Sunday, August 14, 2016

The Boole-Galois Games

Filed under: Uncategorized — 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.

Tuesday, November 25, 2014

Euclidean-Galois Interplay

Filed under: Uncategorized — 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)).


   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.

Friday, September 17, 2010

The Galois Window

Filed under: Uncategorized — 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—


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.

Saturday, February 17, 2018

The Binary Revolution

Filed under: Uncategorized — m759 @ 5:00 PM

Michael Atiyah on the late Ron Shaw

Phrases by Atiyah related to the importance in mathematics
of the two-element Galois field GF(2) —

  • "The digital revolution based on the 2 symbols (0,1)"
  • "The algebra of George Boole"
  • "Binary codes"
  • "Dirac's spinors, with their up/down dichotomy"

These phrases are from the year-end review of Trinity College,
Cambridge, Trinity Annual Record 2017 .

I prefer other, purely geometric, reasons for the importance of GF(2) —

  • The 2×2 square
  • The 2x2x2 cube
  • The 4×4 square
  • The 4x4x4 cube

See Finite Geometry of the Square and Cube.

See also today's earlier post God's Dice and Atiyah on the theology of 
(Boolean) algebra vs. (Galois) geometry:

Thursday, April 20, 2017

Stone Logic

Filed under: Uncategorized — m759 @ 9:48 PM

See also "Romancing the Omega" —

Image- Josefine Lyche work (with 1986 figures by Cullinane) in a 2009 exhibition in Oslo

Related mathematics — Guitart in this journal —

From 'Moving Logic, from Boole to Galois,' by René Guitart, 2005

See also Weyl + Palermo in this journal —


Sunday, April 16, 2017

Art Space Paradigm Shift

Filed under: Uncategorized — m759 @ 1:00 AM

This post's title is from the tags of the previous post


The title's "shift" is in the combined concepts of

Space and Number

From Finite Jest (May 27, 2012):

IMAGE- History of Mathematics in a Nutshell

The books pictured above are From Discrete to Continuous ,
by Katherine Neal, and Geometrical Landscapes , by Amir Alexander.

For some details of the shift, see a Log24 search for Boole vs. Galois.
From a post found in that search —

"Benedict Cumberbatch Says
a Journey From Fact to Faith
Is at the Heart of Doctor Strange

io9 , July 29, 2016

" 'This man comes from a binary universe
where it’s all about logic,' the actor told us
at San Diego Comic-Con . . . .

'And there’s a lot of humor in the collision
between Easter [ sic ] mysticism and
Western scientific, sort of logical binary.' "

[Typo now corrected, except in a comment.]

Sunday, January 29, 2017

Lottery Hermeneutics

Filed under: Uncategorized — Tags: — m759 @ 1:00 PM

For some backstory, see Lottery in this journal,
esp. a post of June 28, 2007:

Real Numbers: An Object Lesson.

One such number, 8775, is suggested by 
a Heinlein short story in a Jan. 25 post.

A search today for that number —

That Jan. 25 post, "For Your Consideration," also mentions logic.

Logic appears as well within a post from the above "8775" date,
August 16, 2016 —

Update of 10 am on August 16, 2016 —

See also Atiyah on the theology of 
(Boolean) algebra vs. (Galois) geometry:

Related:  Remarks by Charles Altieri on Wittgenstein in
today's previous post.

For remarks by  Wittgenstein related to geometry and logic, see 
(for instance) "Logical space" in "A Wittgenstein Dictionary," by
Hans-Johann Glock (Wiley-Blackwell, 1996).

Friday, November 25, 2016


Filed under: Uncategorized — Tags: , — m759 @ 12:00 AM

Before the monograph "Diamond Theory" was distributed in 1976,
two (at least) notable figures were published that illustrate
symmetry properties of the 4×4 square:

Hudson in 1905 —

Golomb in 1967 —

It is also likely that some figures illustrating Walsh functions  as
two-color square arrays were published prior to 1976.

Update of Dec. 7, 2016 —
The earlier 1950's diagrams of Veitch and Karnaugh used the
1's and 0's of Boole, not those of Galois.

Tuesday, August 16, 2016

Midnight Narrative

Filed under: Uncategorized — m759 @ 12:00 AM

The images in the previous post do not lend themselves
to any straightforward narrative. Two portions of the
large image search are, however, suggestive —

Boulez and Boole      and


Cross and Boolean lattice.

