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—


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.

Thursday, December 17, 2020

In Memoriam

Filed under: General — m759 @ 1:29 PM

Composer Harold Budd reportedly died at 84 on December 8
in Arcadia, California.

“The way I work is that
I focus entirely on a small thing
and try to milk that for all it’s worth,
to find everything in it
that makes musical sense,”
Budd explained in a 1997 interview….

Elegy for Budd at NPR

See related remarks in posts now tagged Quartet,
as well as posts now tagged Galois Window.

Wednesday, December 16, 2020

Kramer’s Cross

Filed under: General — Tags: — m759 @ 12:21 AM

See Crucial Kramer and Galois Window.

Tuesday, November 20, 2018


Filed under: General,Geometry — Tags: — m759 @ 12:21 PM


Musical accompaniment from Sunday morning

'The Eddington Song'

Update of Nov. 21 —

The reader may contrast the above Squarespace.com logo
(a rather serpentine version of the acronym SS) with a simpler logo
for a square space (the Galois window ):

Monday, June 26, 2017

Four Dots

Analogies — “A : B  ::  C : D”  may be read  “A is to B  as  C is to D.”

Gian-Carlo Rota on Heidegger…

“… The universal as  is given various names in Heidegger’s writings….

The discovery of the universal as  is Heidegger’s contribution to philosophy….

The universal ‘as‘ is the surgence of sense in Man, the shepherd of Being.

The disclosure of the primordial as  is the end of a search that began with Plato….
This search comes to its conclusion with Heidegger.”

— “Three Senses of ‘A is B’ in Heideggger,” Ch. 17 in Indiscrete Thoughts
See also Four Dots in this journal.

Some context:  McLuhan + Analogy.

Saturday, April 29, 2017

For the Church of Synchronology*

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

A book cover from Amazon.com —

See also this journal on the above date, September 27, 2016 —

Chomsky and Levi-Strauss in China,
Or: Philosophy for Jews

Some other remarks related to the figure on the book cover —

Field Theology and Galois Window.

* See Synchronology in this journal.

Saturday, February 18, 2012


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

Pentagram design agency on the new Windows 8 logo

"… the logo re-imagines the familiar four-color symbol
as a modern geometric shape"—


Sam Moreau, Principal Director of User Experience for Windows,

On Redesigning the Windows Logo

"To see what is in front of one's nose
needs a constant struggle."
George Orwell

That is the feeling we had when Paula Scher
(from the renowned Pentagram design agency)
showed us her sketches for the new Windows logo.

Related material:


Tuesday, January 24, 2012

The Infinity Point

Filed under: General,Geometry — m759 @ 2:20 PM

From Labyrinth of the Line (March 2, 2011)—

"… construct the Golay code by taking the 24 points
to be the points of the projective line F23 ∪ {}…."

— Robert A. Wilson

A simpler projective line— a Galois geometry
model of the line F2 ∪ {}—

Image- The Three-Point Line: A Finite Projective Geometry

Here we may consider  to be modeled*
by the third square above— the Galois window .

* Update of about 1 AM Jan. 25, 2012—
  This infinity-modeling is of course a poetic conceit,
  not to be taken too seriously. For a serious 
  discussion of points at infinity and finite fields,
  see (for instance) Daniel Bump's "The Group GL(2)."

Friday, November 25, 2011

Window Actions

Filed under: General,Geometry — m759 @ 4:25 PM

A post by Gowers today on group actions suggests a review.

See WindowWindow Continued,  and The Galois Window.

Tuesday, July 6, 2010

What “As” Is

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

or:  Combinatorics (Rota) as Philosophy (Heidegger) as Geometry (Me)

“Dasein’s full existential structure is constituted by
the ‘as-structure’ or ‘well-joined structure’ of the rift-design*…”

— Gary Williams, post of January 22, 2010


Gian-Carlo Rota on Heidegger…

“… The universal as  is given various names in Heidegger’s writings….

The discovery of the universal as  is Heidegger’s contribution to philosophy….

The universal ‘as‘ is the surgence of sense in Man, the shepherd of Being.

The disclosure of the primordial as  is the end of a search that began with Plato….
This search comes to its conclusion with Heidegger.”

— “Three Senses of ‘A is B’ in Heideggger,” Ch. 17 in Indiscrete Thoughts

… and projective points as separating rifts

Image-- The Three-Point Line: A Finite Projective Space

    Click image for details.

* rift-design— Definition by Deborah Levitt

Rift.  The stroke or rending by which a world worlds, opening both the ‘old’ world and the self-concealing earth to the possibility of a new world. As well as being this stroke, the rift is the site— the furrow or crack— created by the stroke. As the ‘rift design‘ it is the particular characteristics or traits of this furrow.”

— “Heidegger and the Theater of Truth,” in Tympanum: A Journal of Comparative Literary Studies, Vol. 1, 1998

Window, continued

Filed under: General,Geometry — Tags: , , — m759 @ 10:31 AM
“Simplicity, simplicity, simplicity!  I say, let your affairs
be as two or three,
and not a hundred or a thousand;
instead of a million count half a dozen,
and keep your accounts on your thumb-nail.”
— Henry David Thoreau, Walden
This quotation is the epigraph to Section 1.1 of
Alexandre V. Borovik’s
Mathematics Under the Microscope:

Notes on Cognitive Aspects of Mathematical Practice
(American Mathematical Society, Jan. 15, 2010, 317 pages).

From Peter J. Cameron’s review notes for
his new course in group theory


From Log24 on June 24

Geometry Simplified

Image-- The Four-Point Plane: A Finite Affine Space
(an affine  space with subsquares as points
and sets  of subsquares as hyperplanes)

Image-- The Three-Point Line: A Finite Projective Space
(a projective  space with, as points, sets
of line segments that separate subsquares)


Show that the above geometry is a model
for the algebra discussed by Cameron.

Monday, July 5, 2010


Filed under: General — Tags: , , — m759 @ 9:00 AM

“Examples are the stained-glass
windows of knowledge.” — Nabokov

Image-- Example of group actions on the set Omega of three partitions of a 4-set into two 2-sets

Related material:

Thomas Wolfe and the
Kernel of Eternity

Thursday, June 24, 2010

Midsummer Noon

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

Geometry Simplified

Image-- The Three-Point Line: A Finite Projective Space
(a projective space)

The above finite projective space
is the simplest nontrivial example
of a Galois geometry (i.e., a finite
geometry with coordinates in a
finite (that is, Galois) field.)

The vertical (Euclidean) line represents a
(Galois) point, as does the horizontal line
and also the vertical-and-horizontal
cross that represents the first two points’
binary sum (i.e., symmetric difference,
if the lines are regarded as sets).

Homogeneous coordinates for the
points of this line —

(1,0), (0,1), (1,1).

Here 0 and 1 stand for the elements
of the two-element Galois field GF(2).

The 3-point line is the projective space
corresponding to the affine space
(a plane, not a line) with four points —

(an affine space)

The (Galois) points of this affine plane are
not the single and combined (Euclidean)
line segments that play the role of
points in the 3-point projective line,
but rather the four subsquares
that the line segments separate.

For further details, see Galois Geometry.

There are, of course, also the trivial
two-point affine space and the corresponding
trivial one-point projective space —


Here again, the points of the affine space are
represented by squares, and the point of the
projective space is represented by a line segment
separating the affine-space squares.

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