Log24

Sunday, June 4, 2023

The Galois Core

Filed under: General — Tags: , — m759 @ 9:24 pm
 

  Rubik core:

 

Swarthmore Cube Project, 2008


Non- Rubik core:

Illustration for weblog post 'The Galois Core'

Central structure from a Galois plane

    (See image below.)

Some small Galois spaces (the Cullinane models)

Wednesday, October 26, 2022

The Hunt for Galois October

Filed under: General — Tags: , , — m759 @ 10:04 am

"… Évariste was born on October 25, 1811."
— Eric Temple Bell, Men of Mathematics

Related material — 

https://math.stackexchange.com/questions/1793035/
galois-field-of-order-2-constituting-a-boolean-algebra
 .

But seriously . . .

Tuesday, March 22, 2022

Esprit for Pascal and Galois: Finesse vs. Geometrie

Filed under: General — Tags: — m759 @ 9:19 pm

Finesse —

Sunday December 10, 2006  m759 @ 9:00 PM

A Miniature Rosetta Stone:

The image “http://www.log24.com/log/pix05B/grid3x3med.bmp” cannot be displayed, because it contains errors.

“Function defined form, expressed in a pure geometry
that the eye could easily grasp in its entirety.”

– J. G. Ballard on Modernism
(The Guardian , March 20, 2006)

“The greatest obstacle to discovery is not ignorance –
it is the illusion of knowledge.”

— Daniel J. Boorstin,
Librarian of Congress, quoted in Beyond Geometry

Geometrie —

http://www.log24.com/log/pix11/110107-Aleph-Sm.jpg

Saturday, December 18, 2021

Desargues, Galois, Veblen, and Young

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

This post was suggested by a review in the Jan. 2022
Notices of the American Mathematical Society :

My own sympathies are with Veblen.

Sunday, November 22, 2020

The Galois-Fano Plane

Filed under: General — Tags: , — m759 @ 9:52 pm

A figure adapted from “Magic Fano Planes,” by
Ben Miesner and David Nash, Pi Mu Epsilon Journal
Vol. 14, No. 1, 1914, CENTENNIAL ISSUE 3 2014
(Fall 2014), pp. 23-29 (7 pages) —

Related material — The Eightfold Cube.

Update at 10:51 PM ET the same day —

Essentially the same figure as above appears also in
the second arXiv version (11 Jan. 2016) of . . .

DAVID A. NASH, and JONATHAN NEEDLEMAN.
“When Are Finite Projective Planes Magic?”
Mathematics Magazine, vol. 89, no. 2, 2016, pp. 83–91.
JSTOR, www.jstor.org/stable/10.4169/math.mag.89.2.83.

The arXiv versions

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.

Monday, June 1, 2020

A Graveyard Smash: Galois Geometry Meets Nordic Aliens

Filed under: General — Tags: — m759 @ 4:02 pm

See also Vril Chick.

Friday, April 3, 2020

Galois Space

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

"Galois space" is now a domain name:  galois.space.

Saturday, March 2, 2019

Schoolgirls for Galois

Filed under: General — m759 @ 12:09 pm

"My God, it's
full of numbers!"

Saturday, May 5, 2018

Galois Imaginary

Filed under: General,Geometry — m759 @ 9:00 pm

" Lying at the axis of everything, zero is both real and imaginary. Lovelace was fascinated by zero; as was Gottfried Leibniz, for whom, like mathematics itself, it had a spiritual dimension. It was this that let him to imagine the binary numbers that now lie at the heart of computers: 'the creation of all things out of nothing through God's omnipotence, it might be said that nothing is a better analogy to, or even demonstration of such creation than the origin of numbers as here represented, using only unity and zero or nothing.' He also wrote, 'The imaginary number is a fine and wonderful recourse of the divine spirit, almost an amphibian between being and nonbeing.' "

— A footnote from page 229 of Sydney Padua's
    April 21, 2015, book on Lovelace and Babbage

A related passage —

From The French Mathematician
by Tom Petsinis (Nov. 30, 1998) —

0

I had foreseen it all in precise detail.
One step led inevitably to the next,
like the proof of a shining theorem,
down to the conclusive shot that still echoes
through time and space. 
Facedown in the damp pine needles,
I embraced that fatal sphere
with my whole body. Dreams, memories,
even the mathematics I had cherished
and set down in my last will and testament–
all receded. I am reduced to
a singular point; in an instant
I am transformed to .

i = an imaginary being

Here, on this complex space,
i  am no longer the impetuous youth
who wanted to change the world
first with a formula and then with a flame.
Having learned the meaning of infinite patience,
i  now rise to the text whenever anyone reads 
about Evariste Galois, preferring to remain 
just below the surface, 
like a goldfish nibbling the fringe of a floating leaf.
Ink is more mythical than blood
(unless some ancient poet slit his 
vein and wrote an epic in red):
The text is a two-way mirror 
that allows me to look into
the life and times of the reader. 
Who knows, someday i  may rise
to a text that will compel me 
to push through to the other side.
Do you want proof that i  exist? Where am ?
Beneath every word, behind each letter, 
on the side of a period that will never see the light.

Wednesday, May 2, 2018

Galois’s Space

Filed under: General,Geometry — Tags: , , — m759 @ 2:20 pm

(A sequel to Foster's Space and Sawyer's Space)

See posts now tagged Galois's Space.

Sunday, November 19, 2017

Galois Space

Filed under: General,Geometry — Tags: — m759 @ 8:00 pm

This is a sequel to yesterday's post Cube Space Continued.

Saturday, July 8, 2017

Desargues and Galois in Japan

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

Related material now available online —

A less business-oriented sort of virtual reality —

Link to 'Desargues via Galois' in Japan

For example, "A very important configuration is obtained by
taking the plane section of a complete space five-point." 
(Veblen and Young, 1910, p. 39)—

'Desargues via Galois' in Japan (via Pinterest)

Saturday, May 20, 2017

van Lint and Wilson Meet the Galois Tesseract*

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

Click image to enlarge.

The above 35 projective lines, within a 4×4 array —


The above 15 projective planes, within a 4×4 array (in white) —

* See Galois Tesseract  in this journal.

Sunday, August 14, 2016

The Boole-Galois Games

Filed under: General,Geometry — Tags: — m759 @ 5:01 pm

Continued from earlier posts on Boole vs. Galois.

From a Google image search today for “Galois Boole.”
Click the image to enlarge it.

Thursday, June 30, 2016

Rubik vs. Galois: Preconception vs. Pre-conception

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

From Psychoanalytic Aesthetics: The British School ,
by Nicola Glover, Chapter 4  —

In his last theoretical book, Attention and Interpretation  (1970), Bion has clearly cast off the mathematical and scientific scaffolding of his earlier writings and moved into the aesthetic and mystical domain. He builds upon the central role of aesthetic intuition and the Keats's notion of the 'Language of Achievement', which

… includes language that is both
a prelude to action and itself a kind of action;
the meeting of psycho-analyst and analysand
is itself an example of this language.29.

Bion distinguishes it from the kind of language which is a substitute  for thought and action, a blocking of achievement which is lies [sic ] in the realm of 'preconception' – mindlessness as opposed to mindfulness. The articulation of this language is possible only through love and gratitude; the forces of envy and greed are inimical to it..

This language is expressed only by one who has cast off the 'bondage of memory and desire'. He advised analysts (and this has caused a certain amount of controversy) to free themselves from the tyranny of the past and the future; for Bion believed that in order to make deep contact with the patient's unconscious the analyst must rid himself of all preconceptions about his patient – this superhuman task means abandoning even the desire to cure . The analyst should suspend memories of past experiences with his patient which could act as restricting the evolution of truth. The task of the analyst is to patiently 'wait for a pattern to emerge'. For as T.S. Eliot recognised in Four Quartets , 'only by the form, the pattern / Can words or music reach/ The stillness'.30. The poet also understood that 'knowledge' (in Bion's sense of it designating a 'preconception' which blocks  thought, as opposed to his designation of a 'pre -conception' which awaits  its sensory realisation), 'imposes a pattern and falsifies'

For the pattern is new in every moment
And every moment is a new and shocking
Valuation of all we have ever been.31.

The analyst, by freeing himself from the 'enchainment to past and future', casts off the arbitrary pattern and waits for new aesthetic form to emerge, which will (it is hoped) transform the content of the analytic encounter.

29. Attention and Interpretation  (Tavistock, 1970), p. 125

30. Collected Poems  (Faber, 1985), p. 194.

31. Ibid., p. 199.

See also the previous posts now tagged Bion.

Preconception  as mindlessness is illustrated by Rubik's cube, and
"pre -conception" as mindfulness is illustrated by n×n×n Froebel  cubes
for n= 1, 2, 3, 4. 

Suitably coordinatized, the Froebel  cubes become Galois  cubes,
and illustrate a new approach to the mathematics of space .

Tuesday, May 31, 2016

Galois Space —

Filed under: General,Geometry — Tags: — m759 @ 7:00 pm

A very brief introduction:

Seven is Heaven...

Tuesday, January 12, 2016

Harmonic Analysis and Galois Spaces

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

The above sketch indicates, in a vague, hand-waving, fashion,
a connection between Galois spaces and harmonic analysis.

For more details of the connection, see (for instance) yesterday
afternoon's post Space Oddity.