The improvised cross in the second pair of images
is perhaps being wielded to counteract the
Boole of the first pair of images. See the heading
of the webpage that is the source of the lattice
diagram toward which the cross is directed —

Update of 10 am on August 16, 2016 —

See also Atiyah on the theology of 
(Boolean) algebra vs. (Galois) geometry:

Saturday, July 30, 2016


Filed under: Uncategorized — m759 @ 9:48 AM

"Benedict Cumberbatch Says a Journey
From Fact to Faith Is at the Heart of Doctor Strange

— io9 yesterday

" 'This man comes from a binary universe where it’s all about logic,' 
the actor told us at San Diego Comic-Con . . . .

'And there’s a lot of humor in the collision between Easter [sic ]
mysticism and Western scientific, sort of logical binary.' "

Related material — Strange Awards, April 14, 2016.

I prefer a different sort of journey. See Boole vs. Galois.

Wednesday, July 20, 2016

Weather for Jews

Filed under: Uncategorized — m759 @ 5:00 AM

Somewhere Over the Rainbow

"What I'm aiming for are moments of strong sensation —
moments of total physical experience of the landscape,
when weather just reaches out and sucks you in."

The late Jane Wilson —

See also the previous post and, from the date of Wilson's death,

Geometry for Jews (Continued) —

Thursday, April 14, 2016

One Funeral at a Time

Filed under: Uncategorized — m759 @ 1:37 PM

On this date in 2005, mathematician Saunders Mac Lane died at 95.

Related material —

Max Planck quotations:

Mac Lane on Boolean algebra:

Mac Lane's summary chart (note the absence of Galois geometry ):

I disagree with Mac Lane's assertion that "the finite models of
Boolean algebra are dull."  See Boole vs. Galois in this journal.

Wednesday, January 13, 2016

Geometry for Jews

Filed under: Uncategorized — m759 @ 7:45 AM

(Continued from previous episodes)

'Games Played by Boole and Galois'

Boole and Galois also figure in the mathematics of space
i.e. , geometry.  See Boole + Galois in this journal.

Related material, according to Jung's notion of synchronicity —

Monday, January 11, 2016

Space Oddity

Filed under: Uncategorized — m759 @ 3:15 PM

It is an odd fact that the close relationship between some
small Galois spaces and small Boolean spaces has gone 
unremarked by mathematicians.  

A Google search today for "Galois spaces" + "Boolean spaces"
yielded, apart from merely terminological sources, only some
introductory material I have put on the Web myself. 

Some more sophisticated searches, however led to a few
documents from the years 1971 – 1981 …

"Harmonic Analysis of Switching Functions" ,
by Robert J. Lechner, Ch. 5 in A. Mukhopadhyay, editor,
Recent Developments in Switching Theory , Academic Press, 1971.

"Galois Switching Functions and Their Applications,"
by B. Benjauthrit and I. S. Reed,
JPL Deep Space Network Progress Report 42-27 , 1975

D.K. Pradhan, “A Theory of Galois Switching Functions,”
IEEE Trans. Computers , vol. 27, no. 3, pp. 239-249, Mar. 1978

"Switching functions constructed by Galois extension fields,"
by Iwaro Takahashi, Information and Control ,
Volume 48, Issue 2, pp. 95–108, February 1981

An illustration from the Lechner paper above —

"There is  such a thing as harmonic analysis of switching functions."

— Saying adapted from a young-adult novel

Monday, December 28, 2015

ART WARS Continues

Filed under: Uncategorized — m759 @ 9:00 PM

Combining two headlines from this morning's
New York Times  and Washington Post , we have

Deceptively Simple Geometries
on a Bold Scale

   Voilà —

Click image for details.

More generally, see
Boole vs. Galois.

Friday, December 25, 2015

Dark Symbol

Filed under: Uncategorized — Tags: — m759 @ 12:00 PM

Related material:

The previous post (Bright Symbol) and
a post from Wednesday
December 23, 2015, that links to posts
on Boolean algebra vs. Galois geometry.

"An analogy between mathematics and religion is apposite."

— Harvard Magazine  review by Avner Ash of
     Mathematics without Apologies
(Princeton University Press, January 18, 2015)

Wednesday, December 23, 2015

Splitting Apart

Filed under: Uncategorized — m759 @ 1:01 PM

Bleecker Street logo —

Click image for some background.

