Log24

Sunday, January 22, 2023

“Preform an affine transformation”

Filed under: General — Tags: , — m759 @ 4:42 pm

"Before time began, there was the Cube." — Transformers  (2007)

Related literature:

Thursday, September 22, 2022

Affine Dürer

Filed under: General — Tags: — m759 @ 6:21 pm

The previous post's image illustrating the
ancient Lo Shu  square as an affine transformation
suggests a similar view of Dürer's square.

That view illustrates the structural principle
underlying the diamond theorem


Thursday, August 25, 2022

Affine I Ching

Filed under: General — Tags: — m759 @ 10:45 pm

'Affine I Ching' image search

Affine Lo Shu

Filed under: General — Tags: , — m759 @ 5:37 pm

'Affine Lo Shu' Google search

Monday, March 18, 2024

Ekphrasis for Lily

Filed under: General — m759 @ 8:31 pm

Sunday, March 10, 2024

Permutations of Congruent Subarrays

The groups generated as above are affine groups in finite geometries.

What other results are known from this area of research,
which might be called "groups generated by permutations of
congruent subarrays"? (Search phrase: "congruent subarrays")

Sunday, March 3, 2024

Deep Blue Research: A Report by You.com AI

Filed under: General — Tags: , — m759 @ 12:34 pm
 

Cullinane Diamond Theorem Research Report

by https://you.com/?chatMode=research on March 3, 2024

Overview of the Cullinane Diamond Theorem

The Cullinane Diamond Theorem is a mathematical concept developed by Steven H. Cullinane that explores the symmetrical properties of certain geometric patterns. It is particularly concerned with the structure of finite projective geometry as it relates to the 35 square patterns found in R. T. Curtis's Miracle Octad Generator (MOG).

The theorem asserts that every G-image of a diamond figure D possesses some form of ordinary or color-interchange symmetry.

Symmetry and Group Theory

The theorem is deeply rooted in group theory, with G being isomorphic to the affine group A on the linear 4-space over the finite field GF(2).

This group has a significant order of 322,560 and underlies the symmetry of both the diamond-theorem figures and the square patterns of the MOG . The 35 structures of the 840 G-images of D are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2), which explains the symmetry of the line diagrams and the two-color patterns.

The Diamond Theorem in Various Dimensions

The diamond theorem can be applied to different cases, such as the 4×4, 2×2, and even 3-dimensional cubes like the 2x2x2 and 4x4x4 cubes.

The generalized statement of the theorem involves inscribing a white diamond in a black square and dividing it into four quadrants, each with a diagonal split into black and white halves.

Algebraic Implications

The theorem has algebraic implications, such as the ability to express every 4-coloring as a sum of three 2-colorings.

This algebraic formulation is useful for demonstrating that certain four-colored graphic images can form a ring under multiplication. There is also an infinite family of such 'diamond' rings, isomorphic to rings of matrices over GF(4).

Philosophical and Cultural Context

Steven H. Cullinane has also explored the philosophical implications of the diamond theorem, referring to it in the context of Plato's Meno dialogue and opposing it to the 'Story Theory of truth' as discussed by Richard J. Trudeau.

The theorem is seen as a correct statement of the relation between geometry and the physical universe.

Clarification on Cullinan Diamonds

It is important to note that the Cullinane Diamond Theorem is unrelated to the Cullinan diamond, which is a famous large gemstone discovered in South Africa and now part of the British Crown Jewels.

The Cullinan diamond has its own historical and cultural significance, with its various cut shapes and ownership history.

In summary, the Cullinane Diamond Theorem is a mathematical concept that reveals the symmetrical properties of certain geometric patterns, with applications in group theory and algebra. It also has philosophical implications, drawing connections to classical ideas about truth and geometry.

Instagram ad for You.com AI in research mode

"Show me ALL  your sources, babe."

— Line adapted from Leonardo DiCaprio

Monday, February 19, 2024

Mythspace: Its Logic, Poetry, and Geometry

Filed under: General — Tags: — m759 @ 12:42 pm

For logicians

http://www.log24.com/log/pix09A/090815-Grid8x8.gif

For poets

The 1955 Levi-Strauss 'canonic formula' in its original context of permutation groups

For geometers

Affine groups on small binary spaces

Sunday, February 4, 2024

Microsoft Copilot Doggerel

Filed under: General — Tags: — m759 @ 4:31 am

A poem written yesterday by Microsoft Copilot —

The prompt: "Write a poem about the Cullinane diamond theorem."

In the mind of the mathematician,
Among the figures and forms,
Lies the Cullinane diamond theorem,
A structure that transforms.

Four-diamond figure, a 4×4 array,
Of two-color diagonally-divided square tiles,
Permutations of these sixteen,
In rows, columns, and quadrants compiles.

Every G-image of the four-diamond figure,
Has some ordinary or color-interchange symmetry,
A pattern, a structure, a mathematical picture,
A testament to geometry's clarity.

The symmetry of the line diagrams,
Accounts for the symmetry of the two-color patterns,
A proof shows that a 2nx2n two-color triangular half-squares pattern,
Must have a 2×2 center with a symmetry that matters.

And this symmetry must be shared by the entire pattern,
A statement, a theorem, a mathematical lantern,
G is isomorphic to the affine group A,
On the linear 4-space over GF (2), they say.

The 35 structures of the 840 = 35 x 24 G-images,
Are isomorphic to the 35 lines,
In the 3-dimensional projective space over GF (2),
A concept that shines.

So here lies the beauty of the Cullinane diamond theorem,
In the mind of the mathematician, it's a gem,
A structure, a pattern, a mathematical diagram,
A testament to the beauty of STEM.

Actually, the complete  prompt, suggested by a Log24 post of
12:12 PM ET yesterday, was . . .

"Write a poem in the style of Wallace Stevens
about the Cullinane diamond theorem."

Copilot still has a thing or two to learn about poetic style.

Saturday, January 20, 2024

Needful Things: Faustus at the Magic Mountain

Filed under: General — Tags: , — m759 @ 6:55 am

Image from a Sunday, January 7th, 2024, post now tagged "A Seventh Seal" —

Related image from a "Mathematics for Davos" post of
Thursday, January 18, 2024 —

Thursday, January 18, 2024

But Seriously: Mathematics for Davos

Filed under: General — Tags: , — m759 @ 1:03 pm

Click image to enlarge.

Wednesday, January 10, 2024

The Crimson Abyss . . .

Filed under: General — Tags: — m759 @ 6:58 am

Continues . . .

"And as the characters in the meme twitch into the abyss
that is the sky, this meme will disappear into whatever
internet abyss swallowed MySpace."

—Staff writer Kamila Czachorowski, Harvard Crimson , March 29, 2017

Myspace.com (today) —

See also  this  journal on New Year's Eve 2005
and other remarks from that date . . .

Mytruth.com —

NOTE: Do not try to view the current  version of mytruth.com.
It was blocked by my antivirus program due to a possible trojan.

Tuesday, December 19, 2023

Copilot Report

Filed under: General — Tags: , , — m759 @ 12:16 am
 
From Microsoft Copilot (with "Precise" setting)
on Dec. 18, 2023 —
 
How is the 1910 work of G. M. Conwell on PG(3,2) related to the Miracle Octad Generator of R. T. Curtis?

Searching for: Relation between G. M. Conwell work on PG(3,2) and Miracle Octad Generator of R. T. Curtis

Searching for: Relation between G. M. Conwell work on PG(3,2) and Miracle Octad Generator of R. T. Curtis explained

Generating answers for you…

The work of George M. Conwell on the finite projective 3-space PG(3,2) and the Miracle Octad Generator (MOG) of R. T. Curtis are both significant contributions to the field of combinatorial mathematics12.

Conwell’s work on PG(3,2) provided a detailed study of the structure of this finite projective space1On the other hand, the MOG is a tool for exploring the binary Golay code and its automorphism group M242.

The connection between these two works lies in the underlying combinatorial structures they explore. The Rosenhain and Göpel tetrads, which are key structures in the finite geometry PG(3,2), play a significant role in the Curtis Miracle Octad Generator3These structures provide a link between the finite geometric structures studied by Conwell and the combinatorial structures generated by the MOG3.

For a more detailed understanding of this connection, I recommend checking out the resources mentioned in the references123. If you have any specific questions about the concepts in these works, feel free to ask! I’ll do my best to help.

Learn more

5 archive.org
_________________________________________________________

__________________________________________________________


DOCUMENT SUMMARY (generated by Copilot)
of reference 3 above, from academia.edu —
"Rosenhain and Göpel Tetrads in PG(3,2),"
by Steven H. Cullinane on March 17, 2013

Here is a summary of the document:

Reference:
 
Copilot suggestions following document summary —
 



Correction:

"Subsets of four points in a projective 3-space" above is a Copilot error, and not
from the document being summarized. It should be "subsets of four points in an
affine 4-space."

Sunday, December 17, 2023

Scholastic Stochastic

Filed under: General — Tags: — m759 @ 1:14 pm

"Let G be the group generatied by arbitrarily  mixing random
permutations of rows and of columns with random  permutations
of the four 2×2 quadrants…. G is isomorphic to the affine group
on V (GF(2))." — Steven H. Cullinane, October 1978.

Compare and contrast with another discussion of randomness and
affine groups, from American Mathematical Monthly, November 1995 —

Related material not cited by Poole in 1995 —

J. E. Johnson, "Markov-type Lie groups in GL(n, R),"
J. Math. Phys. 26, 252-257 (1985).

The above Johnson citation is from an article that also discusses
the work of Poole —

Irene Paniello, "On Actions on Cubic Stochastic Matrices,"
arXiv preprint dated April 22, 2017. Published in
Markov Processes and Related Fields, 2017, v.23, Issue 2, 325-348.

Monday, December 4, 2023

Latin and Latin Squares

Filed under: General — Tags: , — m759 @ 8:37 pm

"… it is not just its beauty that has made Mathematics so attractive.
Thirty or so years ago, a philosopher friend of mine remarked
rather dolefully, 'I am afraid that Latin, the knowledge of which
used to be the mark of a civilised person, will be replaced by
Mathematics as the universally accepted mark of learning.'
This was probably the most prescient statement he ever made,
as the importance of Mathematics is now recognised in fields
as diverse as medicine, linguistics, and even literature."

Address by mathematician Dominic Welsh on June 16, 2006

Some Latin-square  images from pure mathematics

Some related Latin from this  journal on June 16, 2006 —

AD PULCHRITUDINEM TRIA REQUIRUNTUR:
INTEGRITAS, CONSONANTIA, CLARITAS.

St. Thomas Aquinas

For some remarks on Latin-square structure,
see other posts tagged Affine Squares.

Monday, November 27, 2023

Birkhoff-von Neumann Symmetry* over Finite Fields

Filed under: General — Tags: — m759 @ 8:09 pm

See David G. Poole, "The Stochastic Group,"
American Mathematical Monthly,  volume 102, number 9
(November, 1995), pages 798–801.

* This post was suggested by the phrase "The Diamond Theorem,
also known as the von Neumann-Birkhoff conjecture" in a
ChatGPT-3.5 hallucination today.

That phrase suggests a look at the Birkhoff-von Neumann theorem:

The B.-von N. theorem suggests a search for analogous results
over finite fields. That search yields the Poole paper above,
which is  related to my own "diamond theorem" via affine groups.

Saturday, November 11, 2023

A Star for David

Filed under: General — Tags: — m759 @ 3:27 pm

In memory of poet David Ferry, who reportedly died
at 99 last Sunday — Guy Fawkes Day —
an image linked to here  on that day . . .

Sunday, November 5, 2023

For the Old Guy

Filed under: General — Tags: , — m759 @ 12:51 am

https://www.instagram.com/p/CeB2rA5O4ey/

Saturday, September 16, 2023

A Cube for Casaubon

Filed under: General — Tags: , — m759 @ 1:25 pm

The 1955 Levi-Strauss 'canonic formula' in its original context of permutation groups

Later . . .

Affine groups on small binary spaces

Thursday, June 15, 2023

Michaelmas 2019

Filed under: General — Tags: — m759 @ 1:06 pm

Transcribed from a PDF:

Received September 29, 2019, accepted October 15, 2019,
date of publication October 24, 2019, date of current version
November 7, 2019.

