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Notes on Finite Geometry
by Steven H. Cullinane...

Chronological index of typewritten notes closely related to Diamond Theory

76-08-??... Diamond theory cover page  From the author's 1976 booklet. See for the meaning of the cover illustration.
78-??-??... Research announcement (4x4 case of diamond theorem and algebraic generalization)  This research announcement was the basis for an abstract (79T-A37) in the Feb. 1979 AMS Notices.
78-12-??... Orthogonality of Latin squares viewed as skewness of lines  Structural diagrams of 4x4 arrays play the role of lines in PG(3,2). Orthogonality of arrays corresponds to skewness of lines.
82-05-12... Map systems
82-06-12... Inscapes  A new combinatorial way of illustrating symplectic polarities in PG(3,2).
82-09-22... Inscapes II  The concept in "Inscapes" is generalized.
83-06-21... An invariance of symmetry The diamond theorem on a 4x4x4 cube, and a sketch of the proof.
83-10-01... Portrait of O  A table of the octahedral group O using the 24 patterns from the 2x2 case of the diamond theorem.
83-10-16... Study of O  A different way of looking at the octahedral group, using cubes that illustrate the 2x2x2 case of the diamond theorem.
84-09-15... Diamonds and whirls  Block designs of a different sort -- graphic figures on cubes. See also the University of Exeter page on the octahedral group O.
84-09-25... Affine groups on small binary spaces Six ways to slice a cube, and the resulting affine groups. For details, see the author's 1984 paper Binary Coordinate Systems.
85-03-26... Visualizing GL(2, p)
85-04-28... Generating the octad generator  The Miracle Octad Generator (MOG) of R. T. Curtis -- A correspondence between the 35 partitions of an 8-set into two 4-sets and the 35 lines of PG(3,2).
85-08-22... Symmetry invariance under M12  A generalization of the two-color plane patterns, made up of all-black and all-white squares, that underlie plane patterns, made up of two-color diagonally-divided squares, of diamond theory.
86-02-04... Inscapes III: PG(2,4) from PG(3,2)
The relativity problem in finite geometry  "This is the relativity problem: to fix objectively a class of  equivalent coordinatizations and to ascertain the group of transformations S mediating between them." -- Hermann Weyl, The  Classical Groups
86-04-26... Picturing the smallest projective 3-space  A symplectic polarity, the Conwell-Curtis correspondence, and the large Mathieu group.
86-05-08... A linear complex related to M24  Anatomy of the polarity pictured in the 86-04-06 note.
86-05-26... The 2-subsets of a 6-set are the points of a PG(3,2)
Beutelspacher's model of the 15 points of PG(3,2) compared with a 15-line complex in PG(3,2).
86-06-06... Twenty-one projective partitions The author's model of the 21-point projective plane PG(2,4).
For a general method of constructing such models, see Modeling the 21-point plane with outer automorphisms of S6.
The following problem was suggested by the above.
PROBLEM: Let M be an nxn square (0,1) matrix with exactly k 1's in each row and each column. What conditions are necessary and sufficient for there to exist row/column permutations that make M a symmetric matrix?
For more on this problem, see Duality and Symmetry.
86-06-11... An outer automorphism of S6 related to M24  An application of the "inscape" model of a generalized quadrangle to the construction of the large Mathieu group.
86-07-03... Picturing outer automorphisms of S6  An application of the "inscape" model to S6.
86-07-11... Inscapes IV  An outer automorphism that is literally outer.