Related remarks on mathematics:

Boole vs. Galois

Sunday, December 13, 2015

The Monster as Big as the Ritz

Filed under: Uncategorized — Tags: , — m759 @ 11:30 AM

"The colorful story of this undertaking begins with a bang."

— Martin Gardner on the death of Évariste Galois

Monday, November 2, 2015

The Devil’s Offer

Filed under: Uncategorized — Tags: — m759 @ 11:09 AM

This is a sequel to the previous post and to the Oct. 24 post
Two Views of Finite Space.  From the latter —

" 'All you need to do is give me your soul:  
give up geometry and you will have this 
marvellous machine.' (Nowadays you
can think of it as a computer!) "

George Boole in image posted on All Souls' Day 2015

Colorful Story

Filed under: Uncategorized — Tags: , — m759 @ 10:00 AM

"The office of color in the color line
is a very plain and subordinate one.
It simply advertises the objects of
oppression, insult, and persecution.
It is not the maddening liquor, but
the black letters on the sign
telling the world where it may be had."

— Frederick Douglass, "The Color Line,"
The North American Review , Vol. 132,
No. 295, June 1881, page 575

Or gold letters.

From a search for Seagram in this  journal —

Seagram VO ad, image posted on All Souls's Day 2015

"The colorful story of this undertaking begins with a bang."

— Martin Gardner on the death of Évariste Galois

Saturday, October 31, 2015

Raiders of the Lost Crucible

Filed under: Uncategorized — m759 @ 10:15 AM

Stanford Encyclopedia of Philosophy
on the date Friday, April 5, 2013 —

Paraconsistent Logic

"First published Tue Sep 24, 1996;
substantive revision Fri Apr 5, 2013"

This  journal on the date Friday, April 5, 2013 —

The object most closely resembling a "philosophers' stone"
that I know of is the eightfold cube .

For some related philosophical remarks that may appeal 
to a general Internet audience, see (for instance) a website
by I Ching  enthusiast Andreas Schöter that displays a labeled
eightfold cube in the form of a lattice diagram —

Related material by Schöter —

A 20-page PDF, "Boolean Algebra and the Yi Jing."
(First published in The Oracle: The Journal of Yijing Studies ,
Vol 2, No 7, Summer 1998, pp. 19–34.)

I differ with Schöter's emphasis on Boolean algebra.
The appropriate mathematics for I Ching  studies is,
I maintain, not Boolean algebra  but rather Galois geometry.

See last Saturday's post Two Views of Finite Space.
Unfortunately, that post is, unlike Schöter's work, not  
suitable for a general Internet audience.

Saturday, October 24, 2015

Two Views of Finite Space

Filed under: Uncategorized — Tags: — m759 @ 10:00 AM

The following slides are from lectures on "Advanced Boolean Algebra" —

The small Boolean  spaces above correspond exactly to some small
Galois  spaces. These two names indicate approaches to the spaces
via Boolean algebra  and via Galois geometry .

A reading from Atiyah that seems relevant to this sort of algebra
and this sort of geometry —

" 'All you need to do is give me your soul:  give up geometry 
and you will have this marvellous machine.' (Nowadays you
can think of it as a computer!) "

Related material — The article "Diamond Theory" in the journal
Computer Graphics and Art , Vol. 2 No. 1, February 1977.  That
article, despite the word "computer" in the journal's title, was
much less about Boolean algebra  than about Galois geometry 

For later remarks on diamond theory, see finitegeometry.org/sc.

Wednesday, October 21, 2015

Algebra and Space

Filed under: Uncategorized — m759 @ 7:59 AM

"Perhaps an insane conceit …."    Perhaps.

Related remarks on algebra and space —

"The Quality Without a Name" (Log24, August 26, 2015).

Sunday, September 6, 2015

Elementally, My Dear Watson

Filed under: Uncategorized — m759 @ 9:45 AM

Sarah Larson in the online New Yorker  on Sept. 3, 2015,
discussed Google's new parent company, "Alphabet"—

"… Alphabet takes our most elementally wonderful
general-use word—the name of the components of
language itself*—and reassigns it, like the words
tweet, twitter, vine, facebook, friend, and so on,
into a branded realm."