Wednesday, January 6, 2016

Galois.io

Filed under: General,Geometry — m759 @ 8:35 am

The title is a new URL.

Midrash on the URL suffix —

" 'I/O' is a computer term of very long standing
that means 'input/output,' i.e. the means by which
a computer communicates with the outside world.
In a domain name, it's a shibboleth that implies
that the intended audience for a site is other
programmers."

Phil Darnowsky on Dec. 18, 2014

Remarks for a wider audience —

See some Log24 posts related to Dec. 18, 2014.

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.

Tuesday, November 25, 2014

Euclidean-Galois Interplay

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

For previous remarks on this topic, as it relates to
symmetry axes of the cube, see previous posts tagged Interplay.

The above posts discuss, among other things, the Galois
projective plane of order 3, with 13 points and 13 lines.

Oxley's 2004 drawing of the 13-point projective plane

These Galois points and lines may be modeled in Euclidean geometry
by the 13 symmetry axes and the 13 rotation planes
of the Euclidean cube. They may also be modeled in Galois geometry
by subsets of the 3x3x3 Galois cube (vector 3-space over GF(3)).

http://www.log24.com/log/pix11A/110427-Cube27.jpg

   The 3×3×3 Galois Cube 

Exercise: Is there any such analogy between the 31 points of the
order-5 Galois projective plane and the 31 symmetry axes of the
Euclidean dodecahedron and icosahedron? Also, how may the
31 projective points  be naturally pictured as lines  within the 
5x5x5 Galois cube (vector 3-space over GF(5))?

Update of Nov. 30, 2014 —

For background to the above exercise, see
pp. 16-17 of A Geometrical Picture Book ,
by Burkard Polster (Springer, 1998), esp.
the citation to a 1983 article by Lemay.

Saturday, October 18, 2014

Elementary Galois Geometry

Filed under: General,Geometry — m759 @ 9:29 pm

The image search (2.26 MB).

Tuesday, June 3, 2014

Galois Matrices

Filed under: General,Geometry — m759 @ 1:00 pm

The webpage Galois.us, on Galois matrices , has been created as
a starting point for remarks on the algebra  (as opposed to the geometry)
underlying the rings of matrices mentioned in AMS abstract 79T-A37,
Symmetry invariance in a diamond ring.”

See also related historical remarks by Weyl and by Atiyah.

Thursday, March 20, 2014

Classical Galois

Filed under: General,Geometry — Tags: , , — m759 @ 12:26 pm

IMAGE- The large Desargues configuration and Desargues's theorem in light of Galois geometry

Click image for more details.

To enlarge image, click here.

Sunday, November 24, 2013

Galois Groups and Harmonic Analysis

Filed under: General,Geometry — Tags: , , — m759 @ 9:29 am

“In 1967, he [Langlands] came up with revolutionary
insights tying together the theory of Galois groups
and another area of mathematics called harmonic
analysis. These two areas, which seem light years
apart
, turned out to be closely related.”

— Edward Frenkel, Love and Math, 2013

“Class field theory expresses Galois groups of
abelian extensions of a number field F
in terms of harmonic analysis on the
multiplicative group of [a] locally compact
topological ring, the adèle ring, attached to F.”

— Michael Harris in a description of a Princeton
mathematics department talk of October 2012

Related material: a Saturday evening post.

See also Wikipedia on the history of class field theory.
For greater depth, see Tate’s [1950] thesis and the book
Fourier Analysis on Number Fields .

Tuesday, August 6, 2013

Desargues via Galois

Filed under: General,Geometry — Tags: , — m759 @ 5:12 pm

The following image gives a brief description
of the geometry discussed in last spring's
Classical Geometry in Light of Galois Geometry.

IMAGE- The large Desargues configuration in light of Galois geometry

Update of Aug. 7, 2013:  See also an expanded PDF version.

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.

Saturday, April 13, 2013

Veblen and Young in Light of Galois

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

Another in a series of figures illustrating
Desargues's theorem in light of Galois geometry:

IMAGE- Veblen and Young,1910 Desargues illustration, with 2013 Galois-geometry version

See also the original Veblen-Young figure in context.

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).)

Monday, March 4, 2013

Occupy Galois Space

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

Continued from February 27, the day Joseph Frank died

"Throughout the 1940s, he published essays
and criticism in literary journals, and one,
'Spatial Form in Modern Literature'—
a discussion of experimental treatments
of space and time by Eliot, Joyce, Proust,
Pound and others— published in
The Sewanee Review  in 1945, propelled him
to prominence as a theoretician."

— Bruce Weber in this morning's print copy
of The New York Times  (p. A15, NY edition)

That essay is reprinted in a 1991 collection
of Frank's work from Rutgers University Press:

See also Galois Space and Occupy Space in this journal.

Frank was best known as a biographer of Dostoevsky.
A very loosely related reference… in a recent Log24 post,
Freeman Dyson's praise of a book on the history of
mathematics and religion in Russia:

"The intellectual drama will attract readers
who are interested in mystical religion
and the foundations of mathematics.
The personal drama will attract readers
who are interested in a human tragedy
with characters who met their fates with
exceptional courage."

Frank is survived by, among others, his wife, a mathematician.

Thursday, February 21, 2013

Galois Space

Filed under: General,Geometry — Tags: — m759 @ 6:00 pm

(Continued)

The previous post suggests two sayings:

"There is  such a thing as a Galois space."

— Adapted from Madeleine L'Engle

"For every kind of vampire, there is a kind of cross."

Thomas Pynchon

Illustrations—

(Click to enlarge.)

Wednesday, January 23, 2013

DNA and a Galois Field

Filed under: General,Geometry — m759 @ 9:29 pm

From Ewan Birney's weblog today:

WEDNESDAY, 23 JANUARY 2013

Using DNA as a digital archive media

Today sees the publication in Nature  of “Toward practical high-capacity low-maintenance storage of digital information in synthesised DNA,” a paper spearheaded by my colleague Nick Goldman and in which I played a major part, in particular in the germination of the idea.

Birney appeared in Log24 on Dec. 30, 2012, quoted as follows:

"It is not often anyone will hear the phrase 'Galois field' and 'DNA' together…."

— Birney's weblog on July 3, 2012, "Galois and Sequencing."

Birney's widespread appearance in news articles today about the above Nature  publication suggests a review of the "Galois-field"-"DNA" connection.

See, for instance, the following papers:

  • Gail Rosen and Jeff Moore. "Investigation of Coding Structure in DNA," IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), Hong Kong, April 2003. [pdf]
  • Gail Rosen. "Finding Near-Periodic DNA Regions using a Finite-Field Framework," 2nd IEEE Genomic Signal Processing Workshop (GENSIPS), Baltimore, MD, May 2004. [pdf]
  • Gail Rosen. "Examining Coding Structure and Redundancy in DNA," IEEE Engineering in Medicine and Biology Magazine, Volume 25, Issue 1, January/February 2006. [pdf]

A  Log24 post of Sept. 17, 2012, also mentions the phrases "Galois field" and "DNA" together.

Sunday, October 14, 2012

Cameron vs. Galois

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

IMAGE- Peter J. Cameron discusses 'ambiguity' in poety and in mathematics.

See Galois Ambiguity in this  journal.

Sunday, July 29, 2012

The Galois Tesseract

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

(Continued)

The three parts of the figure in today's earlier post "Defining Form"—

IMAGE- Hyperplanes (square and triangular) in PG(3,2), and coordinates for AG(4,2)

— share the same vector-space structure:

   0     c     d   c + d
   a   a + c   a + d a + c + d
   b   b + c   b + d b + c + d
a + b a + b + c a + b + d   a + b + 
  c + d

   (This vector-space a b c d  diagram is from  Chapter 11 of 
    Sphere Packings, Lattices and Groups , by John Horton
    Conway and N. J. A. Sloane, first published by Springer
    in 1988.)

The fact that any  4×4 array embodies such a structure was implicit in
the diamond theorem (February 1979). Any 4×4 array, regarded as
a model of the finite geometry AG(4, 2), may be called a Galois tesseract.
(So called because of the Galois geometry involved, and because the
16 cells of a 4×4 array with opposite edges identified have the same
adjacency pattern as the 16 vertices of a tesseract (see, for instance,
Coxeter's 1950 "Self-Dual Configurations and Regular Graphs," figures
5 and 6).)

A 1982 discussion of a more abstract form of AG(4, 2):

Source:

The above 1982 remarks by Brouwer may or may not have influenced
the drawing of the above 1988 Conway-Sloane diagram.

Thursday, July 12, 2012

Galois Space

Filed under: General,Geometry — Tags: , — m759 @ 6:01 pm

An example of lines in a Galois space * —

The 35 lines in the 3-dimensional Galois projective space PG(3,2)—

(Click to enlarge.)

There are 15 different individual linear diagrams in the figure above.
These are the points of the Galois space PG(3,2).  Each 3-set of linear diagrams
represents the structure of one of the 35  4×4 arrays and also represents a line
of the projective space.

The symmetry of the linear diagrams accounts for the symmetry of the
840 possible images in the kaleidoscope puzzle.

* For further details on the phrase "Galois space," see
Beniamino Segre's "On Galois Geometries," Proceedings of the
International Congress of Mathematicians, 1958  
[Edinburgh].
(Cambridge U. Press, 1960, 488-499.)

(Update of Jan. 5, 2013— This post has been added to finitegeometry.org.)