Digital Object Identifier 10.1109/ACCESS.2019.2949310

A Method for Determining
the Affine Equivalence of Boolean Functions

ZIYU WANG1 , XIAO ZENG1 , JINZHAO WU2,3, AND
GUOWU YANG1

1Big Data Research Center, School of Computer Science
and Engineering, University of Electronic Science and Technology
of China, Chengdu 611731, China

2Guangxi Key Laboratory of Hybrid Computation and
IC Design Analysis, Guangxi University for Nationalities,
Nanning 530006, China

3School of Computer and Electronic Information,
Guangxi University, Nanning 530004, China

Corresponding authors:
Jinzhao Wu (gxmdwjzh@aliyun.com) and
Guowu Yang (ygwuestc@163.com)

This work was supported in part by the National Natural Science Foundation
of China under Grant 61772006 and Grant 61572109, in part by the
State Key Laboratory of Information Security, Institute of Information Engineering, 
Chinese Academy of Sciences, Beijing, in part by the Science and Technology
Program of Guangxi  under Grant AB17129012, in part by the Science and
Technology Major Project of Guangxi under Grant AA17204096, in part by 
the Special Fund for Scientific and Technological Bases and Talents
of Guangxi under Grant 2016AD05050, and in part by the Special Fund for
Bagui Scholars of Guangxi, in part by the Open fund of State Key Laboratory 
of Information Security.

ABSTRACT 
Determining the affine equivalence of Boolean functions
has significant applications in circuit and cryptography.
Previous methods for determining this require a large
amount of computation when Boolean functions are bent
functions or when the truth table is sparse. This paper
presents a new method to determine the affine equivalence
based on matrix algebra. By transforming Boolean function
to the corresponding matrix representation, we first propose
and prove the congruent standard form of Boolean function.
It lays the foundation for the determination of equivalence
because affine Boolean functions must have the same
standard form. Then we find the generators of orthogonal
matrix group and symplectic matrix group, which greatly
reduce the search space. The computation complexity of
our method is o (2r2/2+n∗(nr) ), where is the number of
bit operations, and r  is the rank of the matrix, which is
the product of Boolean-1 matrix of the test Boolean function
and its transposition. The experimental results show that our
method is useful when the test Boolean function is no more
than 7 bits and r  is greater than 2.

INDEX TERMS  Logic synthesis, Boolean functions,
affine equivalence, matrix group, algorithm.

Meanwhile . . .

Monday, May 15, 2023

Chatbot Review

Filed under: General — Tags: , , — m759 @ 9:49 pm
 

By ChatGPT* — 

What is the Cullinane diamond theorem? 
5/10/2023, revised as below on 5/15/2023

The Cullinane Diamond Theorem, also known as the Diamond Theorem, is a result in combinatorial mathematics proposed by Steven H. Cullinane. The theorem explores connections between finite geometries, algebraic structures, and combinatorial designs. 

The theorem is primarily concerned with four related structures:
 
1. Finite geometries: Specifically, the theorem relates to finite projective geometries over finite fields, such as projective planes and projective spaces. 

2. Vector spaces: The theorem connects finite vector spaces over finite fields to the aforementioned finite geometries. 

3. Boolean functions:* The theorem establishes relationships between the finite geometries and certain Boolean functions derived from them. 

4. Symmetry groups: The theorem involves symmetry groups associated with the finite geometries.
 
The precise details and implications of the Cullinane Diamond Theorem are beyond the scope of this response. It involves abstract algebra, group theory, combinatorial designs, and finite fields. The theorem has been the subject of various publications and research discussions in mathematical literature. If you are interested in delving deeper into the Cullinane Diamond Theorem, I recommend referring to the original works by Steven H. Cullinane and related research articles on the topic.

* With one revision by Cullinane: "Boolean functions" replaces "Boolean algebras." See (for instance) M. A. Harrison, “On the classification of Boolean functions by the general linear and affine groups,” Journal of the Society for Industrial and Applied Mathematics  1964 12:2, 285-299.

Tuesday, May 9, 2023

Plan 9: The Center

Filed under: General — Tags: , , — m759 @ 12:00 am

The Lo Shu as a Finite Space

Wednesday, May 3, 2023

Art Lesson for Doktor Faustus

Filed under: General — Tags: , , — m759 @ 9:16 pm

Exercise Show that Dürer's 1514 "magic" square is an affine automorphism.

For a solution, see other posts now tagged Affine Squares.

Friday, April 28, 2023

The Small Space Model

Filed under: General — Tags: , , , — m759 @ 6:28 pm

From the previous post, "The Large Language Model,"
a passage from Wikipedia —

"… sometimes large models undergo a 'discontinuous phase shift' 
where the model suddenly acquires substantial abilities not seen
in smaller models. These are known as 'emergent abilities,' and
have been the subject of substantial study." — Wikipedia

Compare and contrast 
this with the change undergone by a "small space model,"
that of the finite affine 4-space A  with 16 points (a Galois tesseract ), 
when it is augmented by an eight-point "octad." The 30 eight-point
hyperplanes of A  then have a natural extension within the new
24-point set to 759 eight-point octads, and the 322,560 affine
automorphisms of the space expand to the 244,823,040 Mathieu
automorphisms of the 759-octad set — a (5, 8, 24) Steiner system.

For a visual analogue of the enlarged 24-point space and some remarks
on analogy by Simone Weil's brother, a mathematician, see this journal
on September 8 and 9, 2022.

Sunday, February 26, 2023

TOE

Filed under: General — Tags: — m759 @ 2:24 am


 

Related images from The Crimson Abyss —

1984 —

IMAGE- 'Affine Groups on Small Binary Spaces,' illustration

2010 —

Logo design for Stack Exchange Math by Jin Yang

Saturday, January 21, 2023

Dead-Poet Witcraft

Filed under: General — Tags: , , — m759 @ 2:40 pm

"Death is the mother of beauty." — Wallace Stevens

From the 2020 Feast of St. Wallace Stevens,
who reportedly died in 1955 on August 2 —

Related material —

Durer magic square as an affine transformation

Exercise Can each  order-4 magic square be obtained by some
transformation like the one above (i.e., preserving affine hyperplanes)?
If not, why not?

Update of 31 Jan. 2023 — Spoiler Remarks by Tilman Piesk.

Friday, January 13, 2023

The “Diamond Space” of Mazzola

Filed under: General — Tags: , , — m759 @ 11:30 am

The Source —

Some similar notions from my own work . . .

The "Digraph" of Mazzola might correspond to a directed graph
indicating the structure of a permutation, as at right below —

Mazzola's "Formula" might correpond to a matrix and translation that
transform the above "Space" of eight coordinates, and his "Gesture"
to a different way of generating affine transformations of that space . . . 
as in my webpage "Cube Space, 1984-2003."

Thursday, October 6, 2022

From Mysticism to Mathematics

Filed under: General — Tags: , — m759 @ 2:48 pm

[Klein, 1983] S. Klein.
"Analogy and Mysticism and the Structure of Culture
(and Comments & Reply)
"
Current Anthropology , 24 (2):151–180, 1983.

The citation above is from a 2017 paper —

"Analogy-preserving Functions:
A Way to Extend Boolean Samples
,"
by M. Couceiro, N. Hug, H. Prade, G, Richard.
26th International Joint Conference on Artificial Intelligence
(IJCAI 2017), Aug. 2017, Melbourne, Australia. pp.1-7, ff.

That 2017 paper discusses Boolean functions .

Some more-recent remarks on these functions
as pure  mathematics —

"On the Number of Affine Equivalence Classes
of Boolean Functions,
" by Xiang-dong Hou,
arXiv:2007.12308v2 [math.CO]. Rev. Aug. 18, 2021.

See also other posts now tagged Analogy and Mysticism.

Thursday, September 22, 2022

De Colores

Filed under: General — Tags: — m759 @ 8:47 am

See as well . . .

Three-color patterns from 1964,
  rendered as shades of grey —

A rather different approach —

Saturday, September 10, 2022

Orthogonal Latin Triangles

Filed under: General — Tags: , — m759 @ 1:38 am

From a 1964 recreational-mathematics essay —

Note that the first two triangle-dissections above are analogous to
mutually orthogonal Latin squares . This implies a connection to
affine transformations within Galois geometry. See triangle graphics
in this  journal.

Affine transformation of 'magic' squares and triangles: the triangle Lo Shu 

Update of 4:40 AM ET —

Other mystical figures —

Magic cube and corresponding hexagram, or Star of David, with faces mapped to lines and edges mapped to points

"Before time began, there was the Cube."

— Optimus Prime in "Transformers" (Paramount, 2007)

Wednesday, June 22, 2022

Occupy Space  Continues.

Filed under: General — Tags: — m759 @ 8:33 am

Alternate Title —

Types of Ambiguity:
The Circle in the Triangle,
the Singer in the Song.

From an excellent June 17 Wall Street Journal  review of a new
Isaac Bashevis Singer book from Princeton University Press

" 'Old Truths and New Clichés,' a collection of 19
prose articles, most appearing in English for the
first time, reveals that Singer was as consummate
an essayist as he was a teller of tales." — Benjamin Balint 

From a search in this  journal for Singer

Related material —

From a post of June 2, "Self-Enclosing" —

"… the self-enclosing processes by which late 20th-century
American academics established and secured their status
(you painfully develop a thesis in competition with your peers,
then you keep on elaborating it until you die)."

— Colin Burrow in the June 9, 2022 issue 
of London Review of Books

Affine transformation of 'magic' squares and triangles: the triangle Lo Shu 

From the December 14, 2021, post Notes on Lines —

Triangle (percussion instrument)

The triangle, a percussion instrument that was
featured prominently in the Tom Stoppard play
"Every Good Boy Deserves Favour."

Tuesday, June 14, 2022

A Triangle of Sadness

Filed under: General — Tags: — m759 @ 7:58 am

 The title refers to this year's
 Cannes Film Festival winner.

Houghton, Mifflin, Harcourt, and Yehoshua

Related material:

From a post of June 2, "Self-Enclosing" —

"… the self-enclosing processes by which late 20th-century
American academics established and secured their status
(you painfully develop a thesis in competition with your peers,
then you keep on elaborating it until you die)."

— Colin Burrow in the June 9, 2022 issue 
of London Review of Books

From a post of June 13, "The Theater Game" —

From a post of June 12, "Triangle.graphics, 2012-2022" —

Affine transformation of 'magic' squares and triangles: the triangle Lo Shu 

Sunday, June 12, 2022

Triangle.graphics, 2012-2022

Filed under: General — Tags: , , — m759 @ 2:13 am

Affine transformation of 'magic' squares and triangles: the triangle Lo Shu

Saturday, June 4, 2022

Old Dog, New Trick

Filed under: General — Tags: — m759 @ 3:04 am

Affine transformation of 'magic' squares and triangles: the triangle Lo Shu

Thursday, May 26, 2022

Mystical Mathematicks

Filed under: General — Tags: , , — m759 @ 10:17 am

Affine transformation of 'magic' squares and triangles: the triangle Lo Shu

Friday, May 20, 2022

Squares to Triangles

Filed under: General — Tags: , , , , — m759 @ 11:24 am

(Continued)

Related concepts: Steiner system, Affine transformation, Square triangle.

Monday, May 16, 2022

Sketch for a Magic Triangle

'Magic Triangle' by Steven H. Cullinane, 16 May 2022

Updates from later the same day —

Related affine structures —

'Magic Triangle' affine structure

See also "Square+Triangles" in this journal.

 

The fishlike shapes within three of the above
ninefold colored triangles suggest some . . .

Related Entertainment —

Wednesday, April 27, 2022

Ennead  (Pace Moon Knight)

Filed under: General — Tags: , , — m759 @ 1:33 pm

Putting the graphic  in lexicographic

'The 3x3 Magic Square as an Affine Transformation'

Sunday, April 10, 2022

Plan 9 Continues . . .

Filed under: General — Tags: , , — m759 @ 10:26 pm

A meditation on Coxeter's Aleph

'The 3x3 Magic Square as an Affine Transformation'

Saturday, March 26, 2022

Box Geometry: Space, Group, Art  (Work in Progress)

Filed under: General — Tags: — m759 @ 2:06 am

Many structures of finite geometry can be modeled by
rectangular or cubical arrays ("boxes") —
of subsquares or subcubes (also "boxes").