Emma Watson in "The Bling Ring"

This journal, also on September 3 —

Thursday, September 3, 2015

Rings of August

Filed under: Uncategorized — m759 @ 7:20 AM

For the title, see posts from August 2007
tagged Gyges.

Related theological remarks:

Boolean  spaces (old)  vs. Galois  spaces (new)  in 
"The Quality Without a Name" . . . .

* Actually, Sarah, that would be "phonemes."

Friday, September 4, 2015

Space Program

Filed under: Uncategorized — Tags: — m759 @ 12:00 PM

Galois via Boole

(Courtesy of Intel)

Thursday, September 3, 2015

Rings of August

Filed under: Uncategorized — m759 @ 7:20 AM

For the title, see posts from August 2007 tagged Gyges.

Related theological remarks:

Boolean  spaces (old) vs. Galois  spaces  (new) in 
"The Quality Without a Name"
(a post from August 26, 2015) and the

Related literature:  A search for Borogoves in this journal will yield
remarks on the 1943 tale underlying the above film.

Wednesday, August 26, 2015

“The Quality Without a Name”

Filed under: Uncategorized — Tags: — m759 @ 8:00 AM

The title phrase, paraphrased without quotes in
the previous post, is from Christopher Alexander's book
The Timeless Way of Building  (Oxford University Press, 1979).

A quote from the publisher:

"Now, at last, there is a coherent theory
which describes in modern terms
an architecture as ancient as
human society itself."

Three paragraphs from the book (pp. xiii-xiv):

19. Within this process, every individual act
of building is a process in which space gets
differentiated. It is not a process of addition,
in which preformed parts are combined to
create a whole, but a process of unfolding,
like the evolution of an embryo, in which
the whole precedes the parts, and actualy
gives birth to then, by splitting.

20. The process of unfolding goes step by step,
one pattern at a time. Each step brings just one
pattern to life; and the intensity of the result
depends on the intensity of each one of these
individual steps.

21. From a sequence of these individual patterns,
whole buildings with the character of nature
will form themselves within your thoughts,
as easily as sentences.

Compare to, and contrast with, these illustrations of "Boolean space":

(See also similar illustrations from Berkeley and Purdue.)

Detail of the above image —

Note the "unfolding," as Christopher Alexander would have it.

These "Boolean" spaces of 1, 2, 4, 8, and 16 points
are also Galois  spaces.  See the diamond theorem —

Monday, December 29, 2014

Dodecahedron Model of PG(2,5)

Filed under: Uncategorized — Tags: , — m759 @ 2:28 PM

Recent posts tagged Sagan Dodecahedron 
mention an association between that Platonic
solid and the 5×5 grid. That grid, when extended
by the six points on a "line at infinity," yields
the 31 points of the finite projective plane of
order five.  

For details of how the dodecahedron serves as
a model of this projective plane (PG(2,5)), see
Polster's A Geometrical Picture Book , p. 120:

For associations of the grid with magic rather than
with Plato, see a search for 5×5 in this journal.

Thursday, December 18, 2014

Platonic Analogy

Filed under: Uncategorized — Tags: , , — m759 @ 2:23 PM

(Five by Five continued)

As the 3×3 grid underlies the order-3 finite projective plane,
whose 13 points may be modeled by
the 13 symmetry axes of the cube,
so the 5×5 grid underlies the order-5 finite projective plane,
whose 31 points may be modeled by
the 31 symmetry axes of the dodecahedron.

See posts tagged Galois-Plane Models.

Wednesday, December 3, 2014

Pyramid Dance

Filed under: Uncategorized — Tags: , — m759 @ 10:00 AM

Oslo artist Josefine Lyche has a new Instagram post,
this time on pyramids (the monumental kind).

My response —

Wikipedia's definition of a tetrahedron as a
"triangle-based pyramid"

and remarks from a Log24 post of August 14, 2013 :

Norway dance (as interpreted by an American)

IMAGE- 'The geometry of the dance' is that of a tetrahedron, according to Peter Pesic

I prefer a different, Norwegian, interpretation of "the dance of four."

Related material:
The clash between square and tetrahedral versions of PG(3,2).

See also some of Burkard Polster's triangle-based pyramids
and a 1983 triangle-based pyramid in a paper that Polster cites —

(Click image below to enlarge.)

Some other illustrations that are particularly relevant
for Lyche, an enthusiast of magic :

From On Art and Magic (May 5, 2011) —





The Fano plane block design



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


(Updated at about 7 PM ET on Dec. 3.)