Tuesday, July 10, 2012

Euclid vs. Galois

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

(Continued)

Euclidean square and triangle

Galois square and triangle

Background—

This journal on the date of Hilton Kramer's death,
The Galois Tesseract, and The Purloined Diamond.

Wednesday, October 26, 2011

Erlanger and Galois

Filed under: General,Geometry — Tags: , , , — m759 @ 8:00 pm

Peter J. Cameron yesterday on Galois—

"He was killed in a duel at the age of 20…. His work languished for another 14 years until Liouville published it in his Journal; soon it was recognised as the foundation stone of modern algebra, a position it has never lost."

Here Cameron is discussing Galois theory, a part of algebra. Galois is known also as the founder* of group theory, a more general subject.

Group theory is an essential part of modern geometry as well as of modern algebra—

"In der Galois'schen Theorie, wie hier, concentrirt sich das Interesse auf Gruppen von Änderungen. Die Objecte, auf welche sich die Änderungen beziehen, sind allerdings verschieden; man hat es dort mit einer endlichen Zahl discreter Elemente, hier mit der unendlichen Zahl von Elementen einer stetigen Mannigfaltigkeit zu thun."

— Felix Christian Klein, Erlanger Programm , 1872

("In the Galois theory, as in ours, the interest centres on groups of transformations. The objects to which the transformations are applied are indeed different; there we have to do with a finite number of discrete elements, here with the infinite number of elements in a continuous manifoldness." (Translated by M.W. Haskell, published in Bull. New York Math. Soc. 2, (1892-1893), 215-249))

Related material from Hermann Weyl, Symmetry , Princeton University Press, 1952 (paperback reprint of 1982, pp. 143-144)—

"A field is perhaps the simplest algebraic structure we can invent. Its elements are numbers…. Space is another example of an entity endowed with a structure. Here the elements are points…. What we learn from our whole discussion and what has indeed become a guiding principle in modern mathematics is this lesson: Whenever you have to do with a structure-endowed entity  Σ try to determine is group of automorphisms , the group of those element-wise transformations which leave all structural relations undisturbed. You can expect to gain a deep insight into the constitution of Σ in this way."

For a simple example of a group acting on a field (of 8 elements) that is also a space (of 8 points), see Generating the Octad Generator and Knight Moves.

* Joseph J. Rotman, An Introduction to the Theory of Groups , 4th ed., Springer, 1994, page 2

For Galois

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

Tue Oct 25, 2011 08:26 AM [London time]
from the weblog of Peter Cameron

Today is Évariste Galois’ 200th birthday.

The event will be celebrated with the publication of a new transcription
and translation of Galois’ works (edited by Peter M. Neumann)
by the European Mathematical Society. The announcement is here.

Cameron's further remarks are also of interest.

Friday, September 9, 2011

Galois vs. Rubik

(Continued from Abel Prize, August 26)

IMAGE- Elementary Galois Geometry over GF(3)

The situation is rather different when the
underlying Galois field has two rather than
three elements… See Galois Geometry.

Image-- Sugar cube in coffee, from 'Bleu'

The coffee scene from “Bleu”

Related material from this journal:

The Dream of
the Expanded Field

Image-- 4x4 square and 4x4x4 cube

Saturday, September 3, 2011

The Galois Tesseract (continued)

A post of September 1, The Galois Tesseract, noted that the interplay
of algebraic and geometric properties within the 4×4 array that forms
two-thirds of the Curtis Miracle Octad Generator (MOG) may first have
been described by Cullinane (AMS abstract 79T-A37, Notices , Feb. 1979).

Here is some supporting material—

http://www.log24.com/log/pix11B/110903-Carmichael-Conway-Curtis.jpg

The passage from Carmichael above emphasizes the importance of
the 4×4 square within the MOG.

The passage from Conway and Sloane, in a book whose first edition
was published in 1988, makes explicit the structure of the MOG's
4×4 square as the affine 4-space over the 2-element Galois field.

The passage from Curtis (1974, published in 1976) describes 35 sets
of four "special tetrads" within the 4×4 square of the MOG. These
correspond to the 35 sets of four parallel 4-point affine planes within
the square. Curtis, however, in 1976 makes no mention of the affine
structure, characterizing his 140 "special tetrads" rather by the parity
of their intersections with the square's rows and columns.

The affine structure appears in the 1979 abstract mentioned above—

IMAGE- An AMS abstract from 1979 showing how the affine group AGL(4,2) of 322,560 transformations acts on a 4x4 square

The "35 structures" of the abstract were listed, with an application to
Latin-square orthogonality, in a note from December 1978

IMAGE- Projective-space structure and Latin-square orthogonality in a set of 35 square arrays

See also a 1987 article by R. T. Curtis—

Further elementary techniques using the miracle octad generator, by R. T. Curtis. Abstract:

“In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M24, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure* underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was mis-named as simply an ‘octad generator’; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code.”

(Received July 20 1987)

Proceedings of the Edinburgh Mathematical Society (Series 2) (1989), 32: 345-353

* For instance:

Algebraic structure in the 4x4 square, by Cullinane (1985) and Curtis (1987)

Update of Sept. 4— This post is now a page at finitegeometry.org.

Thursday, September 1, 2011

The Galois Tesseract

Filed under: General,Geometry — Tags: , — m759 @ 7:11 pm

Click to enlarge

IMAGE- The Galois Tesseract, 1979-1999

IMAGE- Review of Conway and Sloane's 'Sphere Packings...' by Rota

Monday, June 27, 2011

Galois Cube Revisited

Filed under: General,Geometry — m759 @ 1:00 pm

http://www.log24.com/log/pix11A/110427-Cube27.jpg
   The 3×3×3 Galois Cube

    See Unity and Multiplicity.

   This cube, unlike Rubik's, is a
    purely mathematical structure.

    Its properties may be compared
    with those of the order-2  Galois
    cube (of eight subcubes, or
    elements ) and the order-4  Galois
    cube (of 64 elements). The
    order-3  cube (of 27 elements)
    lacks, because it is based on
    an odd  prime, the remarkable
    symmetry properties of its smaller
    and larger cube neighbors.

Saturday, November 6, 2010

Galois Field of Dreams, continued

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

Hollywood Reporter Exclusive

Martin Sheen Caught in
Spider-Man's Web

King's Moves

lux in tenebris lucet…"

Sally Field is in early talks
to play Aunt May.

Related material:

Birthdays in this journal,
Galois Field of Dreams,
and Class of 64.

Friday, September 17, 2010

The Galois Window

Filed under: General,Geometry — Tags: , , , — m759 @ 5:01 am

Yesterday's excerpt from von Balthasar supplies some Catholic aesthetic background for Galois geometry.

That approach will appeal to few mathematicians, so here is another.

Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace  is a book by Leonard Mlodinow published in 2002.

More recently, Mlodinow is the co-author, with Stephen Hawking, of The Grand Design  (published on September 7, 2010).

A review of Mlodinow's book on geometry—

"This is a shallow book on deep matters, about which the author knows next to nothing."
— Robert P. Langlands, Notices of the American Mathematical Society,  May 2002

The Langlands remark is an apt introduction to Mlodinow's more recent work.

It also applies to Martin Gardner's comments on Galois in 2007 and, posthumously, in 2010.

For the latter, see a Google search done this morning—

http://www.log24.com/log/pix10B/100917-GardnerGalois.jpg

Here, for future reference, is a copy of the current Google cache of this journal's "paged=4" page.

Note the link at the bottom of the page in the May 5, 2010, post to Peter J. Cameron's web journal. Following the link, we find…

For n=4, there is only one factorisation, which we can write concisely as 12|34, 13|24, 14|23. Its automorphism group is the symmetric group S4, and acts as S3 on the set of three partitions, as we saw last time; the group of strong automorphisms is the Klein group.

This example generalises, by taking the factorisation to consist of the parallel classes of lines in an affine space over GF(2). The automorphism group is the affine group, and the group of strong automorphisms is its translation subgroup.

See also, in this  journal, Window and Window, continued (July 5 and 6, 2010).

Gardner scoffs at the importance of Galois's last letter —

"Galois had written several articles on group theory, and was
merely annotating and correcting those earlier published papers."
Last Recreations, page 156

For refutations, see the Bulletin of the American Mathematical Society  in March 1899 and February 1909.

Monday, June 14, 2010

Birkhoff on the Galois “Theory of Ambiguity”

Filed under: General,Geometry — m759 @ 9:48 pm

The Principle of Sufficient Reason

by George David Birkhoff

from "Three Public Lectures on Scientific Subjects,"
delivered at the Rice Institute, March 6, 7, and 8, 1940

EXCERPT 1—

My primary purpose will be to show how a properly formulated
Principle of Sufficient Reason plays a fundamental
role in scientific thought and, furthermore, is to be regarded
as of the greatest suggestiveness from the philosophic point
of view.2

In the preceding lecture I pointed out that three branches
of philosophy, namely Logic, Aesthetics, and Ethics, fall
more and more under the sway of mathematical methods.
Today I would make a similar claim that the other great
branch of philosophy, Metaphysics, in so far as it possesses
a substantial core, is likely to undergo a similar fate. My
basis for this claim will be that metaphysical reasoning always
relies on the Principle of Sufficient Reason, and that
the true meaning of this Principle is to be found in the
Theory of Ambiguity” and in the associated mathematical
“Theory of Groups.”