Here is a draft for a table of related material, arranged
as internet URL labels.

Finite Geometry Notes — Summary Chart
 

Name Tag .Space .Group .Art
Box4

2×2 square representing the four-point finite affine geometry AG(2,2).

(Box4.space)

S4 = AGL(2,2)

(Box4.group)

 

(Box4.art)

Box6 3×2 (3-row, 2-column) rectangular array
representing the elements of an arbitrary 6-set.
S6  
Box8 2x2x2 cube or  4×2 (4-row, 2-column) array. S8 or Aor  AGL(3,2) of order 1344, or  GL(3,2) of order 168  
Box9 The 3×3 square. AGL(2,3) or  GL(2,3)  
Box12 The 12 edges of a cube, or  a 4×3  array for picturing the actions of the Mathieu group M12. Symmetries of the cube or  elements of the group M12  
Box13 The 13 symmetry axes of the cube. Symmetries of the cube.  
Box15 The 15 points of PG(3,2), the projective geometry
of 3 dimensions over the 2-element Galois field.
Collineations of PG(3,2)  
Box16 The 16 points of AG(4,2), the affine geometry
of 4 dimensions over the 2-element Galois field.

AGL(4,2), the affine group of 
322,560 permutations of the parts
of a 4×4 array (a Galois tesseract)

 
Box20 The configuration representing Desargues's theorem.    
Box21 The 21 points and 21 lines of PG(2,4).    
Box24 The 24 points of the Steiner system S(5, 8, 24).    
Box25 A 5×5 array representing PG(2,5).    
Box27 The 3-dimensional Galois affine space over the
3-element Galois field GF(3).
   
Box28 The 28 bitangents of a plane quartic curve.    
Box32 Pair of 4×4 arrays representing orthogonal 
Latin squares.
Used to represent
elements of AGL(4,2)
 
Box35 A 5-row-by-7-column array representing the 35
lines in the finite projective space PG(3,2)
PGL(3,2), order 20,160  
Box36 Eurler's 36-officer problem.    
Box45 The 45 Pascal points of the Pascal configuration.    
Box48 The 48 elements of the group  AGL(2,3). AGL(2,3).  
Box56

The 56 three-sets within an 8-set or
56 triangles in a model of Klein's quartic surface or
the 56 spreads in PG(3,2).

   
Box60 The Klein configuration.    
Box64 Solomon's cube.    

— Steven H. Cullinane, March 26-27, 2022

Thursday, February 3, 2022

Four-Color Structures (Review)

Filed under: General — Tags: , , — m759 @ 1:30 pm

Four-color decomposition applied to the 8-point binary affine space

Miracle Octad Generator — Analysis of Structure

For those who prefer art that is less abstract — Heartland Sutra.

Saturday, January 29, 2022

On the Diamond-Theorem Group* of Order 322,560

Filed under: General — Tags: , , , , , — m759 @ 2:06 am

Taormina and Wendland have often discussed this group, which they
call "overarching" within the context of their Mathieu-moonshine research.

This seems to be the first time they have attempted to explore its geometric
background as an affine group, apart from its role as "the octad group" in the
researches of R. T. Curtis and John Conway on the large Mathieu group M24.

* See a Log24 post of June 1, 2013.

Friday, December 31, 2021

Aesthetics in Academia

Filed under: General — Tags: — m759 @ 9:33 am

Related art — The non-Rubik 3x3x3 cube —

The above structure illustrates the affine space of three dimensions
over the three-element finite (i.e., Galois) field, GF(3). Enthusiasts
of Judith Brown's nihilistic philosophy may note the "radiance" of the
13 axes of symmetry within the "central, structuring" subcube.

I prefer the radiance  (in the sense of Aquinas) of the central, structuring 
eightfold cube at the center of the affine space of six dimensions over
the two-element field GF(2).

Friday, December 10, 2021

Dance of the Lo Shu

Filed under: General — Tags: , , — m759 @ 4:46 am

The ancient Chinese matrix known as the Lo Shu
is one of 432 matrices equivalent under the action of . . .

The Lo Shu Group:

For related material, see (for instance) AGL(2,3) in . . .

"Let be be finale of seem.
The only emperor is the emperor of ice-cream."

— Wallace Stevens

Thursday, December 9, 2021

Lo Shu Space . . .

Filed under: General — Tags: — m759 @ 12:43 am

. . . is now at loshu.space. (Update on 10 Dec. — See also loshu.group.)

See as well GL(2,3) in this journal.

The Lo Shu as a Finite Space

Tuesday, December 7, 2021

Tortoise Variations

Filed under: General — Tags: , — m759 @ 2:42 am

IMAGE- Herbert John Ryser, 'Combinatorial Mathematics' (1963), page 1

Fanciful version —

Less fanciful versions . . . 

Unmagic Squares

Consecutive positive integers:

1   2   3
4   5   6
7   8   9

Consecutive nonnegative integers:

0   1   2
3   4   5
6   7   8

Consecutive nonnegative integers
written in base 3:

00  01  02
10  11  12
20  21  22

This last square may be viewed as
coordinates, in the 3-element Galois
field GF(3), of the ninefold square.

Note that the ninefold square so viewed
embodies the 12 lines of the two-dimensional
affine space over GF(3)

As does, similarly, the ancient Chinese
"magic" square known as the "Lo Shu."

These squares are therefore equivalent under
affine transformations.

This method generalizes.

— Steven H. Cullinane, Nov. 20, 2021

 

The Lo Shu as a Finite Space

Saturday, November 20, 2021

The Unmagicking

Filed under: General — Tags: — m759 @ 11:51 am
 

Unmagic Squares

Consecutive positive integers:

1   2   3
4   5   6
7   8   9

Consecutive nonnegative integers:

0   1   2
3   4   5
6   7   8

Consecutive nonnegative integers
written in base 3:

00  01  02
10  11  12
20  21  22

This last square may be viewed as
coordinates, in the 3-element Galois
field GF(3), of the ninefold square.

Note that the ninefold square so viewed
embodies the 12 lines of the two-dimensional
affine space over GF(3)

As does, similarly, the ancient Chinese
"magic" square known as the "Lo Shu."

These squares are therefore equivalent under
affine transformations.

This method generalizes.

— Steven H. Cullinane, Nov. 20, 2021

Monday, July 12, 2021

Educational Series

Filed under: General — Tags: , — m759 @ 11:06 am

(Continued from St. Luke's Day, 2014)


 

Tablet:

 

The Lo Shu as a Finite Space
 

Cube:

 

IMAGE- A Galois cube: model of the 27-point affine 3-space

Saturday, May 8, 2021

A Tale of Two Omegas

Filed under: General — Tags: , — m759 @ 5:00 am

The Greek capital letter Omega, Ω, is customarily
used to denote a set that is acted upon by a group.
If the group is the affine group of 322,560
transformations of the four-dimensional
affine space over the two-element Galois field,
the appropriate Ω is the 4×4 grid above.

See the Cullinane diamond theorem .

If the group is the large Mathieu group  of
244,823,040 permutations of 24 things,
the appropriate Ω  is the 4×6 grid below.

See the Miracle Octad Generator  of R. T. Curtis.

Sunday, December 6, 2020

The Undoing

Filed under: General — m759 @ 10:16 pm

Today’s earlier post “Binary Coordinates” discussed a Dec. 6
revision to the Wikipedia article on PG(3,2), the projective
geometry of 3 dimensions over the 2-element field GF(2).

The revision, which improved the article, was undone later today
by a clueless retired academic, one William “Bill” Cherowitzo,
a professor emeritus of mathematics at U. of Colorado at Denver.
(See his article “Adventures of a Mathematician in Wikipedia-land,”
MAA Focus , December 2014/January 2015.)

See my earlier remarks on this topic . . . specifically, on this passage —

“A 3-(16,4,1) block design has 140 blocks
of size 4 on 16 points, such that each triplet
of points is covered exactly once. Pick any
single point, take only the 35 blocks
containing that point, and delete that point.
The 35 blocks of size 3 that remain comprise
a PG(3,2) on the 15 remaining points.”

As I noted on November 17, this is bullshit. Apparently Cherowitzo
never bothered to find out that an arbitrary  “3-(16,4,1) block design”
(an example of a Steiner quadruple system ) does not  yield a PG(3,2).

PG(3,2) is derived from the classical  3-(16,4,1) block design formed by the affine
space of 4 dimensions over GF(2).  That  design has 322,560 automorphisms.
In contrast, see a 3-(16,4,1) block design that is  automorphism-free.

Wednesday, November 11, 2020

Qube

Filed under: General — Tags: , — m759 @ 8:30 pm

The new domain qube.link  forwards to . . .
http://finitegeometry.org/sc/64/solcube.html .

More generally, qubes.link  forwards to this post,
which defines qubes .

Definition: A qube  is a positive integer that is
a prime-power cube , i.e. a cube that is the order
of a Galois field. (Galois-field orders in general are
customarily denoted by the letter q .)

Examples:  8, 27, 64.  See qubes.site.

Update on Nov. 18, 2020, at about 9:40 PM ET —

Problem:

For which qubes, visualized as n×n×n arrays,
is it it true that the actions of the two-dimensional
galois-geometry affine group on each n×n face, extended
throughout the whole array, generate the affine group
on the whole array? (For the cases 8 and 64, see Binary
Coordinate Systems and  Affine Groups on Small
Binary Spaces.)

Saturday, October 24, 2020

The Galois Tesseract

Filed under: General — Tags: — m759 @ 9:32 am

Stanley E. Payne and J. A. Thas in 1983* (previous post) —

“… a 4×4 grid together with
the affine lines on it is AG(2,4).”

Payne and Thas of course use their own definition
of affine lines on a grid.

Actually, a 4×4 grid together with the affine lines on it
is, viewed in a different way, not AG(2,4) but rather AG(4,2).

For AG(4,2) in the proper context, see
Affine Groups on Small Binary Spaces and
The Galois Tesseract.

* And 26 years later,  in 2009.

Grids

Filed under: General — Tags: — m759 @ 9:00 am

Wikipedia on what has been called “the doily” —

“The smallest non-trivial generalized quadrangle
is GQ(2,2), whose representation* has been dubbed
‘the doily’ by Stan Payne in 1973.”

A later publication relates the doily to grids.

From Finite Generalized Quadrangles , by Stanley E. Payne
and J. A. Thas, December 1983, at researchgate.net, pp. 81-82—

“Then the lines … define a 3×3 grid G  (i.e. a grid
consisting of 9 points and 6 lines).”
. . . .
“So we have shown that the grid G  can completed [sic ]
in a unique way to a grid with 8 lines and 16 points.”
. . . .
“A 4×4 grid defines a linear subspace
of  the 2−(64,4,1) design, i.e. a 4×4 grid
together with the affine lines on it is AG(2,4).”

A more graphic approach from this journal —

Seven is Heaven...

Click the image for further details.

* This wording implies that GQ(2,2) has a unique
visual representation. It does not. See inscape .

Sunday, August 2, 2020

Zero-Sum Theorem

Filed under: General — Tags: — m759 @ 6:41 am

Durer Magic Square as an affine transformation

Monday, April 27, 2020

The Cracked Nut

Filed under: General — Tags: , , — m759 @ 1:25 pm

“At that instant he saw, in one blaze of light, an image of unutterable
conviction, the reason why the artist works and lives and has his being –
the reward he seeks –the only reward he really cares about, without which
there is nothing. It is to snare the spirits of mankind in nets of magic,
to make his life prevail through his creation, to wreak the vision of his life,
the rude and painful substance of his own experience, into the congruence
of blazing and enchanted images that are themselves the core of life, the
essential pattern whence all other things proceed, the kernel of eternity.”

— Thomas Wolfe, Of Time and the River

“… the stabiliser of an octad preserves the affine space structure on its
complement, and (from the construction) induces AGL(4,2) on it.
(It induces A8 on the octad, the kernel of this action being the translation
group of the affine space.)”

— Peter J. Cameron,
The Geometry of the Mathieu Groups (pdf)

“The yarns of seamen have a direct simplicity, the whole meaning
of which lies within the shell of a cracked nut. But Marlow was not
typical (if his propensity to spin yarns be excepted), and to him the
meaning of an episode was not inside like a kernel but outside…."