Sunday, November 30, 2014

Two Physical Models of the Fano Plane

Filed under: Uncategorized — Tags: , — m759 @ 1:23 AM

The Regular Tetrahedron

The seven symmetry axes of the regular tetrahedron
are of two types: vertex-to-face and edge-to-edge.
Take these axes as the "points" of a Fano plane.
Each of the tetrahedron's six reflection planes contains 
two vertex-to-face axes and one edge-to-edge axis.
Take these six planes as six of the "lines" of a Fano
plane. Then the seventh line is the set of three 
edge-to-edge axes.

(The Fano tetrahedron is not original with me.
See Polster's 1998 A Geometrical Picture Book pp. 16-17.)

The Cube

There are three reflection planes parallel to faces
of the cube. Take the seven nonempty subsets of
the set of these three planes as the "points" of a
Fano plane. Define the Fano "lines" as those triples
of these seven subsets in which each member of
the triple is the symmetric-difference sum of the 
other two members.

(This is the eightfold cube  discussed at finitegeometry.org.)

Wednesday, November 26, 2014

A Tetrahedral Fano-Plane Model

Filed under: Uncategorized — Tags: — m759 @ 5:30 PM

Update of Nov. 30, 2014 —

It turns out that the following construction appears on
pages 16-17 of A Geometrical Picture Book , by 
Burkard Polster (Springer, 1998).

"Experienced mathematicians know that often the hardest
part of researching a problem is understanding precisely
what that problem says. They often follow Polya's wise
advice: 'If you can't solve a problem, then there is an
easier problem you can't solve: find it.'"

—John H. Conway, foreword to the 2004 Princeton
Science Library edition of How to Solve It , by G. Polya

For a similar but more difficult problem involving the
31-point projective plane, see yesterday's post
"Euclidean-Galois Interplay."

The above new [see update above] Fano-plane model was
suggested by some 1998 remarks of the late Stephen Eberhart.
See this morning's followup to "Euclidean-Galois Interplay" 
quoting Eberhart on the topic of how some of the smallest finite
projective planes relate to the symmetries of the five Platonic solids.

Update of Nov. 27, 2014: The seventh "line" of the tetrahedral
Fano model was redefined for greater symmetry.

Class Act

Filed under: Uncategorized — Tags: — m759 @ 7:18 AM

Update of Nov. 30, 2014 —

For further information on the geometry in
the remarks by Eberhart below, see
pp. 16-17 of A Geometrical Picture Book ,
by Burkard Polster (Springer, 1998). Polster
cites a different article by Lemay.

A search for background to the exercise in the previous post
yields a passage from the late Stephen Eberhart:

The first three primes p = 2, 3, and 5 therefore yield finite projective planes with 7, 13, and 31 points and lines, respectively. But these are just the numbers of symmetry axes of the five regular solids, as described in Plato's Timaeus : The tetrahedron has 4 pairs of face planes and comer points + 3 pairs of opposite edges, totalling 7 axes; the cube has 3 pairs of faces + 6 pairs of edges + 4 pairs of comers, totalling 13 axes (the octahedron simply interchanges the roles of faces and comers); and the pentagon dodecahedron has 6 pairs of faces + 15 pairs of edges + 10 pairs of comers, totalling 31 axes (the icosahedron again interchanging roles of faces and comers). This is such a suggestive result, one would expect to find it dealt with in most texts on related subjects; instead, while "well known to those who well know such things" (as Richard Guy likes to quip), it is scarcely to be found in the formal literature [9]. The reason for the common numbers, it turns out, is that the groups of symmetry motions of the regular solids are subgroups of the groups of collineations of the respective finite planes, a face axis being different from an edge axis of a regular solid but all points of a projective plane being alike, so the latter has more symmetries than the former.

[9] I am aware only of a series of in-house publications by Fernand Lemay of the Laboratoire de Didactique, Faculté des Sciences de I 'Éducation, Univ. Laval, Québec, in particular those collectively titled Genèse de la géométrie  I-X.