If I were a Leibnizian mystic, believing in his “preestablished
harmony,” and the “best possible world” so
satirized by Voltaire in “Candide,” I would say that the
metaphysical importance of the Principle of Sufficient Reason
and the cognate Theory of Groups arises from the fact that
God thinks multi-dimensionally3 whereas men can only
think in linear syllogistic series, and the Theory of Groups is

2 As far as I am aware, only Scholastic Philosophy has fully recognized and ex-
ploited this principle as one of basic importance for philosophic thought

3 That is, uses multi-dimensional symbols beyond our grasp.
______________________________________________________________________

the appropriate instrument of thought to remedy our deficiency
in this respect.

The founder of the Theory of Groups was the mathematician
Evariste Galois. At the end of a long letter written in
1832 on the eve of a fatal duel, to his friend Auguste
Chevalier, the youthful Galois said in summarizing his
mathematical work,4 “You know, my dear Auguste, that
these subjects are not the only ones which I have explored.
My chief meditations for a considerable time have been
directed towards the application to transcendental Analysis
of the theory of ambiguity. . . . But I have not the time, and
my ideas are not yet well developed in this field, which is
immense.” This passage shows how in Galois’s mind the
Theory of Groups and the Theory of Ambiguity were
interrelated.5

Unfortunately later students of the Theory of Groups
have all too frequently forgotten that, philosophically
speaking, the subject remains neither more nor less than the
Theory of Ambiguity. In the limits of this lecture it is only
possible to elucidate by an elementary example the idea of a
group and of the associated ambiguity.

Consider a uniform square tile which is placed over a
marked equal square on a table. Evidently it is then impossible
to determine without further inspection which one
of four positions the tile occupies. In fact, if we designate
its vertices in order by A, B, C, D, and mark the corresponding
positions on the table, the four possibilities are for the
corners A, B, C, D of the tile to appear respectively in the
positions A, B, C, D;  B, C, D, A;  C, D, A, B; and D, A, B, C.
These are obtained respectively from the first position by a

4 My translation.
5 It is of interest to recall that Leibniz was interested in ambiguity to the extent
of using a special notation v (Latin, vel ) for “or.” Thus the ambiguously defined
roots 1, 5 of x2-6x+5=0 would be written x = l v 5 by him.
______________________________________________________________________

null rotation ( I ), by a rotation through 90° (R), by a rotation
through 180° (S), and by a rotation through 270° (T).
Furthermore the combination of any two of these rotations
in succession gives another such rotation. Thus a rotation R
through 90° followed by a rotation S through 180° is equivalent
to a single rotation T through 270°, Le., RS = T. Consequently,
the "group" of four operations I, R, S, T has
the "multiplication table" shown here:

http://www.log24.com/log/pix10A/100614-BirkhoffTable.jpg
This table fully characterizes the group, and shows the exact
nature of the underlying ambiguity of position.
More generally, any collection of operations such that
the resultant of any two performed in succession is one of
them, while there is always some operation which undoes
what any operation does, forms a "group."
__________________________________________________

EXCERPT 2—

Up to the present point my aim has been to consider a
variety of applications of the Principle of Sufficient Reason,
without attempting any precise formulation of the Principle
itself. With these applications in mind I will venture to
formulate the Principle and a related Heuristic Conjecture
in quasi-mathematical form as follows:

PRINCIPLE OF SUFFICIENT REASON. If there appears
in any theory T a set of ambiguously determined ( i e .
symmetrically entering) variables, then these variables can themselves
be determined only to the extent allowed by the corresponding
group G. Consequently any problem concerning these variables
which has a uniquely determined solution, must itself be
formulated so as to be unchanged by the operations of the group
G ( i e . must involve the variables symmetrically).

HEURISTIC CONJECTURE. The final form of any
scientific theory T is: (1) based on a few simple postulates; and
(2) contains an extensive ambiguity, associated symmetry, and
underlying group G, in such wise that, if the language and laws
of the theory of groups be taken for granted, the whole theory T
appears as nearly self-evident in virtue of the above Principle.

The Principle of Sufficient Reason and the Heuristic Conjecture,
as just formulated, have the advantage of not involving
excessively subjective ideas, while at the same time
retaining the essential kernel of the matter.

In my opinion it is essentially this principle and this
conjecture which are destined always to operate as the basic
criteria for the scientist in extending our knowledge and
understanding of the world.

It is also my belief that, in so far as there is anything
definite in the realm of Metaphysics, it will consist in further
applications of the same general type. This general conclu-
sion may be given the following suggestive symbolic form:

Image-- Birkhoff diagram relating Galois's theory of ambiguity to metaphysics

While the skillful metaphysical use of the Principle must
always be regarded as of dubious logical status, nevertheless
I believe it will remain the most important weapon of the
philosopher.

___________________________________________________________________________

A more recent lecture on the same subject —

"From Leibniz to Quantum World:
Symmetries, Principle of Sufficient Reason
and Ambiguity in the Sense of Galois
"

by Jean-Pierre Ramis (Johann Bernoulli Lecture at U. of Groningen, March 2005)

Monday, May 31, 2010

Memorial for Galois

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

… and for Louise Bourgeois

Image-- Louise Bourgeois, sculptor of giant spiders, dies at 98

"The épateurs  were as boring as the bourgeois,
two halves of one dreariness."

— D. H. Lawrence, The Plumed Serpent

Image-- Google 5/31/2010 search for 'eightfold geometry' yields page on mother goddess as spider figure, also pages on some actual geometry

Sunday, May 30, 2010

A Post for Galois

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

Evariste Galois, 1811-1832 (Vita Mathematica, V. 11)

  • Paperback: 168 pages
  • Publisher: Birkhäuser Basel; 1 edition (December 6, 1996)
  • Language: English
  • ISBN-10: 3764354100
  • ISBN-13: 978-3764354107
  • Product Dimensions: 9.1 x 6 x 0.4 inches
  • Shipping Weight: 9.1 ounces
  • Average Customer Review: 5.0 out of 5 stars  (1 customer review)
  • Amazon Bestsellers Rank: #933,939 in Books

Awarded 5 stars by Christopher G. Robinson (Cambridge, MA USA).
See also other reviews by Robinson.

Galois was shot in a duel on today's date, May 30, in 1832. Related material for those who prefer entertainment to scholarship—

"It is a melancholy pleasure that what may be [Martin] Gardner’s last published piece, a review of Amir Alexander’s Duel at Dawn: Heroes, Martyrs & the Rise of Modern Mathematics, will appear next week in our June issue." —Roger Kimball of The New Criterion, May 23, 2010.

Today is, incidentally, the feast day of St. Joan of Arc, Die Jungfrau von Orleans. (See "against stupidity" in this journal.)

Sunday, March 21, 2010

Galois Field of Dreams

Filed under: General,Geometry — Tags: , — m759 @ 10:01 am

It is well known that the seven (22 + 2 +1) points of the projective plane of order 2 correspond to 2-point subspaces (lines) of the linear 3-space over the two-element field Galois field GF(2), and may be therefore be visualized as 2-cube subsets of the 2×2×2 cube.

Similarly, recent posts* have noted that the thirteen (32 + 3 + 1) points of the projective plane of order 3 may be seen as 3-cube subsets in the 3×3×3 cube.

The twenty-one (42 + 4 +1) points of the (unique) projective plane of order 4 may also be visualized as subsets of a cube– in this case, the 4×4×4 cube. This visualization is somewhat more complicated than the 3×3×3 case, since the 4×4×4 cube has no central subcube, and each projective-plane point corresponds to four, not three, subcubes.

These three cubes, with 8, 27, and 64 subcubes, thus serve as geometric models in a straightforward way– first as models of finite linear spaces, hence as models for small Galois geometries derived from the linear spaces. (The cubes with 8 and 64 subcubes also serve in a less straightforward, and new, way as finite-geometry models– see The Eightfold Cube, Block Designs, and Solomon's Cube.)

A group of collineations** of the 21-point plane is one of two nonisomorphic simple groups of order 20,160. The other is the linear group acting on the linear 4-space over the two-element Galois field  GF(2). The 1899 paper establishing the nonisomorphism notes that "the expression Galois Field is perhaps not yet in general use."

Coordinates of the 4×4×4 cube's subcubes can, of course, be regarded as elements of the Galois field GF(64).

The preceding remarks were purely mathematical. The "dreams" of this post's title are not. See…

Number and Time, by Marie-Louise von Franz

See also Geometry of the I Ching and a search in this journal for "Galois + Ching."

* February 27 and March 13

** G20160 in Mitchell 1910,  LF(3,22) in Edge 1965

— Mitchell, Ulysses Grant, "Geometry and Collineation Groups
   of the Finite Projective Plane PG(2,22),"
   Princeton Ph.D. dissertation (1910)

— Edge, W. L., "Some Implications of the Geometry of
   the 21-Point Plane," Math. Zeitschr. 87, 348-362 (1965)

Tuesday, March 26, 2024

Pour les petites

Filed under: General — m759 @ 5:07 pm

"This is by way of a pep talk, mes petites . . ."

Musical and theatrical accompaniment . . .

Musical:  Valse pour les petites , reportedly recorded on March 2, 2019;

Theatrical:  A Log24 post from that same date, Schoolgirls for Galois.
 