— Joseph Conrad in Heart of Darkness

Monday, April 13, 2020

Tribute

Filed under: General — m759 @ 2:46 pm

The May 2020 Notices of the American Mathematical Society  has a
memorial tribute article on Goro Shimura, who died on May 3, 2019.

See also this  journal on May 3, 2019 in posts now tagged Wondertale.

Related ethnic remark:  “As a Chinese jar…” — T. S.  Eliot

 

“Things can get muddled further….”

— Webpage on the Japanese word

manji.

Indeed they can :

Saturday, March 7, 2020

The “Octad Group” as Symmetries of the 4×4 Square

From "Mathieu Moonshine and Symmetry Surfing" —

(Submitted on 29 Sep 2016, last revised 22 Jan 2018)
by Matthias R. Gaberdiel (1), Christoph A. Keller (2),
and Hynek Paul (1)

(1)  Institute for Theoretical Physics, ETH Zurich
(2)  Department of Mathematics, ETH Zurich

https://arxiv.org/abs/1609.09302v2 —

"This presentation of the symmetry groups Gi  is
particularly well-adapted for the symmetry surfing
philosophy. In particular it is straightforward to
combine them into an overarching symmetry group G
by combining all the generators. The resulting group is
the so-called octad group

G = (Z2)4  A8 .

It can be described as a maximal subgroup of M24
obtained by the setwise stabilizer of a particular
'reference octad' in the Golay code, which we take
to be O= {3,5,6,9,15,19,23,24} ∈ 𝒢24. The octad
subgroup is of order 322560, and its index in M24
is 759, which is precisely the number of
different reference octads one can choose."

This "octad group" is in fact the symmetry group of the affine 4-space over GF(2),
so described in 1979 in connection not with the Golay code but with the geometry
of the 4×4 square.* Its nature as an affine group acting on the Golay code was
known long before 1979, but its description as an affine group acting on
the 4×4 square may first have been published in connection with the
Cullinane diamond theorem and Abstract 79T-A37, "Symmetry invariance in a
diamond ring
," by Steven H. Cullinane in Notices of the American Mathematical
Society
, February 1979, pages A-193, 194.

* The Galois tesseract .

Update of March 15, 2020 —

Conway and Sloane on the "octad group" in 1993 —

Tuesday, January 28, 2020

Very Stable Kool-Aid

Filed under: General — Tags: , , — m759 @ 2:16 pm

Two of the thumbnail previews
from yesterday's 1 AM  post

"Hum a few bars"

"For 6 Prescott Street"

Further down in the "6 Prescott St." post, the link 5 Divinity Avenue
leads to

A Letter from Timothy Leary, Ph.D., July 17, 1961

Harvard University
Department of Social Relations
Center for Research in Personality
Morton Prince House
5 Divinity Avenue
Cambridge 38, Massachusetts

July 17, 1961

Dr. Thomas S. Szasz
c/o Upstate Medical School
Irving Avenue
Syracuse 10, New York

Dear Dr. Szasz:

Your book arrived several days ago. I've spent eight hours on it and realize the task (and joy) of reading it has just begun.

The Myth of Mental Illness is the most important book in the history of psychiatry.

I know it is rash and premature to make this earlier judgment. I reserve the right later to revise and perhaps suggest it is the most important book published in the twentieth century.

It is great in so many ways–scholarship, clinical insight, political savvy, common sense, historical sweep, human concern– and most of all for its compassionate, shattering honesty.

. . . .

The small Morton Prince House in the above letter might, according to
the above-quoted remarks by Corinna S. Rohse, be called a "jewel box."
Harvard moved it in 1978 from Divinity Avenue to its current location at
6 Prescott Street.

Related "jewel box" material for those who
prefer narrative to mathematics —

"In The Electric Kool-Aid Acid Test , Tom Wolfe writes about encountering 
'a young psychologist,' 'Clifton Fadiman’s nephew, it turned out,' in the
waiting room of the San Mateo County jail. Fadiman and his wife were
'happily stuffing three I-Ching coins into some interminable dense volume*
of Oriental mysticism' that they planned to give Ken Kesey, the Prankster-
in-Chief whom the FBI had just nabbed after eight months on the lam.
Wolfe had been granted an interview with Kesey, and they wanted him to
tell their friend about the hidden coins. During this difficult time, they
explained, Kesey needed oracular advice."

— Tim Doody in The Morning News  web 'zine on July 26, 2012**

Oracular advice related to yesterday evening's
"jewel box" post …

A 4-dimensional hypercube H (a tesseract ) has 24 square
2-dimensional faces
.  In its incarnation as a Galois  tesseract
(a 4×4 square array of points for which the appropriate transformations
are those of the affine 4-space over the finite (i.e., Galois) two-element
field GF(2)), the 24 faces transform into 140 4-point "facets." The Galois 
version of H has a group of 322,560 automorphisms. Therefore, by the
orbit-stabilizer theorem, each of the 140 facets of the Galois version has
a stabilizer group of  2,304 affine transformations.

Similar remarks apply to the I Ching  In its incarnation as  
a Galois hexaract , for which the symmetry group — the group of
affine transformations of the 6-dimensional affine space over GF(2) —
has not 322,560 elements, but rather 1,290,157,424,640.

* The volume Wolfe mentions was, according to Fadiman, the I Ching.

** See also this  journal on that date — July 26, 2012.

Friday, August 16, 2019

Stanza Romanza

Filed under: General — Tags: — m759 @ 7:49 pm

Wallace Stevens's 'a million diamonds' quote in Woodman's 'Stanza My Stone'

For those who prefer greater clarity than is offered by Stevens . . .

The A section —

The B section —

"A paper from Helsinki in 2005 says there are more than a million
3-(16,4,1) block designs, of which only one has an automorphism
group of order 322,560. This is the affine 4-space over GF(2)."

Thursday, August 15, 2019

On Steiner Quadruple Systems of Order 16

Filed under: General — Tags: , — m759 @ 4:11 am

An image from a Log24 post of March 5, 2019

Cullinane's 1978  square model of PG(3,2)

The following paragraph from the above image remains unchanged
as of this morning at Wikipedia:

"A 3-(16,4,1) block design has 140 blocks of size 4 on 16 points,
such that each triplet of points is covered exactly once. Pick any
single point, take only the 35 blocks containing that point, and
delete that point. The 35 blocks of size 3 that remain comprise
a PG(3,2) on the 15 remaining points."

Exercise —

Prove or disprove the above assertion about a general "3-(16,4,1) 
block design," a structure also known as a Steiner quadruple system
(as I pointed out in the March 5 post).

Relevant literature —

A paper from Helsinki in 2005* says there are more than a million
3-(16,4,1) block designs, of which only one has an automorphism
group of order 322,560. This is the affine 4-space over GF(2),
from which PG(3,2) can be derived using the well-known process
from finite geometry described in the above Wikipedia paragraph.

* "The Steiner quadruple systems of order 16," by Kaski et al.,
   Journal of Combinatorial Theory Series A  
Volume 113, Issue 8, 
   November 2006, pages 1764-1770.

Thursday, June 27, 2019

Group Actions on the 4x4x4 Cube

Filed under: General — Tags: — m759 @ 6:23 am

For affine  group actions, see Ex Fano Appollinis  (June 24)
and Solomon's Cube.

For one approach to Mathieu  group actions on a 24-cube subset
of the 4x4x4 cube, see . . .

For a different sort of Mathieu cube, see Aitchison.

Friday, May 3, 2019

“As a Chinese jar” — T. S. Eliot

Filed under: General — Tags: , , , , — m759 @ 1:06 pm

 

Tuesday, March 5, 2019

A Block Design 3-(16,4,1) as a Steiner Quadruple System:

Filed under: General — Tags: , , , — m759 @ 11:19 am

A Midrash for Wikipedia 

Midrash —

Related material —


________________________________________________________________________________

The Miracle Octad Generator (MOG), the affine 4-space over GF(2), and the Cullinane diamond theorem

Friday, March 1, 2019

Wikipedia Scholarship (Continued)

Filed under: General — Tags: , , , , — m759 @ 12:45 pm

This post continues a post from yesterday on the square model of
PG(3,2) that apparently first appeared (presented as such*) in . . .

Cullinane, "Symmetry invariance in a diamond ring,"
Notices of the AMS , pp. A193-194, Feb. 1979.

The Cullinane diamond theorem, AMS Notices, Feb. 1979, pp. A-193-194

Yesterday's Wikipedia presentation of the square model was today
revised by yet another anonymous author —

Revision history accounting for the above change from yesterday —

The jargon "rm OR" means "remove original research."

The added verbiage about block designs is a smokescreen having
nothing to do with the subject, which is square  representation
of the 35 points and lines.

* The 35 squares, each consisting of four 4-element subsets, appeared earlier
   in the Miracle Octad Generator (MOG) of R. T. Curtis (published in 1976).
  They were not at that time  presented as constituting a finite geometry, 
  either affine (AG(4,2)) or projective (PG(3,2)).

Friday, February 22, 2019

Back Issues of AMS Notices

Filed under: General — Tags: , — m759 @ 3:04 pm

From the online home page of the new March issue —

Feb. 22, 2019 — AMS Notices back issues now available.

For instance . . .

The Cullinane diamond theorem, AMS Notices, Feb. 1979, pp. A-193-194

Related material now at Wikipedia

The Miracle Octad Generator (MOG), the affine 4-space over GF(2), and the Cullinane diamond theorem

Friday, February 15, 2019

The Gifts Reserved for Age

Filed under: General — m759 @ 3:30 pm

"But, as the passage now presents no hindrance
     To the spirit unappeased and peregrine
     Between two worlds become much like each other…."

T. S. Eliot, Four Quartets

Related obituary:

https://www.nytimes.com/2019/02/15/obituaries/tom-cade-dead.html

Related date:

"as of Feb. 6, 2019" (from a post at 12 AM ET Feb. 7) —

"There is  such a thing as a four-dimensional finite affine space."
— Saying adapted from a 1962 young-adult novel by Madeleine L'Engle

Thursday, February 7, 2019

Geometry of the 4×4 Square: The Kummer Configuration

Filed under: General — Tags: , , , — m759 @ 12:00 am

From the series of posts tagged Kummerhenge

A Wikipedia article relating the above 4×4 square to the work of Kummer —

A somewhat more interesting aspect of the geometry of the 4×4 square
is its relationship to the 4×6 grid underlying the Miracle Octad Generator
(MOG) of R. T. Curtis.  Hudson's 1905 classic Kummer's Quartic Surface
deals with the Kummer properties above and also foreshadows, without
explicitly describing, the finite-geometry properties of the 4×4 square as
a finite affine 4-space — properties that are of use in studying the Mathieu
group M24  with the aid of the MOG.

Tuesday, December 25, 2018

Simply

Filed under: General — Tags: — m759 @ 11:42 am
 

"So to obtain the isomorphism from L2(7) onto L3(2) we simply
multiply any given permutation of L2(7) by the affine translation
that restores to its rightful place."

— Sphere Packings, Lattices and Groups ,
by John H. Conway and N. J. A. Sloane.
First edition, 1988, published by Springer-Verlag New York, Inc.
Chapter 11 (by J. H. Conway), "The Golay Codes and the Mathieu Groups," 
Section 12, "The trio group 26:(S3×L2(7))"

Compare and contrast —

Why PSL(2,7) is isomorphic to GL(3.2)

This post was suggested by a New York Times  headline today —

Sunday, December 9, 2018

Quaternions in a Small Space

Filed under: G-Notes,General,Geometry — Tags: , , — m759 @ 2:00 pm

The previous post, on the 3×3 square in ancient China,
suggests a review of group actions on that square
that include the quaternion group.

Click to enlarge

Three links from the above finitegeometry.org webpage on the
quaternion group —

Related material —

Iain Aitchison on the 'symmetric generation' of R. T. Curtis

See as well the two Log24 posts of December 1st, 2018 —

Character and In Memoriam.

Sunday, September 16, 2018

Husserl for Beginners

Filed under: General — m759 @ 10:31 pm

http://www.log24.com/log/pix18/180916-Hustvedt-Auster-covers.jpg

See also Trivial + Affine and Eidetic Reduction.

Friday, September 14, 2018

Denkraum

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 1:00 am

http://www.log24.com/log/pix18/180914-Warburg_Denkraum-Google-result.jpg

I Ching Geometry search result

Underlying the I Ching  structure  is the finite affine space
of six dimensions over the Galois field with two elements.