— Stephen Eberhart, Dept. of Mathematics,
California State University, Northridge, 
"Pythagorean and Platonic Bridges between
Geometry and Algebra," in BRIDGES: Mathematical
Connections in Art, Music, and Science 
, 1998,

Eberhart died of bone cancer in 2003. A memorial by his
high school class includes an Aug. 7, 2003, transcribed
letter from Eberhart to a classmate that ends…

… I earned MA’s in math (UW, Seattle) and history (UM, Missoula) where a math/history PhD program had been announced but canceled.  So 1984 to 2002 I taught math (esp. non-Euclidean geometry) at C.S.U. Northridge.  It’s been a rich life.  I’m grateful. 

See also another informative BRIDGES paper by Eberhart
on mathematics and the seven traditional liberal arts.

Tuesday, February 26, 2013


Filed under: Uncategorized — Tags: — m759 @ 4:00 PM

"I’ve had the privilege recently of being a Harvard University
professor, and there I learned one of the greatest of Harvard
jokes. A group of rabbis are on the road to Golgotha and 
Jesus is coming by under the cross. The young rabbi bursts
into tears and says, 'Oh, God, the pity of it!' The old rabbi says,
'What is the pity of it?' The young rabbi says, 'Master, Master,
what a teacher he was.'

'Didn’t publish!'

That cold tenure- joke at Harvard contains a deep truth.
Indeed, Jesus and Socrates did not publish."

— George Steiner, 2002 talk at York University

Related material

See also Steiner on Galois.

Les Miserables  at the Academy Awards

Tuesday, February 5, 2013


Filed under: Uncategorized — Tags: — m759 @ 1:06 PM

The previous post discussed some fundamentals of logic.

The name "Boole" in that post naturally suggests the
concept of Boolean algebra . This is not  the algebra
needed for Galois geometry . See below. 

IMAGE- Logic related to 'the arsenal of algebraic analysis tools for fields'

Some, like Dan Brown, prefer to interpret symbols using
religion, not logic. They may consult Diamond Mandorla,
as well as Blade and Chalice, in this journal.

See also yesterday's Universe of Discourse.

Tuesday, May 29, 2012

The Shining of May 29

Filed under: Uncategorized — Tags: — m759 @ 1:00 PM

(Continued from May 29, 2002)

May 29, 1832—


Évariste Galois, Lettre de Galois à M. Auguste Chevalier

Après cela, il se trouvera, j'espère, des gens qui trouveront leur profit à déchiffrer tout ce gâchis.

(Later there will be, I hope, some people who will find it to their advantage to decipher all this mess.)

Martin Gardner on the above letter—

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

The Last Recreations , by Martin Gardner, published by Springer in 2007, page 156.

Commentary from Dec. 2011 on Gardner's word "published" —

(Click to enlarge.)

IMAGE- Peter M. Neumann, 'Galois and His Groups,' EMS Newsletter, Dec. 2011

Sunday, October 30, 2011


Filed under: Uncategorized — Tags: — m759 @ 11:07 AM

Part I: Timothy Gowers on equivalence relations

Part II: Martin Gardner on normal subgroups

Part III: Evariste Galois on normal subgroups

"In all the history of science there is no completer example
 of the triumph of crass stupidity over untamable genius…."

— Eric Temple Bell, Men of Mathematics

See also an interesting definition and Weyl on Galois.

Update of 6:29 PM EDT Oct. 30, 2011—

For further details, see Herstein's phrase
"a tribute to the genius of Galois."

Wednesday, October 20, 2010

Celebration of Mind

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"Why the Celebration?"

"Martin Gardner passed away on May 22, 2010."

IMAGE-- Imaginary movie poster- 'The Galois Connection'- from stoneship.org

Imaginary movie poster from stoneship.org

Context— The Gardner Tribute.

Monday, September 27, 2010

The Social Network…

Filed under: Uncategorized — Tags: — m759 @ 9:29 AM

… In the Age of Citation

Social network analysis is focused on the patterning of the social
relationships that link social actors. Typically, network data take the
form of a square-actor by actor-binary adjacency matrix, where
each row and each column in the matrix represents a social actor. A
cell entry is 1 if and only if a pair of actors is linked by some social
relationship of interest (Freeman 1989).

