Finite Geometry Notes

7608??...  Diamond theory cover page From the author's 1976 booklet. See Math16.com 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 (79TA37) in the Feb. 1979 AMS Notices. 
7812??...  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. 
820512...  Map systems 
820612...  Inscapes A new combinatorial way of illustrating symplectic polarities in PG(3,2). 
820922...  Inscapes II The concept in "Inscapes" is generalized. 
830621...  An invariance of symmetry The diamond theorem on a 4x4x4 cube, and a sketch of the proof. 
831001...  Portrait of O A table of the octahedral group O using the 24 patterns from the 2x2 case of the diamond theorem. 
831016...  Study of O A different way of looking at the octahedral group, using cubes that illustrate the 2x2x2 case of the diamond theorem. 
840915...  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. 
840925...  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. 
850326...  Visualizing GL(2, p) 
850428...  Generating the octad generator The Miracle Octad Generator (MOG) of R. T. Curtis  A correspondence between the 35 partitions of an 8set into two 4sets and the 35 lines of PG(3,2). 
850822...  Symmetry invariance under M_{12} A generalization of the twocolor plane patterns, made up of allblack and allwhite squares, that underlie plane patterns, made up of twocolor diagonallydivided squares, of diamond theory. 
860204...  Inscapes III: PG(2,4) from PG(3,2) 
860220... 

860426...  Picturing the smallest projective 3space A symplectic polarity, the ConwellCurtis correspondence, and the large Mathieu group. 
860508...  A linear
complex
related to M_{24} Anatomy of the polarity pictured in
the 860406 note. 
860526...  The
2subsets of a 6set are the points of a PG(3,2) Beutelspacher's model of the 15 points of PG(3,2) compared with a 15line complex in PG(3,2). 
860606...  Twentyone
projective partitions The author's model of the 21point projective
plane PG(2,4). For a general method of constructing such models, see Modeling the 21point plane with outer automorphisms of S_{6}. 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. 
860611...  An outer
automorphism of S_{6} related to M_{24} An
application of the "inscape" model of a generalized quadrangle to the
construction of the large Mathieu group. 
860703...  Picturing
outer
automorphisms of S_{6} An application of the
"inscape" model to S_{6}. 
860711...  Inscapes IV An outer automorphism that is literally outer. 