In this field,  "1 + 1 = 0,"  as noted here Wednesday.

See also other posts now tagged  Interstice.

http://www.log24.com/log/pix18/180914-Warburg-Wikipedia.jpg

Saturday, September 8, 2018

Review

Filed under: General,Geometry — Tags: — m759 @ 2:00 am

Affine groups on small binary spaces

Friday, June 29, 2018

Triangles in the Eightfold Cube

From a post of July 25, 2008, “56 Triangles,” on the Klein quartic
and the eightfold cube

Baez’s discussion says that the Klein quartic’s 56 triangles
can be partitioned into 7 eight-triangle Egan ‘cubes’ that
correspond to the 7 points of the Fano plane in such a way
that automorphisms of the Klein quartic correspond to
automorphisms of the Fano plane. Show that the
56 triangles within the eightfold cube can also be partitioned
into 7 eight-triangle sets that correspond to the 7 points of the
Fano plane in such a way that (affine) transformations of the
eightfold cube induce (projective) automorphisms of the Fano plane.”

Related material from 1975 —

More recently

Friday, February 16, 2018

Two Kinds of Symmetry

Filed under: General,Geometry — Tags: — m759 @ 11:29 pm

The Institute for Advanced Study (IAS) at Princeton in its Fall 2015 Letter 
revived "Beautiful Mathematics" as a title:

This ugly phrase was earlier used by Truman State University
professor Martin Erickson as a book title. See below. 

In the same IAS Fall 2015 Letter appear the following remarks
by Freeman Dyson —

". . . a special case of a much deeper connection that Ian Macdonald 
discovered between two kinds of symmetry which we call modular and affine.
The two kinds of symmetry were originally found in separate parts of science,
modular in pure mathematics and affine in physics. Modular symmetry is
displayed for everyone to see in the drawings of flying angels and devils
by the artist Maurits Escher. Escher understood the mathematics and got the
details right. Affine symmetry is displayed in the peculiar groupings of particles
created by physicists with high-energy accelerators. The mathematician
Robert Langlands was the first to conjecture a connection between these and
other kinds of symmetry. . . ." (Wikipedia link added.)

The adjective "modular"  might aptly be applied to . . .

The adjective "affine"  might aptly be applied to . . .

From 'Beautiful Mathematics,' by Martin Erickson, an excerpt on the Cullinane diamond theorem (with source not mentioned)

The geometry of the 4×4 square combines modular symmetry
(i.e., related to theta functions) with the affine symmetry above.

Hudson's 1905 discussion of modular symmetry (that of Rosenhain
tetrads and Göpel tetrads) in the 4×4 square used a parametrization
of that square by the digit 0 and the fifteen 2-subsets of a 6-set, but 
did not discuss the 4×4 square as an affine space.

For the connection of the 15 Kummer modular 2-subsets with the 16-
element affine space over the two-element Galois field GF(2), see my note
of May 26, 1986, "The 2-subsets of a 6-set are the points of a PG(3,2)" —

— and the affine structure in the 1979 AMS abstract
"Symmetry invariance in a diamond ring" —

For some historical background on the symmetry investigations by
Dyson and Macdonald, see Dyson's 1972 article "MIssed Opportunities."

For Macdonald's own  use of the words "modular" and "affine," see
Macdonald, I. G., "Affine Lie algebras and modular forms," 
Séminaire N. Bourbaki , Vol. 23 (1980-1981), Talk no. 577, pp. 258-276.

Saturday, January 6, 2018

Yale News

Filed under: General,Geometry — Tags: — m759 @ 5:24 am

The Yale of the title is not the university, but rather the
mathematician Paul B. Yale. Yale's illustration of the Fano
plane is below.

IMAGE- Triangular models of the 4-point affine plane A and 7-point projective plane PA

A different illustration from a mathematician named Greenberg —

This illustration of the ominous phrase "line at infinity"
may serve as a sort of Deathly Hallows  for Greenberg.
According to the AMS website yesterday, he died on
December 12, 2017:

A search of this  journal for Greenberg yields no mention of
the dead mathematician, but does yield some remarks
on art that are pehaps less bleak than the above illustration.

For instance —

Art adapted from the Google search screen. Discuss.

Wednesday, October 18, 2017

Dürer for St. Luke’s Day

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 1:00 pm

Structure of the Dürer magic square 

16   3   2  13
 5  10  11   8   decreased by 1 is …
 9   6   7  12
 4  15  14   1

15   2   1  12
 4   9  10   7
 8   5   6  11
 3  14  13   0 .

Base 4 —

33  02  01  30
10  21  22  13
20  11  12  23 
03  32  31  00 .

Two-part decomposition of base-4 array
as two (non-Latin) orthogonal arrays

3 0 0 3     3 2 1 0
1 2 2 1     0 1 2 3
2 1 1 2     0 1 2 3
0 3 3 0     3 2 1 0 .

Base 2 –

1111  0010  0001  1100
0100  1001  1010  0111
1000  0101  0110  1011
0011  1110  1101  0000 .

Four-part decomposition of base-2 array
as four affine hyperplanes over GF(2) —

1001  1001  1100  1010
0110  1001  0011  0101
1001  0110  0011  0101
0110  0110  1100  1010 .

— Steven H. Cullinane,
  October 18, 2017

See also recent related analyses of
noted 3×3 and 5×5 magic squares.

Friday, April 14, 2017

Hudson and Finite Geometry

Filed under: General,Geometry — Tags: , — m759 @ 3:00 am

IMAGE- Geometry of the Six-Set, Steven H. Cullinane, April 23, 2013

The above four-element sets of black subsquares of a 4×4 square array 
are 15 of the 60 Göpel tetrads , and 20 of the 80 Rosenhain tetrads , defined
by R. W. H. T. Hudson in his 1905 classic Kummer's Quartic Surface .

Hudson did not  view these 35 tetrads as planes through the origin in a finite
affine 4-space (or, equivalently, as lines in the corresponding finite projective
3-space).

In order to view them in this way, one can view the tetrads as derived,
via the 15 two-element subsets of a six-element set, from the 16 elements
of the binary Galois affine space pictured above at top left.

This space is formed by taking symmetric-difference (Galois binary)
sums of the 15 two-element subsets, and identifying any resulting four-
element (or, summing three disjoint two-element subsets, six-element)
subsets with their complements.  This process was described in my note
"The 2-subsets of a 6-set are the points of a PG(3,2)" of May 26, 1986.

The space was later described in the following —

IMAGE- Dolgachev and Keum, coordinatization of the 4x4 array in 'Birational Automorphisms of Quartic Hessian Surfaces,' AMS Transactions, 2002

Wednesday, March 29, 2017

The Crimson Abyss

"And as the characters in the meme twitch into the abyss
that is the sky, this meme will disappear into whatever
internet abyss swallowed MySpace."

—Staff writer Kamila Czachorowski, Harvard Crimson , March 29

1984

IMAGE- 'Affine Groups on Small Binary Spaces,' illustration

2010

Logo design for Stack Exchange Math by Jin Yang
 

Recent posts now tagged Crimson Abyss suggest
the above logo be viewed in light of a certain page 29

"… as if into a crimson abyss …." —

Update of 9 PM ET March 29, 2017:

Prospero's Children  was first published by HarperCollins,
London, in 1999. A statement by the publisher provides
an instance of the famous "much-needed gap." —

"This is English fantasy at its finest. Prospero’s Children 
steps into the gap that exists between The Lion, the Witch
and the Wardrobe
  and Clive Barker’s Weaveworld , and
is destined to become a modern classic."

Related imagery —

See also "Hexagram 64 in Context" (Log24, March 16, 2017).

Art Space Illustrated

Another view of the previous post's art space  —

IMAGE by Cullinane- 'Solomon's Cube' with 64 identical, but variously oriented, subcubes, and six partitions of these 64 subcubes

More generally, see Solomon's Cube in Log24.

See also a remark from Stack Exchange in yesterday's post Backstory,
and the Stack Exchange math logo below, which recalls the above 
cube arrangement from "Affine groups on small binary spaces" (1984).

IMAGE- Current math.stackexchange.com logo and a 1984 figure from 'Notes on Groups and Geometry, 1978-1986'

Monday, January 2, 2017

Sitcom Theology

Filed under: General,Geometry — Tags: — m759 @ 1:20 pm

The Hollywood Reporter

"William Christopher, best known for playing Father Mulcahy
on the hit sitcom M*A*S*H , died Saturday [Dec. 31, 2016] of
lung cancer, his agent confirmed to The Hollywood Reporter.
He was 84.

Christopher died at his home in Pasadena, with his wife by
his bedside, at 5:10 a.m. on New Year's Eve, according to a
statement from his agent."

— 5:59 PM PST 12/31/2016 by Meena Jang

Image reshown in this journal on the midnight (Eastern time)
preceding Christopher's death —

IMAGE- Triangular models of the 4-point affine plane A and 7-point projective plane PA

Related material —

From a Log24 search for "Deathly Hallows" —

Mathematics

http://www.log24.com/log/pix11A/110505-WikipediaFanoPlane.jpg

The Fano plane block design

Magic

http://www.log24.com/log/pix11A/110505-DeathlyHallows.jpg

The Deathly Hallows symbol—
Two blocks short of  a design.

Those who prefer Latin with their theology
may search this journal for "In Nomine Patris."

Friday, December 23, 2016

Requiem for a Mathematician

Filed under: General,Geometry — m759 @ 2:10 pm

From a Dec. 21 obituary posted by the
University of Tennessee at Knoxville —

"Wade was ordained as a pastor and served
at Oakwood Baptist Church in Knoxville."

Other information —

In a Log24 post, "Seeing the Finite Structure,"
of August 16, 2008, Wade appeared as a co-author
of the Walsh series book mentioned above —

Walsh Series: An Introduction to Dyadic Harmonic Analysis, by F. Schipp et. al.

Walsh Series: An Introduction
to Dyadic Harmonic Analysis
,
by F. Schipp et al.,
Taylor & Francis, 1990

From the 2008 post —

The patterns on the faces of the cube on the cover
of Walsh Series above illustrate both the 
Walsh functions of order 3 and the same structure
in a different guise, subspaces of the affine 3-space 
over the binary field. For a note on the relationship
of Walsh functions to finite geometry, see 
Symmetry of Walsh Functions.

Friday, December 16, 2016

Memory, History, Geometry

Filed under: General,Geometry — Tags: — m759 @ 9:48 am

These are Rothko's Swamps .

See a Log24 search for related meditations.

For all three topics combined, see Coxeter —

" There is a pleasantly discursive treatment 
of Pontius Pilate’s unanswered question
‘What is truth?’ "

— Coxeter, 1987, introduction to Trudeau’s
     The Non-Euclidean Revolution

Update of 10 AM ET —  Related material, with an elementary example:

Posts tagged "Defining Form." The example —

IMAGE- Triangular models of the 4-point affine plane A and 7-point projective plane PA

Monday, September 12, 2016

The Kummer Lattice

The previous post quoted Tom Wolfe on Chomsky's use of
the word "array." 

An example of particular interest is the 4×4  array
(whether of dots or of unit squares) —

      .

Some context for the 4×4 array —

The following definition indicates that the 4×4 array, when
suitably coordinatized, underlies the Kummer lattice .

Further background on the Kummer lattice:

Alice Garbagnati and Alessandra Sarti, 
"Kummer Surfaces and K3 surfaces
with $(Z/2Z)^4$ symplectic action." 
To appear in Rocky Mountain J. Math.

The above article is written from the viewpoint of traditional
algebraic geometry. For a less traditional view of the underlying
affine 4-space from finite  geometry, see the website
Finite Geometry of the Square and Cube.

Some further context

"To our knowledge, the relation of the Golay code
to the Kummer lattice is a new observation."

— Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of
Kummer surfaces in the Mathieu group M24 
"

As noted earlier, Taormina and Wendland seem not to be aware of
R. W. H. T. Hudson's use of the (uncoordinatized*) 4×4 array in his
1905 book Kummer's Quartic Surface.  The array was coordinatized,
i.e. given a "vector space structure," by Cullinane eight years prior to
the cited remarks of Curtis.

* Update of Sept. 14: "Uncoordinatized," but parametrized  by 0 and
the 15 two-subsets of a six-set. See the post of Sept. 13.

