Log24

Saturday, May 9, 2026

Galois’s Birthday, 2014

Filed under: General — Tags: — m759 @ 10:19 am

This post was suggested by a film release date and by a
synchronology check in this journal.

Sunday, October 26, 2025

Minority Opinion:
The Geometry of Logic Is Galois Geometry

Filed under: General — Tags: , , — m759 @ 11:02 pm

See The Righteous Gemstone and Boole vs. Galois.

Update of  9:48 AM EDT Oct.  28 . . .

Related material —

Markdown version uploaded Oct. 28, 2025, to NotebookLM.

Friday, December 20, 2024

For Harlan Kane: The Galois Rectangle

Filed under: General — Tags: , — m759 @ 12:30 pm

Galois's birthday, 1993 —

The title rectangle is featured in a recent sequel to The Galois Tesseract

Saturday, December 7, 2024

Galois.io

Filed under: General — m759 @ 8:44 pm

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.

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