Notes on Finite Geometry

Notes on Finite Geometry

by Steven H. Cullinane

This page summarizes the author's work in the area of finite geometry during the years 1975 through 2006.

Notes listed by size of the underlying structure:

4

Theme and variations
2004 April 28 (html)

See also Timothy A. Smith's web page on Fugue 21, Book II, of Bach's "Well-Tempered Clavier"
2005 (pdf) (Shockwave)

An outer automorphism of S₆ related to M₂₄
1986 June 11 (jpg)

Picturing outer automorphisms of S₆
1986 July 3 (jpg)

Modeling the 21-point plane with outer automorphisms of S₆2001 Jan. 24 (html)
21 projective partitions
1986 June 6 (jpg)

For the seven lines (or seven points) of the smallest projective plane, see "The eightfold cube" under "8" below.

For an example of a Singer 7-cycle, see "Generating the octad generator" under "24" below.

The eightfold cube
2005 May 4 (html)

Movable graphic designs on the 2x2x2 cube
2006 March 7 (JavaScript)

For a finite-geometry model of the eight elements of the quaternion group, see the article on the 3x3 square under "9" below.

Geometry of the 3x3 square
2005 August 28 (html)

Symmetry invariance under M₁₂
1985 August 22 (html)

Inscapes: views of the generalized quadrangle GQ(2,2)
2006 Jan. 19 (html)

Inscapes I,
1982 June 12 (jpg) (pdf)

Inscapes II, query,
1982 Sept. 22 (jpg) (pdf)

Inscapes III: PG(2,4) from PG(3,2),
1986 Feb. 4 (jpg)

Inscapes IV: Inner and outer group actions,
1986 July 11 (jpg)

Picturing the smallest projective 3-space,
1986 April 26 (jpg) (gif) (pdf)

A linear complex related to M₂₄
1986 May 8 (jpg)

An outer automorphism of S₆ related to M₂₄
1986 June 11 (jpg)

Picturing outer automorphisms of S₆
1986 July 3 (jpg)

The smallest projective (3-)space
2002 May 21 (html)

The diamond theorem
1978, 2006 (html)

See also --

Cases of the diamond theorem
2002 May 22 (html)

Research announcement: an invariance of symmetry
1978 (pdf)

Geometry of the 4x4 square
2004 March 18 (html) (pdf)

The Kaleidoscope Puzzle
2005 Aug. 9 (JavaScript)

The Diamond 16 Puzzle
2002 Feb. 12 (JavaScript)

Orthogonality of Latin squares viewed as skewness of lines
1978 December (jpg) (pdf)

For a generalization, see Latin-square geometry: orthogonal Latin squares as skew lines
2005 Feb. 6 (html) (pdf)

Diamond theory in 1937: a problem of Carmichael
2006 March 29 (html)

The finite relativity problem
1986 Feb. 20 (html) (pdf)

Quilt geometry
2004 Aug. 9 (html)

List of problems related to permutations acting on a 16-set in Introduction to the Theory of Groups of Finite Order, by Robert D. Carmichael (1937) (reprinted by Dover Books, 1956)
2001 (html)

Modeling the 21-point plane with outer automorphisms of S₆
2001 Jan. 24 (html)

21 projective partitions
1986 June 6 (jpg)

The Miracle Octad Generator (MOG) of R. T. Curtis
2005 Nov. 30 (html)

See also -- How the MOG works, in "Geometry of the 4x4 square" (html)

Generating the octad generator
1985 April 28 (html) (pdf)

For a finite-geometry version of the 27 lines on a cubic surface, see "Solomon's cube" below.

For a finite-geometry version of the 28 bitangents to a quartic, see "Solomon's cube" below.

For a model of the 35 lines in the projective 3-space over the 2-element field, see the section headed "16" above.