Monday, August 1, 2016

Cube

Filed under: General,Geometry — m759 @ 10:28 pm

From this journal —

See (for instance) Sacred Order, July 18, 2006 —

The finite Galois affine space with 64 points

From a novel published July 26, 2016, and reviewed
in yesterday's (print) New York Times Book Review —

The doors open slowly. I step into a hangar. From the rafters high above, lights blaze down, illuminating a twelve-foot cube the color of gunmetal. My pulse rate kicks up. I can’t believe what I’m looking at. Leighton must sense my awe, because he says, “Beautiful, isn’t it?” It is exquisitely beautiful. At first, I think the hum inside the hangar is coming from the lights, but it can’t be. It’s so deep I can feel it at the base of my spine, like the ultralow-frequency vibration of a massive engine. I drift toward the box, mesmerized.

— Crouch, Blake. Dark Matter: A Novel
(Kindle Locations 2004-2010).
Crown/Archetype. Kindle Edition. 

See also Log24 on the publication date of Dark Matter .

Thursday, July 28, 2016

The Giglmayr Foldings

Filed under: General,Geometry — Tags: — m759 @ 1:44 pm

Giglmayr's transformations (a), (c), and (e) convert
his starting pattern

  1    2   5   6
  3    4   7   8
  9  10 13 14
11  12 15 16

to three length-16 sequences. Putting these resulting
sequences back into the 4×4 array in normal reading
order, we have

  1    2    3    4        1   2   4   3          1    4   2   3
  5    6    7    8        5   6   8   7          7    6   8   5 
  9  10  11  12      13 14 16 15       15 14 16 13
13  14  15  16       9  10 12 11        9  12 10 11

         (a)                         (c)                      (e)

Four length-16 basis vectors for a Galois 4-space consisting
of the origin and 15 weight-8 vectors over GF(2):

0 0 0 0       0 0 0 0       0 0 1 1       0 1 0 1
0 0 0 0       1 1 1 1       0 0 1 1       0 1 0 1 
1 1 1 1       0 0 0 0       0 0 1 1       0 1 0 1
1 1 1 1       1 1 1 1       0 0 1 1       0 1 0 1 .

(See "Finite Relativity" at finitegeometry.org/sc.)

The actions of Giglmayr's transformations on the above
four basis vectors indicate the transformations are part of
the affine group (of order 322,560) on the affine space
corresponding to the above vector space.

For a description of such transformations as "foldings,"
see a search for Zarin + Folded in this journal.

Wednesday, May 4, 2016

Golomb and Symmetry

Filed under: General,Geometry — Tags: — m759 @ 12:00 pm

From the webpage Diamond Theory Bibliography

Golomb, Solomon W. 
Shift register sequences  (Revised edition)
Aegean Park Press, Laguna Hills, CA, 1982
   The fifteen "stencils" in Golomb's Fig. VIII-8, page 219,
   are the same as the fifteen affine hyperplanes that
   account for patterns' symmetry in diamond theory.
   This figure occurs in a discussion of Rademacher-
   Walsh functions.

Elsewhere

Wednesday, April 20, 2016

Symmetric Generation of a Simple Group

The reference in the previous post to the work of Guitart and
The Road to Universal Logic  suggests a fiction involving
the symmetric generation of the simple group of order 168.

See The Diamond Archetype and a fictional account of the road to Hell 

'PyrE' in Bester's 'The Stars My Destination'

The cover illustration below has been adapted to
replace the flames of PyrE with the eightfold cube.

IMAGE- 'The Stars My Destination' (with cover slightly changed)

For related symmetric generation of a much larger group, see Solomon’s Cube.

Friday, April 8, 2016

Space Cross

Filed under: General,Geometry — Tags: — m759 @ 11:00 pm

For George Orwell

Illustration from a book on mathematics —

This illustrates the Galois space  AG(4,2).

For some related spaces, see a note from 1984.

"There is  such a thing as a space cross."
— Saying adapted from a young-adult novel

Tuesday, December 15, 2015

Square Triangles

Filed under: General,Geometry — Tags: , — m759 @ 3:57 pm

Click image for some background.

Exercise:  Note that, modulo color-interchange, the set of 15 two-color
patterns above is invariant under the group of six symmetries of the
equilateral triangle. Are there any other such sets of 15 two-color triangular
patterns that are closed as sets , modulo color-interchange, under the six
triangle symmetries and  under the 322,560 permutations of the 16
subtriangles induced by actions of the affine group AGL(4,2)
on the 16 subtriangles' centers , given a suitable coordinatization?

Friday, August 28, 2015

Art and Space

Filed under: General,Geometry — Tags: , — m759 @ 10:00 am

IMAGE- Spielfeld (1982-83), by Wolf Barth
 

            Observatory scene from "Magic in the Moonlight"

"The sixteen nodes… can be parametrized
by the sixteen points in affine four-space
over the tiny field F2 with two elements."

Wolf Barth

Friday, August 14, 2015

Discrete Space

Filed under: General,Geometry — Tags: — m759 @ 7:24 am

(A review)

Galois space:

Image-- examples from Galois affine geometry

Counting symmetries of  Galois space:
IMAGE - The Diamond Theorem

The reason for these graphic symmetries in affine Galois space —

symmetries of the underlying projective Galois space:

Monday, June 15, 2015

Omega Matrix

Filed under: General,Geometry — Tags: , — m759 @ 12:00 pm

See that phrase in this journal.

See also last night's post.

The Greek letter Ω is customarily used to
denote a set that is acted upon by a group.
If the group is the affine group of 322,560
transformations of the four-dimensional
affine space over the two-element Galois
field, the appropriate Ω is the 4×4 grid above.

See the Cullinane diamond theorem.

Thursday, June 11, 2015

Omega

Filed under: General,Geometry — Tags: — m759 @ 12:00 pm

Omega is a Greek letter, Ω , used in mathematics to denote
a set on which a group acts. 

For instance, the affine group AGL(3,2) is a group of 1,344
actions on the eight elements of the vector 3-space over the
two-element Galois field GF(2), or, if you prefer, on the Galois
field  Ω = GF(8).

Related fiction:  The Eight , by Katherine Neville.

Related non-fiction:  A remark by Werner Heisenberg
in this journal on Saturday, June 6, 2015, the eightfold cube ,
and the illustrations below —

Mathematics

http://www.log24.com/log/pix11A/110505-WikipediaFanoPlane.jpg

The Fano plane block design

Magic

http://www.log24.com/log/pix11A/110505-DeathlyHallows.jpg

The Deathly Hallows symbol—
Two blocks short of  a design.

Wednesday, May 13, 2015

Space

Filed under: General,Geometry — Tags: , — m759 @ 2:00 pm

Notes on space for day 13 of May, 2015 —

The 13 symmetry axes of the cube may be viewed as
the 13 points of the Galois projective space PG(2,3).
This space (a plane) may also be viewed as the nine points
of the Galois affine space AG(2,3) plus the four points on
an added "line at infinity."

Related poetic material:

The ninefold square and Apollo, as well as 

http://www.log24.com/log/pix11A/110426-ApolloAndDionysus.jpg

Tuesday, March 24, 2015

Brouwer on the Galois Tesseract

Filed under: General,Geometry — Tags: , , — m759 @ 12:00 pm

Yesterday's post suggests a review of the following —

Andries Brouwer, preprint, 1982:

"The Witt designs, Golay codes and Mathieu groups"
(unpublished as of 2013)

Pages 8-9:

Substructures of S(5, 8, 24)

An octad is a block of S(5, 8, 24).

Theorem 5.1

Let B0 be a fixed octad. The 30 octads disjoint from B0
form a self-complementary 3-(16,8,3) design, namely 

the design of the points and affine hyperplanes in AG(4, 2),
the 4-dimensional affine space over F2.

Proof….

… (iv) We have AG(4, 2).

(Proof: invoke your favorite characterization of AG(4, 2) 
or PG(3, 2), say 
Dembowski-Wagner or Veblen & Young. 

An explicit construction of the vector space is also easy….)

Related material:  Posts tagged Priority.

Saturday, March 14, 2015

Unicode Diamonds

Filed under: General,Geometry — m759 @ 9:16 pm

The following figure, intended to display as
a black diamond, was produced with
HTML and Unicode characters. Depending
on the technology used to view it, the figure
may contain gaps or overlaps.

◢◣
◥◤

Some variations:

◤◥
◣◢

◤◥
◢◣

◤◣
◢◥

◤◣
◥◢

Such combined Unicode characters —

◢  black lower right triangle,
◣  black lower left triangle,
᭘  black upper left triangle,
᭙  black upper right triangle 

— might be used for a text-only version of the Diamond 16 Puzzle
that is more easily programmed than the current version.

The tricky part would be coding the letter-spacing and
line-height to avoid gaps or overlaps within the figures in
a variety of browsers. The w3.org visual formatting model
may or may not be helpful here.

Update of 11:20 PM ET March 15, 2015 — 
Seekers of simplicity should note that there is
a simple program in the Processing.js  language, not  using
such Unicode characters, that shows many random affine
permutations of a 4×4 diamond-theorem array when the
display window is clicked.

Friday, January 16, 2015

A versus PA

Filed under: General,Geometry — Tags: — m759 @ 8:48 pm

"Reality is the beginning not the end,
Naked Alpha, not the hierophant Omega,
of dense investiture, with luminous vassals."

— “An Ordinary Evening in New Haven” VI

From the series of posts tagged "Defining Form" —

The 4-point affine plane A  and
the 7-point projective plane PA  —

IMAGE- Triangular models of the 4-point affine plane A and 7-point projective plane PA

The circle-in-triangle of Yale's Figure 30b (PA ) may,
if one likes, be seen as having an occult meaning.

For the mathematical  meaning of the circle in PA
see a search for "line at infinity."

A different, cubic, model of PA  is perhaps more perspicuous.

Friday, October 31, 2014

Structure

Filed under: General,Geometry — m759 @ 3:00 am

On Devil’s Night

Introducing a group of 322,560 affine transformations of Dürer’s ‘Magic’ Square

IMAGE- Introduction to 322,560 Affine Transformations of Dürer's 'Magic' Square

The four vector-space substructures of digits in 1st, 2nd, 3rd, 4th place,
together with the diamond theorem, indicate that Dürer’s square “minus one”
can be transformed by permutations of rows, columns, and quadrants to a
square with (decimal) digits in the usual numerical order, increasing from
top left to bottom right. Such permutations form a group of order 322,560.

(Continued from Vector Addition in a Finite Field, Twelfth Night, 2013.)

Saturday, October 25, 2014

Foundation Square

Filed under: General,Geometry — Tags: , , , — m759 @ 2:56 pm

In the above illustration of the 3-4-5 Pythagorean triangle,
the grids on each side may be regarded as figures of
Euclidean  geometry or of Galois  geometry.

In Euclidean geometry, these grids illustrate a property of
the inner triangle.

In elementary Galois geometry, ignoring the connection with
the inner triangle, the grids may be regarded instead as
illustrating vector spaces over finite (i.e., Galois) fields.
Previous posts in this journal have dealt with properties of
the 3×3 and 4×4 grids.  This suggests a look at properties of
the next larger grid, the 5×5 array, viewed as a picture of the
two-dimensional vector space (or affine plane) over the finite
Galois field GF(5) (also known as ℤ5).

The 5×5 array may be coordinatized in a natural way, as illustrated
in (for instance) Matters Mathematical , by I.N. Herstein and
Irving Kaplansky, 2nd ed., Chelsea Publishing, 1978, p. 171:

See Herstein and Kaplansky for the elementary Galois geometry of
the 5×5 array.

For 5×5 geometry that is not so elementary, see…

Hafner's abstract:

We describe the Hoffman-Singleton graph geometrically, showing that
it is closely related to the incidence graph of the affine plane over ℤ5.
This allows us to construct all automorphisms of the graph.

The remarks of Brouwer on graphs connect the 5×5-related geometry discussed
by Hafner with the 4×4 geometry related to the Steiner system S(5,8,24).
(See the Miracle Octad Generator of R. T. Curtis and the related coordinatization
by Cullinane of the 4×4 array as a four-dimensional vector space over GF(2).)

