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Configurations and Squares

by Steven H. Cullinane

PART I -- March 29, 2011

4x4 and 3x3

For a definition of "abstract configuration," see Dolgachev—

  Here (93, 124) is a typographical error for (94, 123).

Note that the two examples Dolgachev mentions here, with 16 points and 9 points, are not unrelated to the geometry of 4×4 and 3×3 square arrays.

The 4×4 Square

For the Kummer and related 16-point configurations, see section 10.3, "The Three Biplanes of Order 4," in Burkard Polster's A Geometrical Picture Book  (Springer, 1998). See also the 4×4 array described by Gordon Royle in an undated web page and in 1980 by Assmus and Sardi

Click to enlarge.

The Kummer-configuration biplane

For the order of the automorphism group of the Kummer (166, 166), see the exercises on this configuration— modeled by a 4×4 array— in R.D. Carmichael's classic Introduction to the Theory of Groups of Finite Order  (1937) on pages 42 and 43. (Note that Carmichael does not count the duality exchanging points and blocks as an automorphism. This makes Carmichael's automorphism group of order 11,520 (=16*(6!)) rather than 23,040.) 

Six-Sets Within 4x4 Arrays-- Two Views

For a connection of this sort of 4×4 geometry to the geometry of the diamond theorem, read "The 2-subsets of a 6-set are the points of a PG(3,2)" (a note from 1986) in light of R.W.H.T. Hudson's 1905 classic Kummer's Quartic Surface , pages 8-9, 16-17, 44-45, 76-77, 78-79, and 80. Here are two examples of  how permutations within the 6-sets of the 1986 note induce automorphisms of the Kummer configuration—

Automorphisms of Kummer configuration induced by 6-set permutations

Dolgachev in "Abstract Configurations" also discusses another configuration, the Cremona-Richmond (153, 153), closely related to properties of 6-element sets. The note Inscapes shows that Figure B in the above 1986 note describes that (153, 153).

The 3×3 Square

For the Hesse configuration, see (for instance) the passage from Coxeter quoted in Quaternions in an Affine Galois Plane

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

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

My own annotation at right above shows one way to picture the
Brianchon-Pascal points and lines— regarded as those of a finite,
purely combinatorial , configuration— as subsets of the nine-point
square array discussed in Part I above. The rearrangement of points
in the square yields lines that are in  accord with those in the usual
square picture of the 9-point affine plane.

A more explicit picture—

The Brianchon-Pascal configuration is better known as Pappus's  configuration,
and a search under that name will give an idea of its importance in geometry.

* Birkhäuser Boston, 1998 2nd printing, p. 145

PART III-- September 8, 2011

Starring the Diamond

"In any geometry satisfying Pappus's Theorem,
the four pairs of opposite points of 83 
are joined by four concurrent lines.
-- H. S. M. Coxeter (see below)

Part II related the the Pappus configuration to the "Diamond Star" figure--

  Stylized version of the
"Diamond Star" in Part I above 

Coxeter, in "Self-Dual Configurations and Regular Graphs," also relates Pappus to this figure.

Some excerpts from Coxeter—

The relabeling uses the 8 superscripts
from the first picture above (plus 0).
The order of the superscripts is from a
Singer 8-cycle in the Galois field GF(9).

The relabeled configuration is used in a discussion of Pappus—

Coxeter here has a note referring to page 335 of  G. A. Miller, H. F. Blichfeldt, and L. E. Dickson, Theory and Applications of Finite Groups, New York, 1916.

Coxeter later uses the the 3×3 array (with center omitted) again to illustrate the Desargues  configuration—

The Desargues configuration is discussed by Gian-Carlo Rota on pp. 145-146 of Indiscrete Thoughts

"The value  of Desargues' theorem and the reason  why the statement of this theorem has survived through the centuries, while other equally striking geometrical theorems have been forgotten, is in the realization that Desargues' theorem opened a horizon of possibilities  that relate geometry and algebra in unexpected ways."