Solomon's cube
2003 May 28 (html)

Movable graphic designs on the 4x4x4 cube
2006 March 8 (JavaScript)

Geometry of the I Ching
2000 Aug. 8 (html)

Notes that are not size-specific:

Binary Coordinate Systems
1984 July (html)

Diamond theory
1976, 2000-2004 (html).

Related:

Brief history of symmetry
(html)

References
(html)

Rough critique of an attempted proof
(html)

Latin-square geometry: orthogonal Latin squares as skew lines
2005 Feb. 6 (html) (pdf)

Generalized definition of orthogonal squares
2006 (html)

Map systems: function decomposition over a finite field
1982 May 12 (html)

Block designs
1984 Sept. 15, 2004 Feb. 1 (html)

Duality and Symmetry
2001 Jan. 16 (html)

Pattern groups
2005 Oct. 2 (html)

Related:

Group topologies
1986 March 31 (jpg)

.. and all of the other notes listed in this site map.

Cases of the diamond theorem
2002 May 22 (html)

Symmetry of Walsh functions
2001 Aug. 31 (html)

Galois geometry
2004 July 16 (html)

Modeling finite geometries
2004 Dec. 2 (html)

Crystal and dragon
2003 Aug. 27 (html)

Chronological index of typewritten notes closely related to Diamond Theory:

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.

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)

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

86-04-26... 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.

Chronological index of misc. typewritten notes

Includes some notes on mathematics unrelated or only loosely related to the above notes on finite geometry--

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

81-12-24... Solid symmetry

82-09-12... A symplectic array

82-12-27... Group scores

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

83-08-04... Group identity algebras

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

83-11-08... Compound groups

83-11-10... Group compounds

83-11-27... Table groups

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

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

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

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

86-03-31... Group topologies

Notes of a more philosophical nature:

Index

A mathematician's aesthetics

The diamond archetype

Jung and the Imago Dei

Elegance: The aesthetics of parallelism

Poetry's bones

Time fold

The diamond theory of truth

To download most of the notes listed above, with their associated files and folders, click the following:

Download NotesOnFiniteGeometry.zip.

This is a large (about 7 MB) zipped folder containing about 280 interconnected HTML and image files and 23 subfolders. It may take about an hour to download on a dial-up connection.

The text lines in blue above are links to these notes on the Web, so that you can view them individually without downloading the whole folder. For those who prefer the pdf format, links on this page to pdf versions of some of the notes will be provided as time goes on. See note below* on viewing links within pdf.

The advantages of downloading the whole large folder:

You avoid annoying delays in viewing image files once the files have been downloaded to your hard disk.
You can view the files at your leisure without an internet connection.
You have a permanent copy of the notes in case some day they are no longer on the Web.

The disadvantages:

The time required to download the folder,
the space required by the folder on your hard disk, and
the difficulty of viewing the downloaded (local) files if you use Internet Explorer with Windows XP Service Pack 2. (Other browsers such as Mozilla and Netscape work fine with local files and SP2.)

-- S. H. Cullinane, January 14, 2005

* (Note on viewing pdf with a browser: Links within the pdf may fail to open new windows if you are viewing the pdf with a browser. This may mean that when you try to go back to the pdf after visiting a link, the browser has to load the pdf all over again. You can avoid this problem by downloading the pdf file and using the Acrobat pdf reader itself, not a browser plugin, to view the pdf file.)

Page created January 14, 2005.
Last modified on June 13, 2006.

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.
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)
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
86-04-26...	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.

81-11-05...	Patterns invariant... two examples
81-12-24...	Solid symmetry
82-09-12...	A symplectic array
82-12-27...	Group scores
83-05-31...	Decompositions of group enveloping algebras
83-08-04...	Group identity algebras
83-08-16...	Transformations over a bridge
83-11-08...	Compound groups
83-11-10...	Group compounds
83-11-27...	Table groups
84-01-05...	Linear operators in geometric function spaces
85-04-05...	Group actions on partitions
85-04-05...	GL(2, 3) actions on a cube
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
86-03-31...	Group topologies