Sunday, September 14, 2014

Sensibility

Filed under: General,Geometry — Tags: , — m759 @ 9:26 am

Structured gray matter:

Graphic symmetries of Galois space:
IMAGE - The Diamond Theorem

The reason for these graphic symmetries in affine  Galois space —

symmetries of the underlying projective  Galois space:

Thursday, July 17, 2014

Paradigm Shift:

Filed under: General,Geometry — Tags: — m759 @ 11:01 am
 

Continuous Euclidean space to discrete Galois space*

Euclidean space:

Point, line, square, cube, tesseract

From a page by Bryan Clair

Counting symmetries in Euclidean space:

Galois space:

Image-- examples from Galois affine geometry

Counting symmetries of  Galois space:
IMAGE - The Diamond Theorem

The reason for these graphic symmetries in affine Galois space —

symmetries of the underlying projective Galois space:

* For related remarks, see posts of May 26-28, 2012.

Thursday, March 27, 2014

Diamond Space

Filed under: General,Geometry — Tags: , , , — m759 @ 2:28 pm

(Continued)

Definition:  A diamond space  — informal phrase denoting
a subspace of AG(6, 2), the six-dimensional affine space
over the two-element Galois field.

The reason for the name:

IMAGE - The Diamond Theorem, including the 4x4x4 'Solomon's Cube' case

Click to enlarge.

Friday, March 21, 2014

Three Constructions of the Miracle Octad Generator

Filed under: General,Geometry — Tags: , , , , — m759 @ 12:24 pm

IMAGE- Two constructions, by Turyn/Curtis, and by Conway, of the Miracle Octad Generator

See also a Log24 post on this subject from Dec. 14, 2013,
especially (scroll down) the update of March 9, 2014.

Related material on the Turyn-Curtis construction
from the University of Cambridge —

— Slide by "Dr. Parker" — Apparently Richard A. Parker —
Lecture 4, "Discovering M24," in slides for lectures 1-8 from lectures
at Cambridge in 2010-2011 on "Sporadic and Related Groups."
See also the Parker lectures of 2012-2013 on the same topic.

A third construction of Curtis's 35  4×6  1976 MOG arrays would use
Cullinane's analysis of the 4×4 subarrays' affine and projective structure,
and point out the fact that Conwell's 1910 correspondence of the 35
4+4-partitions of an 8-set with the 35 lines of the projective 3-space
over the 2-element field, PG(3, 2), is essentially the same correspondence
as that constituting Curtis's 1976 MOG.

See The Diamond Theorem,  Finite RelativityGalois Space,
Generating the Octad Generator, and The Klein Correspondence.

Update of March 22-March 23 —

Adding together as (0,1)-matrices over GF(2) the black parts (black
squares as 1's, all other squares as 0's) of the 35  4×6 arrays of the 1976
Curtis MOG would then reveal*  the symmetric role played in octads
by what Curtis called the heavy brick , and so reveal also the action of
S3 on the three Curtis bricks that leaves invariant the set of all 759
octads of the S(5, 8, 24) constructed from the 35  MOG arrays.  For more
details of this "by-hand" construction, see Geometry of the 4×4 Square.
For the mathematical properties of the S(5, 8, 24), it is convenient to
have a separate construction (such as Turyn's), not  by hand, of the
extended binary Golay code. See the Brouwer preprint quoted above.

* "Then a miracle occurs," as in the classic 1977 Sidney Harris cartoon.

Illustration of array addition from March 23 —

IMAGE- Discovery of the S_3 action on bricks in the Conwell-Cullinane 'by-hand' approach to octad-building

Friday, January 17, 2014

The 4×4 Relativity Problem

Filed under: General,Geometry — Tags: , , , — m759 @ 11:00 pm

The sixteen-dot square array in yesterday’s noon post suggests
the following remarks.

“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 ,
Princeton University Press, 1946, p. 16

The Galois tesseract  appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977—

IMAGE- Hypercube and 4x4 matrix from the 1976 'Diamond Theory' preprint, as excerpted in 'Computer Graphics and Art'

The 1977 matrix Q is echoed in the following from 2002—

IMAGE- Dolgachev and Keum, coordinatization of the 4x4 array in 'Birational Automorphisms of Quartic Hessian Surfaces,' AMS Transactions, 2002

A different representation of Cullinane’s 1977 square model of the
16-point affine geometry over the two-element Galois field GF(2)
is supplied by Conway and Sloane in Sphere Packings, Lattices and Groups   
(first published in 1988) :

IMAGE- The Galois tesseract as a four-dimensional vector space, from a diagram by Conway and Sloane in 'Sphere Packings, Lattices, and Groups'

Here a, b, c, d   are basis vectors in the vector 4-space over GF(2).
(For a 1979 version of this vector space, see AMS Abstract 79T-A37.)

See also a 2011 publication of the Mathematical Association of America —

From 'Beautiful Mathematics,' by Martin Erickson, an excerpt on the Cullinane diamond theorem (with source not mentioned)

Saturday, September 21, 2013

Mathematics and Narrative (continued)

Filed under: General,Geometry — Tags: , , — m759 @ 1:00 am

Mathematics:

A review of posts from earlier this month —

Wednesday, September 4, 2013

Moonshine

Filed under: Uncategorized — m759 @ 4:00 PM

Unexpected connections between areas of mathematics
previously thought to be unrelated are sometimes referred
to as "moonshine."  An example—  the apparent connections
between parts of complex analysis and groups related to the
large Mathieu group M24. Some recent work on such apparent
connections, by Anne Taormina and Katrin Wendland, among
others (for instance, Miranda C.N. Cheng and John F.R. Duncan),
involves structures related to Kummer surfaces .
In a classic book, Kummer's Quartic Surface  (1905),
R.W.H.T. Hudson pictured a set of 140 structures, the 80
Rosenhain tetrads and the 60 Göpel tetrads, as 4-element
subsets of a 16-element 4×4 array.  It turns out that these
140 structures are the planes of the finite affine geometry
AG(4,2) of four dimensions over the two-element Galois field.
(See Diamond Theory in 1937.)

Thursday, September 5, 2013

Moonshine II

Filed under: Uncategorized — Tags:  — m759 @ 10:31 AM

(Continued from yesterday)

The foreword by Wolf Barth in the 1990 Cambridge U. Press
reissue of Hudson's 1905 classic Kummer's Quartic Surface
covers some of the material in yesterday's post Moonshine.

The distinction that Barth described in 1990 was also described, and illustrated,
in my 1986 note "Picturing the smallest projective 3-space."  The affine 4-space
over the the finite Galois field GF(2) that Barth describes was earlier described—
within a 4×4 array like that pictured by Hudson in 1905— in a 1979 American
Mathematical Society abstract, "Symmetry invariance in a diamond ring."

"The distinction between Rosenhain and Goepel tetrads
is nothing but the distinction between isotropic and
non-isotropic planes in this affine space over the finite field."

The 1990 paragraph of Barth quoted above may be viewed as a summary
of these facts, and also of my March 17, 2013, note "Rosenhain and Göpel
Tetrads in PG(3,2)
."

Narrative:

Aooo.

Happy birthday to Stephen King.

Thursday, September 5, 2013

Moonshine II

(Continued from yesterday)

The foreword by Wolf Barth in the 1990 Cambridge U. Press
reissue of Hudson's 1905 classic Kummer's Quartic Surface
covers some of the material in yesterday's post Moonshine.

The distinction that Barth described in 1990 was also described, and illustrated,
in my 1986 note "Picturing the smallest projective 3-space."  The affine 4-space
over the the finite Galois field GF(2) that Barth describes was earlier described—
within a 4×4 array like that pictured by Hudson in 1905— in a 1979 American
Mathematical Society abstract, "Symmetry invariance in a diamond ring."

"The distinction between Rosenhain and Goepel tetrads
is nothing but the distinction between isotropic and
non-isotropic planes in this affine space over the finite field."

The 1990 paragraph of Barth quoted above may be viewed as a summary
of these facts, and also of my March 17, 2013, note "Rosenhain and Göpel
Tetrads in PG(3,2)
."

Wednesday, September 4, 2013

Moonshine

Unexpected connections between areas of mathematics
previously thought to be unrelated are sometimes referred
to as "moonshine."  An example—  the apparent connections
between parts of complex analysis and groups related to the 
large Mathieu group M24. Some recent work on such apparent
connections, by Anne Taormina and Katrin Wendland, among
others (for instance, Miranda C.N. Cheng and John F.R. Duncan),
involves structures related to Kummer surfaces .
In a classic book, Kummer's Quartic Surface  (1905),
R.W.H.T. Hudson pictured a set of 140 structures, the 80
Rosenhain tetrads and the 60 Göpel tetrads, as 4-element
subsets of a 16-element 4×4 array.  It turns out that these
140 structures are the planes of the finite affine geometry
AG(4,2) of four dimensions over the two-element Galois field.
(See Diamond Theory in 1937.) 

A Google search documents the moonshine
relating Rosenhain's and Göpel's 19th-century work
in complex analysis to M24  via the book of Hudson and
the geometry of the 4×4 square.

Tuesday, July 9, 2013

Vril Chick

Filed under: General,Geometry — Tags: , , — m759 @ 4:30 am

Profile picture of "Jo Lyxe" (Josefine Lyche) at Vimeo

Profile picture for "Jo Lyxe" (Josefine Lyche) at Vimeo

Compare to an image of Vril muse Maria Orsitsch.

From the catalog of a current art exhibition
(25 May – 31 August, 2013) in Norway,
I DE LANGE NÆTTER —

Josefine Lyche
Born in 1973 in Bergen, Norway.
Lives and works in Oslo and Berlin.

Keywords (to help place my artwork in the
proper context): Aliens, affine geometry, affine
planes, affine spaces, automorphisms, binary
codes, block designs, classical groups, codes,
coding theory, collineations, combinatorial,
combinatorics, conjugacy classes, the Conwell
correspondence, correlations, Cullinane,
R. T. Curtis, design theory, the diamond theorem,
diamond theory, duads, duality, error correcting
codes, esoteric, exceptional groups,
extraterrestrials, finite fields, finite geometry, finite
groups, finite rings, Galois fields, generalized
quadrangles, generators, geometry, GF(2),
GF(4), the (24,12) Golay code, group actions,
group theory, Hadamard matrices, hypercube,
hyperplanes, hyperspace, incidence structures,
invariance, Karnaugh maps, Kirkman’s schoolgirls
problem, Latin squares, Leech lattice, linear
groups, linear spaces, linear transformations,
Magick, Mathieu groups, matrix theory, Meno,
Miracle Octad Generator, MOG, multiply transitive
groups, occultism, octahedron, the octahedral
group, Orsic, orthogonal arrays, outer automorphisms,
parallelisms, partial geometries,
permutation groups, PG(3,2), Plato, Platonic
solids, polarities, Polya-Burnside theorem, projective
geometry, projective planes, projective
spaces, projectivities, Pythagoras, reincarnation,
Reed-Muller codes, the relativity problem,
reverse engineering, sacred geometry, Singer
cycle, skew lines, Socrates, sporadic simple
groups, Steiner systems, Sylvester, symmetric,
symmetry, symplectic, synthemes, synthematic,
Theosophical Society tesseract, Tessla, transvections,
Venn diagrams, Vril society, Walsh
functions, Witt designs.

(See also the original catalog page.)

Clearly most of this (the non-highlighted parts) was taken
from my webpage Diamond Theory. I suppose I should be
flattered, but I am not thrilled to be associated with the
(apparently fictional) Vril Society.

For some background, see (for instance) 
Conspiracy Theories and Secret Societies for Dummies .

Friday, June 14, 2013

Object of Beauty

Filed under: General,Geometry — Tags: , , — m759 @ 2:01 am

This journal on July 5, 2007 —

The Eightfold Cube and its Inner Structure

“It is not clear why MySpace China will be successful."

— The Chinese magazine Caijing  in 2007, quoted in
Asia Sentinel  on July 12, 2011

This  journal on that same date,  July 12, 2011 —

http://www.log24.com/log/pix11B/110712-ObjectOfBeauty.jpg

See also the eightfold cube and kindergarten blocks
at finitegeometry.org/sc.

Friedrich Froebel, Froebel's Chief Writings on Education ,
Part II, "The Kindergarten," Ch. III, "The Third Play":

"The little ones, who always long for novelty and change,
love this simple plaything in its unvarying form and in its
constant number, even as they love their fairy tales with
the ever-recurring dwarfs…."

