Image Files Home Page of Steven H. Cullinane

Author's mathematical home page
Author's personal home page
Author's personal journal
Author's remarks on the philosophy of mathematics

The following image files are sorted by date, in YY-MM-DD format.

76-08-??... 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 (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.

81-11-05... Patterns invariant... two examples

81-12-24... Solid symmetry

82-05-12... Map systems

82-06-12... Inscapes

A new combinatorial concept that illustrates symplectic polarities in PG(3,2).

82-09-12... A symplectic array

82-09-22... Inscapes II

The concept in "Inscapes" is generalized.

82-12-27... Group scores

83-05-31... Decompositions of group enveloping algebras

83-06-21... An invariance of symmetry

The diamond theorem on a 4x4x4 cube, and a sketch of the proof.

83-08-04... Group identity algebras

83-08-16... Transformations over a bridge

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.

83-11-08... Compound groups

83-11-10... Group compounds

83-11-27... Table groups

84-01-05... Linear operators in geometric function spaces

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)

85-04-05... Group actions on partitions

85-04-05... GL(2, 3) actions on a cube

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).
For more on the MOG, see the preprint by Marston Conder and John McKay, "Markings of the Golay Code" -- either the author's PostScript version or the cached version.

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.

85-11-17... Groups related by a nontrivial identity

85-12-11... Dynamic and algebraic compatibility of groups

86-01-11... Geometry of Partitions II

86-02-04... Inscapes III

An excellent source of exercises related to this note is Introduction to the Theory of Groups of Finite Order, by Robert D. Carmichael (1937), reprinted by Dover Books, 1956. See especially
pp. 42-43, ex. 30 and 31;
p. 73, ex. 32;
p. 165, ex. 20;
p. 304, ex. 3;
pp. 320-321, ex. 7-12;
pp. 336-337, ex. 4-7, 9, 10;
p. 353, ex. 5;
p. 392, ex. 6, part (3);
p. 437, ex. 14-17;
p. 439, ex. 11-12;
p. 440, ex. 20.
These exercises in Carmichael have a simplicity and clarity lacking in many more recent works on finite mathematics.
See also "Generalized Steiner Systems of Type 3-(v,{4,6},1)," by E. F. Assmus, Jr., and J. E. Novillo Sardi, in Finite Geometries and Designs, edited by P. J. Cameron, J. W. P. Hirschfeld, and D. R. Hughes, Cambridge University Press, 1981.

86-02-20... 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

86-03-31... Group topologies

86-04-26... Picturing the smallest projective 3-space

86-05-08... A linear complex related to M₂₄

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 PG(2,4).
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

86-07-03... Picturing outer automorphisms of S₆

86-07-11... Inscapes IV

An outer automorphism that is literally outer.

Page last updated February 1, 2001; created July 2000.

Copyright © 2001 Steven H. Cullinane. All rights reserved.