Tuesday, January 2, 2024

Mathematics Made Absurd:  Domain and Range

Filed under: General — Tags: , — m759 @ 7:54 am

" the dominant discourse limits the range
     of discussion in each domain…."

https://americanaffairsjournal.org/2023/11/
the-stagnant-science-mainstream-economics-in-america/

See as well Boole vs. Galois and

Friday, December 29, 2023

A Wrinkle in Logic

Filed under: General — Tags: , — m759 @ 4:17 pm

Also on March 8, 2018

Barbara Johnson, Nothing Fails Like Success, detail

This post was suggested by the word "entanglement' in the previous post.

See as well "Galois (Xor) addition."

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."

Thursday, December 7, 2023

“Wheel Turnin’ ’Round and ’Round” — Song Lyric

Filed under: General — m759 @ 8:28 pm

Image by A. A. ("Bert") Jagers from a search in this journal for Galois sequence

http://www.log24.com/log/pix11/110209-GaloisStamp.jpg

Monday, November 27, 2023

Problems with the Process

Filed under: General — Tags: — m759 @ 5:52 am

Condensed from Peter J. Cameron's weblog today —

“Words that tear and strange rhymes”

"In his youth, Paul Simon thought of himself as a poet . . . .

And surprisingly often he describes problems with the process:

And the song I was writing is left undone
I don’t know why I spend my time
Writing songs I can’t believe
With words that tear and strange rhymes

For me, things were somewhat similar. Like many people, I wrote poetry in my youth. Julian Jaynes says something like 'Poems are rafts grasped at by men drowning in inadequate minds', but I think I knew from early on that one of the main reasons was to practise my writing, so that when I had something to say I could say it clearly. When Bob Dylan renounced the over-elaborate imagery of Blonde on Blonde  for the clean simplicity of John Wesley Harding, I took that as a role model.

Could Simon’s experience happen in mathematics? It is possible to imagine that an important mathematical truth is expressed in 'words that tear and strange rhymes'. More worryingly, an argument written in the most elegant style could be wrong, and we may be less likely to see the mistake because the writing is so good."

The problem with the process in this  case is Cameron's misheard lyrics.

From https://www.paulsimon.com/track/kathys-song-2/

And a song I was writing is left undone
I don’t know why I spend my time
Writing songs I can’t believe
With words that tear and strain to rhyme

A rather different artist titled a more recent song
"Strange Rhymes Can Change Minds."

See also . . .


 

Saturday, November 18, 2023

“Don’t solicit for your sister,* it’s not nice.” — Tom Lehrer

Filed under: General — Tags: — m759 @ 11:36 am

André Weil to his sister:

From this journal at 1:51 AM  ET Thursday, September 8, 2022

"The pleasure comes from the illusion" . . .

Exercise:

Compare and contrast the following structure with the three
"bricks" of the R. T. Curtis Miracle Octad Generator (MOG).

http://www.log24.com/log/pix11B/110805-The24.jpg

Note that the 4-row-2-column "brick" at left is quite 
different from the other two bricks, which together
show chevron variations within a Galois tesseract —

.

Further Weil remarks . . .

A Slew of Prayers

"The pleasure comes from the illusion
and the far from clear meaning;
once the illusion is dissipated,
and knowledge obtained, one becomes
indifferent at the same time;
at least in the Gitâ there is a slew of prayers
(slokas) on the subject, each one more final
than the previous ones."

*

Friday, November 10, 2023

Logos

Filed under: General — Tags: , , , — m759 @ 12:08 pm

Related art —

(For some backstory, see Geometry of the I Ching
and the history of Chinese philosophy.)

Galois space of six dimensions represented in Euclidean spaces of three and of two dimensions

Tuesday, October 24, 2023

A Bond with Reality:  The Geometry of Cuts

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


Illustrations of object and gestures
from finitegeometry.org/sc/ —

Object

Gestures

An earlier presentation of the above
seven partitions of the eightfold cube:

Seven partitions of the 2x2x2 cube in a book from 1906

Related mathematics:

The use  of binary coordinate systems
as a conceptual tool

Natural physical  transformations of square or cubical arrays
of actual physical cubes (i.e., building blocks) correspond to
natural algebraic  transformations of vector spaces over GF(2).
This was apparently not previously known.

See "The Thing and I."

and . . .

Galois.space .

 

Related entertainment:

Or Matt Helm by way of a Jedi cube.

Thursday, October 19, 2023

Math for Barbie

Filed under: General — Tags: , — m759 @ 2:56 am

Continued from "Barbie at the Space Barn," Oct. 17.

"Open the Space Barn doors, Barbie." —

For those who prefer the Hollywood  part of  L.A.,
there is Barbierella

Friday, October 13, 2023

Turn, Turn, Turn

Filed under: General — Tags: , , — m759 @ 3:06 am

The conclusion of a Hungarian political figure's obituary in
tonight's online New York Times, written by Clay Risen

"A quietly religious man, he spent his last years translating
works dealing with Roman Catholic canon law."

This  journal on the Hungarian's date of death, October 8,
a Sunday, dealt in part with the submission to Wikipedia of
the following brief article . . . and its prompt rejection.

The Cullinane diamond theorem is a theorem
about the Galois geometry underlying
the Miracle Octad Generator of R. T. Curtis.[1]

The theorem also explains symmetry properties of the
sort of chevron or diamond designs often found on quilts.

Reference

1. Cullinane diamond theorem at
the Encyclopedia of Mathematics

Some quotations I prefer to Catholic canon law —

Ludwig Wittgenstein,
Philosophical Investigations  (1953)

97. Thought is surrounded by a halo.
—Its essence, logic, presents an order,
in fact the a priori order of the world:
that is, the order of possibilities * ,
which must be common to both world and thought.
But this order, it seems, must be
utterly simple . It is prior  to all experience,
must run through all experience;
no empirical cloudiness or uncertainty can be
allowed to affect it ——It must rather be of
the purest crystal. But this crystal does not appear
as an abstraction; but as something concrete,
indeed, as the most concrete,
as it were the hardest  thing there is.

* See the post Wittgenstein's Diamond.

Related language in Łukasiewicz (1937)—

http://www.log24.com/log/pix10B/101127-LukasiewiczAdamantine.jpg

See as well Diamond Theory in 1937.

Friday, October 6, 2023

“A Shining” — New Jon Fosse Book, Out on Halloween

Filed under: General — Tags: , , — m759 @ 12:39 pm

Shining-related material from my own life —

Other material related to my own life, also on the date
November 25, 2009, but less personal —

From the current version —

Some background —

Monday, September 18, 2023

The Passage of Time

Filed under: General — Tags: , — m759 @ 9:55 am

The figure above summarizes a new way of looking at 
so-called "figurate numbers." The old  way goes back
at least to the time of Pythagoras.

A more explicit presentation —

Sunday, September 17, 2023

Rhyme and Desire: The Blame Dance

Filed under: General — m759 @ 10:56 pm

A bold dance featuring the art (as paintings and as fabric)
of the talented Marcela Nowak —

A related bold dance —

From this  journal on the above YouTube date — June 15, 2011 —

Clint Eastwood with football in film set in New Orleans

Background: Men in Feminism , edited by Alice Jardine and
published by Taylor & Francis in 1987, "Walking the Tightrope
of Feminism and Male Desire," by Judith Mayne, page 64

Other Hayworth-related chronology —

"Turn to Page Seven."

Sunday, August 6, 2023

Contra Gombrich

Filed under: General — Tags: — m759 @ 10:42 am

A search in this journal for Cornell + Warburg suggests
a review of the concept "iconology of the interval " . . .
Ikonologie des Zwischenraums —

Some small Galois spaces (the Cullinane models)

"Yet if this Denkraum ,  this 'twilight region,'  is where the artist and
emblem-maker invent, then, as Gombrich well knew, Warburg also
constantly regrets the 'loss' of this 'thought-space,' which he also
dubs the Zwischenraum  and Wunschraum ."

— Memory, Metaphor, and Aby Warburg's Atlas of Images ,
     Christopher D. Johnson, Cornell University Press, 2012, p. 56

Monday, June 26, 2023

The Boole Tool  and The XOR Schism

Filed under: General — m759 @ 12:32 pm

'Galois Additions of Space Partitions'

Tuesday, May 30, 2023

Seal 7 . . .

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

Continues.

See John Baez this morning on Galois. Note that Baez's
report of Galois's dies natalis  is in error.

Saturday, May 13, 2023

The Identity of an Entity

Filed under: General — Tags: — m759 @ 4:46 pm

From posts of Walpurgisnacht 2023

Tuesday, May 9, 2023

Alternate Film Title: Something, Somewhere, Sometime

Filed under: General — Tags: — m759 @ 2:16 am

"Long promised road
Flows to the source, gentle force"

— The Beach Boys

From a post of June 10, 2013

Sunday, April 30, 2023

For Harlan Kane: The Walpurgisnacht Hallucination

Note that if the "compact Riemann surface" is a torus formed by
joining opposite edges of a 4×4 square array, and the phrase
"vector bundle" is replaced by "projective line," and so forth,
the above ChatGPT hallucination is not completely unrelated to
the following illustration from the webpage "galois.space" —

See as well the Cullinane  diamond theorem.

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.

Wednesday, April 19, 2023

New Types of Combinatorial Structure

Filed under: General — Tags: , — m759 @ 11:18 am

(For the above title, see the previous post.)