This journal, Group Actions, Nov. 14, 2012:

"Those who insist on vulgarizing their mathematics
may regard linear and affine group actions on the eight
cubes as the dance of  Snow White (representing (0,0,0))
and the Seven Dwarfs—

  ."

Edwin M. Knowles Fine China Company, 1991

Monday, June 10, 2013

Galois Coordinates

Filed under: General,Geometry — Tags: , , — m759 @ 10:30 pm

Today's previous post on coordinate systems
suggests a look at the phrase "Galois coordinates."

A search shows that the phrase, though natural,
has apparently not been used before 2011* for solutions
to what Hermann Weyl called "the relativity problem."

A thorough historical essay on Galois coordinatization
in this sense would require more academic resources
than I have available. It would likely describe a number
of applications of Galois-field coordinates to square
(and perhaps to cubical) arrays that were studied before
1976, the date of my Diamond Theory  monograph.

But such a survey might not  find any such pre-1976
coordinatization of a 4×4 array  by the 16 elements
of the vector 4-space  over the Galois field with two
elements, GF(2).

Such coordinatizations are important because of their
close relationship to the Mathieu group 24 .

See a preprint by Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of Kummer
surfaces in the Mathieu group 24 ," with its remark
denying knowledge of any such coordinatization
prior to a 1989 paper by R. T. Curtis.

Related material: 

Some images related to Galois coordinates, excerpted
from a Google search today (click to enlarge)—

*  A rather abstract  2011 paper that uses the phrase
   "Galois coordinates" may have some implications 
   for the naive form of the relativity problem
   related to square and cubical arrays.

Tuesday, June 4, 2013

Cover Acts

Filed under: General,Geometry — Tags: — m759 @ 11:00 am

The Daily Princetonian  today:

IMAGE- 'How Jay White, a Neil Diamond cover act, duped Princeton'

A different cover act, discussed here  Saturday:

IMAGE- The diamond theorem affine group of order 322,560, published without acknowledgment of its source by the Mathematical Association of America in 2011

See also, in this journal, the Galois tesseract and the Crosswicks Curse.

"There is  such a thing as a tesseract." — Crosswicks saying

Saturday, June 1, 2013

Permanence

Filed under: General,Geometry — Tags: , , , — m759 @ 4:00 pm

"What we do may be small, but it has
  a certain character of permanence."

— G. H. Hardy, A Mathematician's Apology

The diamond theorem  group, published without acknowledgment
of its source by the Mathematical Association of America in 2011—

IMAGE- The diamond-theorem affine group of order 322,560, published without acknowledgment of its source by the Mathematical Association of America in 2011

Tuesday, May 14, 2013

Snakes on a Plane

Filed under: General,Geometry — m759 @ 7:27 am

Continued.

The order-3 affine plane:

Detail from the video in the previous post:

For other permutations of points in the
order-3 affine plane—

See Quaternions in an Affine Galois Plane
and Group Actions, 1984-2009.

See, too, the Mathematics and Narrative post 
from April 28, 2013, and last night's
For Indiana Spielberg.

Sunday, March 10, 2013

Galois Space

Filed under: General,Geometry — Tags: — m759 @ 5:30 pm

(Continued)

The 16-point affine Galois space:

Further properties of this space:

In Configurations and Squares, see the
discusssion of the Kummer 166 configuration.

Some closely related material:

  • Wolfgang Kühnel,
    "Minimal Triangulations of Kummer Varieties,"
    Abh. Math. Sem. Univ. Hamburg 57, 7-20 (1986).

    For the first two pages, click here.

  • Jonathan Spreer and Wolfgang Kühnel,
    "Combinatorial Properties of the 3 Surface:
    Simplicial Blowups and Slicings,"
    preprint, 26 pages. (2009/10) (pdf).
    (Published in Experimental Math. 20,
    issue 2, 201–216 (2011).)

Saturday, January 5, 2013

Vector Addition in a Finite Field

Filed under: General,Geometry — Tags: , — m759 @ 10:18 am

The finite (i.e., Galois) field GF(16),
according to J. J. Seidel in 1974—

The same field according to Steven H. Cullinane in 1986,
in its guise as the affine 4-space over GF(2)—


The same field, again disguised as an affine 4-space,
according to John H. Conway and N.J.A. Sloane in
Sphere Packings, Lattices, and Groups , first published in 1988—

The above figure by Conway and Sloane summarizes, using
a 4×4 array, the additive vector-space structure of the finite
field GF(16).

This structure embodies what in Euclidean space is called
the parallelogram rule for vector addition—

(Thanks to June Lester for the 3D (uvw) part of the above figure.)

For the transition from this colored Euclidean hypercube
(used above to illustrate the parallelogram rule) to the
4×4 Galois space (illustrated by Cullinane in 1979 and
Conway and Sloane in 1988— or later… I do not have
their book’s first edition), see Diamond Theory in 1937,
Vertex Adjacency in a Tesseract and in a 4×4 Array,
Spaces as Hypercubes, and The Galois Tesseract.

For some related narrative, see tesseract  in this journal.

(This post has been added to finitegeometry.org.)

Update of August 9, 2013—

Coordinates for hypercube vertices derived from the
parallelogram rule in four dimensions were better
illustrated by Jürgen Köller in a web page archived in 2002.

Update of August 13, 2013—

The four basis vectors in the 2002 Köller hypercube figure
are also visible at the bottom of the hypercube figure on
page 7 of “Diamond Theory,” excerpts from a 1976 preprint
in Computer Graphics and Art , Vol. 2, No. 1, February 1977.
A predecessor:  Coxeter’s 1950 hypercube figure from
Self-Dual Configurations and Regular Graphs.”

Thursday, November 22, 2012

Finite Relativity

Filed under: General,Geometry — Tags: , — m759 @ 10:48 pm

(Continued from 1986)

S. H. Cullinane
The relativity problem in finite geometry.
Feb. 20, 1986.

This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.

— H. Weyl, The Classical Groups ,
Princeton Univ. Pr., 1946, p. 16

In finite geometry "points" are often defined as ordered n-tuples of a finite (i.e., Galois) field GF(q). What geometric structures ("frames of reference," in Weyl's terms) are coordinatized by such n-tuples? Weyl's use of "objectively" seems to mean that such structures should have certain objective— i.e., purely geometric— properties invariant under each S.

This note suggests such a frame of reference for the affine 4-space over GF(2), and a class of 322,560 equivalent coordinatizations of the frame.

The frame: A 4×4 array.

The invariant structure:

The following set of 15 partitions of the frame into two 8-sets.

Fifteen partitions of a 4x4 array into two 8-sets
 

A representative coordinatization:

 

0000  0001  0010  0011
0100  0101  0110  0111
1000  1001  1010  1011
1100  1101  1110  1111

 

The group: The group AGL(4,2) of 322,560 regular affine transformations of the ordered 4-tuples over GF(2).

S. H. Cullinane
The relativity problem in finite geometry.
Nov. 22, 2012.

This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.

— H. Weyl, The Classical Groups ,
Princeton Univ. Pr., 1946, p. 16

In finite geometry "points" are often defined as ordered n-tuples of a finite (i.e., Galois) field GF(q). What geometric structures ("frames of reference," in Weyl's terms) are coordinatized by such n-tuples? Weyl's use of "objectively" seems to mean that such structures should have certain objective— i.e., purely geometric— properties invariant under each S.

This note suggests such a frame of reference for the affine 4-space over GF(2), and a class of 322,560 equivalent coordinatizations of the frame.

The frame: An array of 16 congruent equilateral subtriangles that make up a larger equilateral triangle.

The invariant structure:

The following set of 15 partitions of the frame into two 8-sets.


Fifteen partitions of an array of 16 triangles into two 8-sets


A representative coordinatization:

 

Coordinates for a triangular finite geometry

The group: The group AGL(4,2) of 322,560 regular affine transformations of the ordered 4-tuples over GF(2).

For some background on the triangular version,
see the Square-Triangle Theorem,
noting particularly the linked-to coordinatization picture.

Wednesday, November 14, 2012

Group Actions

Filed under: General,Geometry — Tags: , , , — m759 @ 4:30 pm

The December 2012 Notices of the American
Mathematical Society  
has an ad on page 1564
(in a review of two books on vulgarized mathematics)
for three workshops next year on “Low-dimensional
Topology, Geometry, and Dynamics”—

(Only the top part of the ad is shown; for further details
see an ICERM page.)

(ICERM stands for Institute for Computational
and Experimental Research in Mathematics.)

The ICERM logo displays seven subcubes of
a 2x2x2 eight-cube array with one cube missing—

The logo, apparently a stylized image of the architecture
of the Providence building housing ICERM, is not unlike
a picture of Froebel’s Third Gift—

 

Froebel's third gift, the eightfold cube

© 2005 The Institute for Figuring

Photo by Norman Brosterman from the Inventing Kindergarten
exhibit at The Institute for Figuring (co-founded by Margaret Wertheim)

The eighth cube, missing in the ICERM logo and detached in the
Froebel Cubes photo, may be regarded as representing the origin
(0,0,0) in a coordinatized version of the 2x2x2 array—
in other words the cube invariant under linear , as opposed to
more general affine , permutations of the cubes in the array.

These cubes are not without relevance to the workshops’ topics—
low-dimensional exotic geometric structures, group theory, and dynamics.

See The Eightfold Cube, A Simple Reflection Group of Order 168, and
The Quaternion Group Acting on an Eightfold Cube.

Those who insist on vulgarizing their mathematics may regard linear
and affine group actions on the eight cubes as the dance of
Snow White (representing (0,0,0)) and the Seven Dwarfs—

.

Sunday, August 5, 2012

Cube Partitions

Filed under: General,Geometry — Tags: , , , — m759 @ 7:59 am

The second Logos  figure in the previous post
summarized affine group actions on partitions
that generate a group of about 1.3 trillion
permutations of a 4x4x4 cube (shown below)—

IMAGE by Cullinane- 'Solomon's Cube' with 64 identical, but variously oriented, subcubes, and six partitions of these 64 subcubes

Click for further details.

Thursday, August 2, 2012

Logos

Filed under: General,Geometry — Tags: — m759 @ 12:00 pm

(Continued from December 26th, 2011)

IMAGE- Current math.stackexchange.com logo and a 1984 figure from 'Notes on Groups and Geometry, 1978-1986'

Some material at math.stackexchange.com related to
yesterday evening's post on Elementary Finite Geometry

Questions on this topic have recently been
discussed at Affine plane of order 4? and at
Turning affine planes into projective planes.

(For a better discussion of the affine plane of order 4,
see Affine Planes and Mutually Orthogonal Latin Squares
at the website of William Cherowitzo, professor at UC Denver.)

Wednesday, August 1, 2012

Elementary Finite Geometry

Filed under: General,Geometry — Tags: , , , — m759 @ 7:16 pm

I. General finite geometry (without coordinates):

A finite affine plane of order has n^2 points.

A finite projective plane of order n  has n^2 + n + 1 

points because it is formed from an order-n finite affine 

plane by adding a line at infinity  that contains n + 1 points.

Examples—

Affine plane of order 3

Projective plane of order 3

II. Galois finite geometry (with coordinates over a Galois field):

A finite projective Galois plane of order n has n^2 + n + 1

points because it is formed from a finite affine Galois 3-space

of order n with n^3 points by discarding the point (0,0,0) and 

identifying the points whose coordinates are multiples of the

(n-1) nonzero scalars.

Note: The resulting Galois plane of order n has 

(n^3-1)/(n-1)= (n^2 + n + 1) points because 

(n^2 + n + 1)(n – 1) =

(n^3 + n^2 + n – n^2 – n – 1) = (n^3 – 1) .
 

III. Related art:

Another version of a 1994 picture that accompanied a New Yorker
article, "Atheists with Attitude," in the issue dated May 21, 2007:

IMAGE- 'Four Gods,' by Jonathan Borofsky

The Four Gods  of Borofsky correspond to the four axes of 
symmetry
  of a square and to the four points on a line at infinity 
in an order-3 projective plane as described in Part I above.

Those who prefer literature to mathematics may, if they like,
view the Borofsky work as depicting

"Blake's Four Zoas, which represent four aspects
of the Almighty God" —Wikipedia

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