry.org/sc/gen/configs_files/110907-HCV-BPconfigSm.jpg
These authors describe the Brianchon-Pascal configuration
of 9 points and 9 lines, with 3 points on each line
and 3 lines through each point, as being
"the most important configuration of all geometry."
Thus it seems worthwhile to relate it to the material
on the 3x3 array in Part I above.
The Encyclopaedia of Mathematics , ed. by Michiel Hazewinkel,
supplies a summary of the configuration apparently
derived from Hilbert and Cohn-Vossen

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