For instance:  "Zero Sum," April 6, 2023 —

'Galois Additions of Space Partitions'

 

Monday, April 10, 2023

Space

Filed under: General — Tags: , , — m759 @ 3:09 am

(Perspective Not  as Symbolic Form)

From a post of June 8, 2014

Some background on the large Desargues configuration

See August 6, 2013 — Desargues via Galois.

Thursday, April 6, 2023

Zero Sum

Filed under: General — Tags: — m759 @ 12:11 pm

'Galois Additions of Space Partitions'

Related elementary mathematics from Google image searches —

Despite the extremely  elementary nature of the above tables,
the difference between the binary addition of Boole and that
of Galois seems not to be widely known.

See "The Hunt for Galois October" and "In Memory of a Mississippi Coach."

Tuesday, March 28, 2023

Onestone Parable

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

“You’re literally looking for like a one in a million thing.
You filter out the 999,999 of the boring ones, then
you’ve got something that’s weird, and then that’s worth
further exploration.”

— Quote from a mathematics story today at Gizmodo

A different "one in a million" mathematics story —

On Steiner Quadruple Systems of Order 16.

See also Galois Tesseract.

Wednesday, March 8, 2023

The Adapter

Filed under: General — Tags: , — m759 @ 12:25 am

See Helen Mirren with a plastic 45-rpm record adapter.

Related Log24 posts — Galois Seals.

Monday, February 27, 2023

For Gen Z: The Mark of Zorro

Filed under: General — m759 @ 12:50 pm

Galois space of six dimensions represented in Euclidean spaces of three and of two dimensions

Thursday, February 23, 2023

“Where Whirled and Well”

Filed under: General — m759 @ 1:32 pm

"Where whirled and well"

— Conrad Aiken, Great Circle

And the light shone in darkness and
Against the Word the unstilled world still whirled
About the centre of the silent Word.

— T. S. Eliot, "Ash Wednesday"
 

About the Centre:

IMAGE- The eight Galois quaternions

See also Dorm Room.

Wednesday, February 8, 2023

Text and Context: The Epiphany Shibboleth

Filed under: General — Tags: , — m759 @ 4:05 pm

The epigraph to Chapter 2 of Category Theory in Context  by Emily Riehl  —

[Maz16] Barry Mazur. Thinking about Grothendieck.
Notices of the AMS, 63(4):404–405, 2016.

The above epigraph in context, in a paper dated
January 6, 2016 (Epiphany) —

Also on Epiphany 2016 —

Wednesday, January 6, 2016

Galois.io

Filed under: General,Geometry — m759 @ 8:35 AM 

The title is a new URL.

Midrash on the URL suffix —

" 'I/O' is a computer term of very long standing
that means 'input/output,' i.e. the means by which
a computer communicates with the outside world.
In a domain name, it's a shibboleth that implies
that the intended audience for a site is other
programmers."

— Phil Darnowsky on Dec. 18, 2014

Remarks for a wider audience —

See some Log24 posts related to Dec. 18, 2014.

Tuesday, February 7, 2023

The Graduate School of Design

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

The above cubic equation may also be written as

x3 – x – 1 = 0.

The equation occurred in my own work in 1985:

An architects' equation appears also in Galois geometry.

An architects' equation that appears also in Galois geometry.

For further details on the plastic number, see an article by
Siobhan Roberts on John Baez  in  The New York Times —

Interstices

Filed under: General — Tags: , , , — m759 @ 10:34 am

Perhaps Crossan should have consulted Galois, not Piaget . . .

From Hermann Weyl's 1952 classic Symmetry —

"Galois' ideas, which for several decades remained
a book with seven seals  but later exerted a more
and more profound influence upon the whole
development of mathematics, are contained in
a farewell letter written to a friend on the eve of
his death, which he met in a silly duel at the age of
twenty-one. This letter, if judged by the novelty and
profundity of ideas it contains, is perhaps the most
substantial piece of writing in the whole literature
of mankind."

Friday, February 3, 2023

Rhyme Time

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

From Wednesday, St. Bridget's Day, 2023

Galois Additions of Space Partitions

Poetic meditation from The New Yorker  today

"If the tendency of rhyme, like that of desire,
is to pull distant things together
and force their boundaries to blur,
then the countervailing force in this book,
the one that makes it go, is the impulse
toward narrative, toward making sense of
the passage of time."

Thursday, February 2, 2023

“Here I Come Again”

Filed under: General — Tags: — m759 @ 1:09 am

From tonight's previous post

"here I come again . . . the square root of minus one,
having terminated my humanities" — 

Samuel Beckett, Stories and Texts for Nothing
(New York: Grove, 1967), 128.

From The French Mathematician
by Tom Petsinis (Nov. 30, 1998) —

0

I had foreseen it all in precise detail.
One step led inevitably to the next,
like the proof of a shining theorem,
down to the conclusive shot that still echoes
through time and space. 
Facedown in the damp pine needles,
I embraced that fatal sphere
with my whole body. Dreams, memories,
even the mathematics I had cherished
and set down in my last will and testament–
all receded. I am reduced to
a singular point; in an instant
I am transformed to .

i = an imaginary being

Here, on this complex space,
i  am no longer the impetuous youth
who wanted to change the world
first with a formula and then with a flame.
Having learned the meaning of infinite patience,
i  now rise to the text whenever anyone reads 
about Evariste Galois, preferring to remain 
just below the surface, 
like a goldfish nibbling the fringe of a floating leaf.
Ink is more mythical than blood
(unless some ancient poet slit his 
vein and wrote an epic in red):
The text is a two-way mirror 
that allows me to look into
the life and times of the reader. 
Who knows, someday i  may rise
to a text that will compel me 
to push through to the other side.
Do you want proof that i  exist? Where am ?
Beneath every word, behind each letter, 
on the side of a period that will never see the light.

Related reading . . .

See also "William Lawvere, Category Theory, Hegel, Mao, and Code."

( https://www.reddit.com/r/socialistprogrammers/comments/m1oe88/
william_lawvere_category_theory_hegel_mao_and_code/ )

Also relating category theory and computation —
the interests of Lawvere and those of Davis — is
an article at something called The Topos Institute (topos.site) —

"Computation and Category Theory," by Joshua Meyers,
Wednesday, 10 Aug., 2022.

Meyers on Davis —

Wednesday, February 1, 2023

Variations in Memory of a Designer

Last updated at 22:46 PM ET on 1 February 2023.

Galois Additions of Space Partitions

Click for a designer's obituary.

Paraphrase for a road-sign collector:

See as well Today's New York Times  obituary
of the Harvard Business School Publishing 
Director of Intellectual Property.

Sunday, January 22, 2023

The Stillwell Dichotomies

Filed under: General — Tags: , , , — m759 @ 1:18 am

Number Space
Arithmetic  Geometry
Discrete  Continuous

Related literature —

IMAGE- History of Mathematics in a Nutshell

Bourbaki on arithmetic and geometry

From a "Finite Fields in 1956" post —

The Nutshell:

    Related Narrative:

Saturday, January 14, 2023

Châtelet on Weil — A “Space of Gestures”

Filed under: General — Tags: , , , — m759 @ 2:21 pm
 

From Gilles Châtelet, Introduction to Figuring Space
(Springer, 1999) —

Metaphysics does have a catalytic effect, which has been described in a very beautiful text by the mathematician André Weil:

Nothing is more fertile, all mathematicians know, than these obscure analogies, these murky reflections of one theory in another, these furtive caresses, these inexplicable tiffs; also nothing gives as much pleasure to the researcher. A day comes when the illusion vanishes: presentiment turns into certainty … Luckily for researchers, as the fogs clear at one point, they form again at another.4

André Weil cuts to the quick here: he conjures these 'murky reflections', these 'furtive caresses', the 'theory of Galois that Lagrange touches … with his finger through a screen that he does not manage to pierce.' He is a connoisseur of these metaphysical 'fogs' whose dissipation at one point heralds their reforming at another. It would be better to talk here of a horizon that tilts thereby revealing a new space of gestures which has not as yet been elucidated and cut out as structure.

4 A. Weil, 'De la métaphysique aux mathématiques', (Oeuvres, vol. II, p. 408.)

For gestures as fogs, see the oeuvre of  Guerino Mazzola.

For some clearer remarks, see . . .


Illustrations of object and gestures
from finitegeometry.org/sc/ —

 

Object

 

Gestures

An earlier presentation
of the above seven partitions
of the eightfold cube:

Seven partitions of the 2x2x2 cube in a book from 1906

Related material: Galois.space .

Friday, December 30, 2022

Bullshit Studies: The View from East Lansing

Filed under: General — Tags: — m759 @ 1:40 pm

Detail of the above screen (click to enlarge) —

See also this  journal on the above date  — June 10, 2021.

From this journal on May 6, 2009

A related picture of images that "reappear metamorphosed
in the coordinate system of the high region" —

(For the backstory, see Geometry of the I Ching
and the history of Chinese philosophy.)