— "Using Galois Lattices to Represent Network Data,"
by Linton C. Freeman and Douglas R. White,
Sociological Methodology,  Vol. 23, pp. 127–146 (1993)

From this paper's CiteSeer page


766  Social Network Analysis: Methods and Applications – WASSERMAN, FAUST – 1994
100 The act of creation – Koestler – 1964
 75 Visual Thinking – Arnheim – 1969

Visual Image of the Problem—

From a Google search today:


Related material—


"It is better to light one candle…"

"… the early favorite for best picture at the Oscars" — Roger Moore

Tuesday, June 22, 2010

Mathematics and Narrative, continued

Filed under: Uncategorized — Tags: — m759 @ 2:14 PM

"By groping toward the light we are made to realize
 how deep the darkness is around us."
  — Arthur Koestler, The Call Girls: A Tragi-Comedy,
      Random House, 1973, page 118

A 1973 review of Koestler's book—

"Koestler's 'call girls,' summoned here and there
 by this university and that foundation
 to perform their expert tricks, are the butts
 of some chilling satire."

Examples of Light—

Felix Christian Klein (1849- June 22, 1925) and Évariste Galois (1811-1832)

Klein on Galois

"… in France just about 1830 a new star of undreamt-of brilliance— or rather a meteor, soon to be extinguished— lighted the sky of pure mathematics: Évariste Galois."

— Felix Klein, Development of Mathematics in the 19th Century, translated by Michael Ackerman. Brookline, Mass., Math Sci Press, 1979. Page 80.

"… um 1830 herum in Frankreich als ein neuer Stern von ungeahntem Glanze am Himmel der reinen Mathematik aufleuchtet, um freilich, einem Meteor gleich, sehr bald zu verlöschen: Évariste Galois."

— Felix Klein, Vorlesungen Über Die Entwicklung Der Mathematick Im 19. Jahrhundert. New York, Chelsea Publishing Co., 1967. (Vol. I, originally published in Berlin in 1926.) Page 88.

Examples of Darkness—

Martin Gardner on Galois

"Galois was a thoroughly obnoxious nerd,
 suffering from what today would be called
 a 'personality disorder.'  His anger was
 paranoid and unremitting."

Gardner was reviewing a recent book about Galois by one Amir Alexander.

Alexander himself has written some reviews relevant to the Koestler book above.

See Alexander on—

The 2005 Mykonos conference on Mathematics and Narrative

A series of workshops at Banff International Research Station for Mathematical Innovation between 2003 and 2006. "The meetings brought together professional mathematicians (and other mathematical scientists) with authors, poets, artists, playwrights, and film-makers to work together on mathematically-inspired literary works."

Wednesday, June 2, 2010

The Harvard Style

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"I wonder if there's just been a critical mass
of creepy stories about Harvard
in the last couple of years…
A kind of piling on of
    nastiness and creepiness…"

Margaret Soltan, October 23, 2006

Harvard University Press
  on Facebook

Harvard University Press Harvard University Press
Martin Gardner on demythologizing mathematicians:
"Galois was a thoroughly obnoxious nerd"
  May 26 at 6:28 pm via Ping.f

The book that the late Gardner was reviewing
was published in April by Harvard University Press.

If Gardner's remark were true,
Galois would fit right in at Harvard. Example—
  The Harvard math department's pie-eating contest

Harvard Math Department Pi Day event

Rite of Passage

Filed under: Uncategorized — Tags: — m759 @ 9:00 AM


"On June 2, Évariste Galois was buried in a common grave of the Montparnasse cemetery whose exact location is unknown."

Évariste Galois, Lettre de Galois à M. Auguste Chevalier

Après cela, il y aura, j'espère, des gens qui trouveront leur profit à déchiffrer tout ce gâchis.

(Later there will be, I hope, some people who will find it to their advantage to decipher all this mess.)

Martin Gardner on the above letter—

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

The Last Recreations, by Martin Gardner, published by Springer in 2007, page 156.

Leonard E. Dickson

Image-- Leonard E. Dickson on the posthumous fundamental memoir of Galois

Tuesday, June 1, 2010

The Gardner Tribute

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"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.

The Gardner piece is now online.  It contains…

Gardner's tribute to Galois

"Galois was a thoroughly obnoxious nerd,
 suffering from what today would be called
 a 'personality disorder.'  His anger was
 paranoid and unremitting."

Friday, June 23, 2006

Friday June 23, 2006

Filed under: Uncategorized — m759 @ 2:56 PM

Binary Geometry

There is currently no area of mathematics named “binary geometry.” This is, therefore, a possible name for the geometry of sets with 2n elements (i.e., a sub-topic of Galois geometry and of algebraic geometry over finite fields– part of Weil’s “Rosetta stone” (pdf)).


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