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.

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 ):

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.

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