Galois space of six dimensions represented in Euclidean spaces of three and of two dimensions

Saturday, December 24, 2022

Window as Matrix

Filed under: General — Tags: , — m759 @ 9:57 am

Grids

Author: Rosalind Krauss
Source: October , Vol. 9 (Summer, 1979), pp. 50-64
Published by: The MIT Press
Stable URL: http://www.jstor.org/stable/778321

From page 59:

"Flowing and freezing; glace  in French means glass, mirror, and ice;
transparency, opacity, and water. In the associative system
of symbolist thought this liquidity points in two directions.
First, towards the flow of birth-the amniotic fluid, the 'source'-
but then, towards the freezing into stasis or death-
the unfecund immobility of the mirror. For Mallarmé, particularly,
the window functioned as this complex, polysemic sign by which
he could also project the 'crystallization  of reality into art.' 5
Mallarmé's Les Fenêtres  dates from 1863;
Redon's most evocative window, Le Jour , appeared in 1891
in the volume Songes . If the window is this matrix of
ambi- or multivalence, and the bars of the windows-the grid-
are what help us to see, to focus on, this matrix, they are
themselves the symbol of the symbolist work of art.
They function as the multilevel representation through which
the work of art can allude, and even reconstitute, the forms of Being."

5 Robert G. Cohn, "Mallarmé's Windows," Yale French Studies ,
no. 54 (1977), 23-31.

Another evocative example — See Galois Window in this  journal.

Thursday, December 22, 2022

Welcome to the Desert of the Real

Filed under: General — Tags: , — m759 @ 10:58 am

IMAGE- Aug. 5, 2005- Galois tesseract, Shakespeherian Rag, Sir Alec Guinness

Wednesday, December 14, 2022

Plan 9 From Moriarty

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

Some related mathematical windmills —

IMAGE- The eight Galois quaternions

For the eight-limbed star at the top of the quaternion array
above, see "Damnation Morning" in this journal —

She drew from her handbag a pale grey gleaming 
implement that looked by quick turns to me like 
a knife, a gun, a slim sceptre, and a delicate 
branding iron—especially when its tip sprouted 
an eight-limbed star of silver wire.

“The test?” I faltered, staring at the thing.

“Yes, to determine whether you can live in 
the fourth dimension or only die in it.”

— Fritz Leiber, short story, 1959

See as well . . .

https://www.thecrimson.com/article/2022/12/13/
harvard-psychedelics-club-signet-society-art-show-2022/
.

Tuesday, December 13, 2022

In Memory of a Mississippi Coach

Filed under: General — Tags: , — m759 @ 1:10 pm

Remarks in memory of football coach Mike Leach of Mississippi State University

The "Boolean exclusive or" is the same as addition 
in the two-element Galois field GF(2).

Friday, November 25, 2022

“Open the pod bay doors, Hal.”

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

"Both Hal and the students were great interlocutors
with lots of ideas and differing perspectives."

Leah Dickerman, MoMA, May 10, 2010, at
https://www.princeton.edu/news/2010/05/10/
class-snapshot-origins-abstraction
 

Later . . .

See as well Desargues + Galois.

Thursday, November 24, 2022

The Drum Machine

Filed under: General — Tags: , , , — m759 @ 2:18 pm

"A struggling music producer sells his soul to a 1970s drum machine."

— Summary of a short film by Kevin Ignatius, "Hook Man."

The music producer pawns his current drum device 
and acquires a demonic 1970s machine.


Artistic symbolism —

The 16-pad device at left may be viewed by enthusiasts of ekphrasis
as a Galois tesseract, and the machine at right as the voice of
Hal Foster, an art theorist who graduated from Princeton in 1977.

For an example of Foster's prose style, see
the current London Review of Books.

Sunday, November 6, 2022

The Paris Review Opening

Filed under: General — Tags: — m759 @ 4:28 pm

And then there is the Paris Review  opening . . . .

See  "The Hunt for Galois October."

Saturday, October 29, 2022

Boolean Halloween

Filed under: General — m759 @ 12:52 pm

From Log24 posts tagged Boole vs. Galois

Kauffman‘s fixation on the work of Spencer-Brown is perhaps in part
due to Kauffman’s familiarity with Boolean algebra and his ignorance of
Galois geometry.  See other posts now tagged Boole vs. Galois.

Detail, 8/14/2016 Google image search for 'Galois Boole'

See also “A Four-Color Epic” (April 16, 2020).

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)

Thursday, September 8, 2022

Analogy in Mathematics: Chevron Variations

André Weil in 1940 on analogy in mathematics —

. "Once it is possible to translate any particular proof from one theory to another, then the analogy has ceased to be productive for this purpose; it would cease to be at all productive if at one point we had a meaningful and natural way of deriving both theories from a single one. In this sense, around 1820, mathematicians (Gauss, Abel, Galois, Jacobi) permitted themselves, with anguish and delight, to be guided by the analogy between the division of the circle (Gauss’s problem) and the division of elliptic functions. Today, we can easily show that both problems have a place in the theory of abelian equations; we have the theory (I am speaking of a purely algebraic theory, so it is not a matter of number theory in this case) of abelian extensions. Gone is the analogy: gone are the two theories, their conflicts and their delicious reciprocal reflections, their furtive caresses, their inexplicable quarrels; alas, all is just one theory, whose majestic beauty can no longer excite us. Nothing is more fecund than these slightly adulterous relationships; nothing gives greater pleasure to the connoisseur, whether he participates in it, or even if he is an historian contemplating it retrospectively, accompanied, nevertheless, by a touch of melancholy. The pleasure comes from the illusion and the far from clear meaning; once the illusion is dissipated, and knowledge obtained, one becomes indifferent at the same time; at least in the Gitâ there is a slew of prayers (slokas) on the subject, each one more final than the previous ones."

"The pleasure comes from the illusion" . . .

Exercise:

Compare and contrast the following structure with the three
"bricks" of the R. T. Curtis Miracle Octad Generator (MOG).

http://www.log24.com/log/pix11B/110805-The24.jpg

Note that the 4-row-2-column "brick" at left is quite 
different from the other two bricks, which together
show chevron variations within a Galois tesseract —

Saturday, September 3, 2022

1984 Revisited

Filed under: General — m759 @ 2:46 pm

Cube Bricks 1984 —

An Approach to Symmetric Generation of the Simple Group of Order 168

Related material

Note the three quadruplets of parallel edges  in the 1984 figure above.

Further Reading

The above Gates article appeared earlier, in the June 2010 issue of
Physics World , with bigger illustrations. For instance —

Exercise: Describe, without seeing the rest of the article,
the rule used for connecting the balls above.

Wikipedia offers a much clearer picture of a (non-adinkra) tesseract —

      And then, more simply, there is the Galois tesseract

For parts of my own  world in June 2010, see this journal for that month.

The above Galois tesseract appears there as follows:

Image-- The Dream of the Expanded Field

See also the Klein correspondence in a paper from 1968
in yesterday's 2:54 PM ET post

Sunday, August 7, 2022

From Coxeter’s Nutshell: Points and Marks

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

Monday, August 1, 2022

Enowning

Filed under: General — Tags: — m759 @ 3:26 pm

Related material — The Eightfold Cube.

See also . . .

"… Mathematics may be art, but to the general public it is
a black art, more akin to magic and mystery. This presents
a constant challenge to the mathematical community: to 
explain how art fits into our subject and what we mean by beauty."

— Sir Michael Atiyah, “The Art of Mathematics”
in the AMS Notices , January 2010

Saturday, July 30, 2022

Modal Diamond Box

Filed under: General — m759 @ 10:54 am
 

A  mnemonic  from a course titled
Galois Connections and Modal Logics“—

“Traditionally, there are two modalities, namely,
possibility and necessity. The basic modal operators
are usually written box (square) for necessarily
and diamond (diamond) for possibly.
Then, for example, diamondP  can be read as
‘it is possibly the case that P .'”

See also Intensional Semantics , lecture notes
by Kai von Fintel and Irene Heim, MIT,
Spring 2007 edition—

“The diamond  symbol for possibility is due to C.I. Lewis, first introduced in Lewis & Langford (1932), but he made no use of a symbol for the dual combination ¬¬. The dual symbol  was later devised by F.B. Fitch and first appeared in print in 1946 in a paper by his doctoral student Barcan (1946). See footnote 425 of Hughes & Cresswell (1968). Another notation one finds is L for necessity and M for possibility, the latter from the German möglich  ‘possible.’”

Barcan, Ruth C.: 1946. “A Functional Calculus of First Order Based on Strict Implication.” Journal of Symbolic Logic, 11(1): 1–16. URL http://www.jstor.org/pss/2269159.

Hughes, G.E. & Cresswell, M.J.: 1968. An Introduction to Modal Logic. London: Methuen.

Lewis, Clarence Irving & Langford, Cooper Harold: 1932. Symbolic Logic. New York: Century.

For less rigorous remarks, search Log24 for Modal Diamond Box.

Tuesday, July 5, 2022

For Ron Howard, Tom Hanks, and Dan Brown — Symbology!

Filed under: General — m759 @ 1:22 am

Thursday, June 23, 2022

The Nutshell Suite

Filed under: General — Tags: , , , — m759 @ 10:35 am

The above is a summary of 
Pythagorean philosophy 
reposted here on . . .

September 10, 2019.
 

Battle of the Nutshells:

IMAGE- History of Mathematics in a Nutshell

From a much larger nutshell
on the above Pythagorean date—

Now let's dig a bit deeper into history . . .

Bourbaki on arithmetic and geometry

Wednesday, June 22, 2022

Code Wars: “Use the Source, Luke.”

