Notes on Finite Geometry (Site Map)

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Notes on Finite Geometry (Site Map)
by Steven H. Cullinane

Home page:

Finite geometry of the square and cube
2006 April (html)

Background:

Collected typewritten notes on groups and finite geometry from 1978 through 1986

"Diamond Theory," by Steven H. Cullinane (about 840 KB), from Computer Graphics and Art, Vol. 2, No. 1, February 1977, pp. 5-7

Some of the above typewritten notes, along with more recent notes, are listed by size of the underlying structure:

3 (and 2, and 1)

Geometry simplified

2010 June 24 (html)

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)

The 2-subsets of a 6-set are the points of a PG(3,2)
1986 May 26 (jpg)

21 projective partitions
1986 June 6 (jpg)

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 S₆1986 December. 4 (html)

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

Coxeter vs. Fano
2010 July 8 (html)

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

The eightfold cube
2005 May 4 (html)

Finite geometry and physical space
2009 April 10 (html)

Non-Euclidean blocks
2009 September 8 (html)

A simple reflection group of order 168
2007 December 7 (html)

Knight moves: geometry of the eightfold cube
2008 January 16 (html)

Coxeter vs. Fano
2010 July 8 (html)

Eightfold geometry (Structure of the 35 partitions of an 8-set into two 4-sets)
2010 April 28 (html)

Diamonds and whirls
1984 September 15 (jpg)

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

Cube space, 1984-2003
2008 October 24 (html)

MAGMA calculation of order-168 group acting on the eightfold cube
2009 September 14 (html)

Symmetry, automorphisms, and Visual Group Theory
2010 March 2 (html)

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

The Quaternion Group Acting on an Eightfold Cube
2011 December 30 (html)

Geometry of the 3x3 square (and the group GL(2,3))
2005 August 28 (html)

A 2008 note, "GL(2,3)," by Nick Wedd with neater versions of the pictures in the above note
2008 December 10 (html)

Group actions, 1984-2009 (Cullinane's 1985 work extended by Ed Pegg Jr.)
2009 August 19 (html)

The Galois quaternion: A story
2010 January 6 (html)

Quaternions in an affine Galois plane
2010 January 6 (html)

Configurations and squares
2011 March 29 (html)

Symmetry invariance under M₁₂
1985 August 22 (html)

Cubist geometries
(The 13-point projective plane within the 3x3x3 cube)
2010 February 27 (html)

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

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

Inscapes II, query,
1982 September 22 (jpg) (html) (pdf)

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

The 2-subsets of a 6-set are the points of a PG(3,2)
1986 May 26 (jpg)

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

(Rosenhain and) Göpel Tetrads in PG(3,2)
2013 March 17 (html)

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 animated diamond theorem
2006 November 18 (html)

The diamond theorem in 1978 (AMS abstract 79T-A37)
1978 October 31 (html)

The diamond theorem
1978 October to 2006 January (html) (pdf) (jpg)

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 August 9 (JavaScript)

The diamond 16 puzzle
2002 February 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 February 6 (html) (pdf)

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

The finite relativity problem
1986 February 20 (html) (pdf)

Finite relativity: The triangular version
2012 November 22 (html)

The geometry of logic
2007 March 10 (html)

Vector logic: excerpt from a paper by Jonathan Westphal and Jim Hardy
2004 February (html) (Copy of paper received Sept. 11, 2007)

The geometry of qubits
2007 August 12 (html)

Quilt geometry
2004 August 9 (html)

Half-circle patterns
2009 December 22 (html with script)

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)

Configurations and squares
2011 March 29 (html)

The order-4 Latin squares
2011 January 22 (html)

The Galois tesseract
2011 September 3 (html)

Vector addition in a finite field
2013 January 5 (html)

Triangles are square (with square-to-triangle mapping problem for the 4x4 square)
1984-1985 (html)

Adjacency of vertices in tesseract and in 4x4 array
2012 August 12 (html)

20 (The number of lines in PG(3,2) that are Rosenhain tetrads
closely related to Baker's figure illustrating Desargues's theorem)

Classical geometry in light of Galois geometry
2013 April 1 (html)

21 projective partitions
1986 June 6 (jpg)

Modeling the 21-point plane with outer automorphisms of S₆
1986 December 4 (html)

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

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

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

Competing definitions of the Miracle Octad Generator
2010 May 14 (html)

The Moore correspondence
2010 August 20 (html)

An exceptional Steiner-triple-systems isomorphism underlying S(5, 8, 24)
(Excerpt from a 2005 article by Klin, Meszka, Reichard, and Rosa)
2013 December 20 (html)

Elements (Stoicheia) of geometry
2007 February 28 (html)

Notes for Epiphany 2009
2009 January 6 (html)

The diamond connection:
background for two papers on constructing S(5,8,24) from PG(3,2)
2009 May 20 (html)

For the 27-point affine 3-space over GF(3), the 3x3x3 cube,
and the 13-point projective plane within the 3x3x3 cube, see
"Cubist geometries"
2010 February 27 (html)

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 sections headed "15" and "16" above.
See also...

Galois space
2012 July 12 (html)

Rosenhain and Göpel tetrads in PG(3,2)
2013 March 17 (html)

The 56 spreads in PG(3,2)
2014 January 2 (html)

Solomon's cube
2003 May 28 (html)

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

Geometry of the I Ching
2000 August 8 (html)

The Klein correspondence, Penrose space-time, and a finite model
2007 May 28 (html)

The Ninth Configuration (Kummer's 16₆ and the 4x4x4 cube)
2012 February 14 (html)

Notes that are not size-specific:

Binary coordinate systems
1984 July (html)

Reflection groups in finite geometry
2007 December 7 (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 February 6 (html) (pdf)

Spaces as Hypercubes
2012 November 3 (html)

The algebra of groups
2008 October 15 (html)

Generalized definition of orthogonal squares
2006 (html)

A four-color theorem: function decomposition over a finite field
1982 May 12 (html)

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

Duality and symmetry
2001 January 16 (html)

Configurations and Squares
2011 March 29 (html)

Pattern groups
2005 October 2 (html)

Related:

Group topologies
1986 March 31 (jpg)

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

Noncontinuous groups
2009 October 16 (html)

Cases of the diamond theorem
2002 May 22 (html)

Symmetry of Walsh functions
2001 August 31 (html)

Galois geometry
2004 July 16 (html)

Mere geometry
2010 April 22 (html)

Modeling finite geometries
2004 December 2 (html)

Crystal and dragon
2003 August 27 (html)

Binary geometry
2006 June 23 (html)

Diamond Theory and Magic Squares
2010 October 14 (html)

The Cube Model and Peano Arithmetic
2012 March 1 (html)

Related websites --

The "Kufi blocks" of Ben Nicholson
2006 July 9 (html)

Applications to physics:

Reflections on symmetry
(finite geometry and quantum information theory)
2004 July 18, plus updates (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 (A four-color theorem)

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

Mathematics of Wallace Stevens's "The Rock"

The diamond archetype

Jung and the Imago Dei

Elegance: The aesthetics of parallelism

Poetry's bones

Time fold

The diamond theory of truth

Mirror-play of the fourfold in Heidegger

Notes for Epiphany 2009

Metadata:

Metadata for selected pages

td valign="top"> 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

Mathematics of Wallace Stevens's "The Rock"

The diamond archetype

Jung and the Imago Dei

Elegance: The aesthetics of parallelism

Poetry's bones

Time fold

The diamond theory of truth

Mirror-play of the fourfold in Heidegger

Notes for Epiphany 2009

Metadata:

Metadata for selected pages

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 (A four-color theorem)
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