Filed under: General — Tags: , , , , — m759 @ 7:13 pm

Click the above galaxy for a larger image.


"O God, I could be bounded in a nutshell
and count myself a king of infinite space,
were it not that I have bad dreams." — Hamlet

Battle of the Nutshells —

IMAGE- History of Mathematics in a Nutshell

From a much larger nutshell
on the above code date—

Tuesday, May 31, 2022

A Mad Night’s Work*

Filed under: General — Tags: , — m759 @ 8:57 am

Last night's posts on triangles, and today's anniversary of the
death of Evariste Galois, suggest a review . . .

"Take triangles, perhaps" . . . as a category.

And then . . . take squares, perhaps, as another category, 
and then . . . find a suitable "translation machine."

See "Square Triangles."

* Title adapted from a 2001 essay by Pierre Cartier.

Saturday, May 28, 2022

Grothendieck at Chapman …

Filed under: General — m759 @ 1:41 pm

Last two days of the conference, May 27 and 28, 2022 —

27th Friday

9:00 – 10:00 Andrés Villaveces (Univ. Nacional de Colombia):
Galoisian model theory:
the role(s) of Grothendieck (à son insu! )

10:00 – 11:00 Olivia Caramello (Univ. of Insubria; by Zoom):
The “unifying notion” of topos 1

1:00 – 11:15 Coffee Break

1:15 – 12:15 Mike Shulman (Univ. of San Diego):
Lifting Grothendieck universes to Grothendieck toposes 

12:15 – 1:15 José Gil-Ferez (Chapman Univ.)
The Isomorphism Theorem of Algebraic Logic:
a Categorical  Perspective

1:15 – 2:30 Lunch

2:30 – 3:30 Oumar Wone (Chapman) :
Vector bundles on Riemann surfaces according to
Grothendieck and his followers

3:30 – 4:30 Claudio Bartocci (Univ. of Genova):
The inception of the theory of moduli spaces:
Grothendieck's Quot scheme

4:30 – 5:30 Christian Houzel (IUFM de Paris):
Riemann surfaces after Grothendieck
[presented by J.J. Szczeciniarz]

28th Saturday

9:00 – 10:00 Silvio Ghilardi (Univ. degli Studi, Milano):
Investigating definability in propositional logic
via Grothendieck topologies and sheaves

10:00 – 11:00 Matteo Viale (Univ. of Turin; by zoom):
The duality between Boolean valuated models and 
topological presheaves

11:00 – 11:15 Coffee Break

11:15 – 12:15 Benjamin Collas (RIMS, Kyoto Univ.):
Galois-Teichmüller: arithmetic geometric principles

12:15 – 1:15 Closing: general discussion
animated by Alex Kurz (Chapman)

Friday, May 27, 2022

Great Escapes

Filed under: General — Tags: — m759 @ 2:12 pm

The above scene from "Hanna" comes from a webpage
dated August 29, 2011. See also

this journal on that date  —

Galois space of six dimensions represented in Euclidean spaces of three and of two dimensions

and today's previous "Escape" post.

Plan 9 from Disney

Filed under: General — m759 @ 3:00 am

 "With the Tablet of Ahkmenrah and the Cube of Rubik,
my power will know no bounds!"
— Kahmunrah in a novelization of Night at the Museum:
Battle of the Smithsonian , Barron's Educational Series

Scholium

Abstracting from narrative to structure, and from structure
to pure number, the Tablet of Ahkmenrah represents the
number 9 and the Cube of Rubik represents the number 27.

Returning from pure abstract numbers to concrete representations,
9 yields the structures in Log24 posts tagged Triangle.graphics,
and 27 yields a Galois  cube .

Tuesday, May 24, 2022

Playing the Palace

Filed under: General — m759 @ 9:54 am

From a Jamestown (NY) Post-Journal  article yesterday on
"the sold-out 10,000 Maniacs 40th anniversary concert at
The Reg Lenna Center Saturday" —

" 'The theater has a special place in our hearts. It’s played
a big part in my life,' Gustafson said.

Before being known as The Reg Lenna Center for The Arts,
it was formerly known as The Palace Theater. He recalled
watching movies there as a child…."

This, and the band's name, suggest some memories perhaps
better suited to the cinematic philosophy behind "Plan 9 from
Outer Space."

IMAGE- The Tablet of Ahkmenrah, from 'Night at the Museum'

 "With the Tablet of Ahkmenrah and the Cube of Rubik,
my power will know no bounds!"
— Kahmunrah in a novelization of Night at the Museum:
Battle of the Smithsonian , Barron's Educational Series

The above 3×3 Tablet of Ahkmenrah  image comes from
a Log24 search for the finite (i.e., Galois) field GF(3) that 
was, in turn, suggested by last night's post "Making Space."

See as well a mysterious document from a website in Slovenia
that mentions a 3×3 array "relating to nine halls of a mythical
palace where rites were performed in the 1st century AD" —

Saturday, May 7, 2022

Interality Meets the Seven Seals

Filed under: General — Tags: , , , — m759 @ 8:41 pm

Related material — Posts tagged Interality and Seven Seals.

From Hermann Weyl's 1952 classic Symmetry —

"Galois' ideas, which for several decades remained
a book with seven seals  but later exerted a more
and more profound influence upon the whole
development of mathematics, are contained in
a farewell letter written to a friend on the eve of
his death, which he met in a silly duel at the age of
twenty-one. This letter, if judged by the novelty and
profundity of ideas it contains, is perhaps the most
substantial piece of writing in the whole literature
of mankind."

Monday, April 25, 2022

Annals of Mathematical History

Filed under: General — Tags: , — m759 @ 11:42 pm

Bourbaki on arithmetic and geometry

Some related remarks —

IMAGE- History of Mathematics in a Nutshell

Monday, April 18, 2022

Iconic Simplicity

Filed under: General — Tags: , — m759 @ 11:38 am

An illustration from posts tagged Holy Field GF(3) —

IMAGE- Elementary Galois Geometry over GF(3)

See also a Log24 search for "Four Gods."

Friday, April 8, 2022

Souls at Stanford

Filed under: General — Tags: — m759 @ 6:00 am

Wednesday, March 30, 2022

Games

Filed under: General — Tags: , , — m759 @ 7:28 am

Click to enlarge.

Related reading — George Steiner's Fields of Force , on chess in Iceland, and . . .

The New Yorker , article by Sam Knight dated March 28, 2022 —

They went to Björk’s house. She cooked salmon.
She had seen “The Witch” and introduced Eggers
to Sjón, who had written a novel about seventeenth-
century witchcraft in Iceland. When he got home,
Eggers read Sjón’s books. “I was, like, this guy’s
a fucking magician,” Eggers said. “He sees all time,
in time, out of time.” 

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

Tuesday, March 15, 2022

The Rosenhain Symmetry

Filed under: General — Tags: , — m759 @ 12:26 pm

See other posts now so tagged.

Hudson's  Rosenhain tetrads,  as 20 of the 35 projective lines in PG(3,2),
illustrate Desargues's theorem as a symmetry within 10 pairs of squares 
under rotation about their main diagonals:

IMAGE- Desargues's theorem in light of Galois geometry

See also "The Square Model of Fano's 1892 Finite 3-Space."

The remaining 15 lines of PG(3,2), Hudson's Göpel tetrads, have their
own symmetries . . . as the Cremona-Richmond configuration.

Sunday, February 20, 2022

4×4 Nomenclature

Filed under: General — Tags: — m759 @ 2:35 am

The geometry of the 4×4 square may be associated with the name
Galois, as in "the Galois tesseract," or similarly with the name Kummer. 
Here is a Google image search using the latter name —

(Click to enlarge.)

 

Thursday, February 17, 2022

Space Memorial

Filed under: General — m759 @ 1:27 pm

"FILE – Retired Sandinista Gen. Hugo Torres poses for portrait
at his home, in Managua, Nicaragua, May 2, 2018."

— Photo caption from a Feb. 12 Washington Post  obituary 

Also on May 2, 2018 —

Related theology —

Tuesday, February 23, 2016

Revolutionary

Filed under: General — m759 @ 7:14 PM 

From New York Times  obituary today —

"The Rev. Fernando Cardenal, a son of privilege
who embraced Latin America’s poor as a revolutionary
priest and brazenly defied Pope John Paul II’s order to
quit Nicaragua’s leftist cabinet in the 1980s, died on
Saturday in Managua. He was 82."

Photo caption from the same obituary —

"Fernando Cardenal in 1990. As education minister of
Nicaragua under the Sandinistas in the 1980s, he
oversaw a sweeping campaign credited with reducing
illiteracy to 13 percent from 51 percent."

This alleged literacy improvement makes him sound like
Protestant  revolutionary.

For a Catholic  view of literacy, see The Gutenberg Galaxy .

See also the post Being Interpreted (Aug. 14, 2015) — 

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

Unhinged Melody

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

The time of the previous post was 4:46 AM ET today.

Fourteen minutes later —

"I'm a groupie, really." — Murray Bartlett in today's online NY Times

The previous post discussed group actions on a 3×3 square array. A tune
about related group actions on a 4×4  square array (a Galois tesseract. . .

'The Eddington Song'

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

Sunday, December 5, 2021

The Venn Lotus

Filed under: General — m759 @ 11:20 pm

Venn Lotus and Galois Tesseract, by Steven H. Cullinane on 5 December 2021.

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