Wednesday, November 29, 2017


Filed under: General,Geometry — m759 @ 11:25 PM

See also Inscape in this journal and, for a related Chapel Hill thesis,
the post Kummer and Dirac.

Wednesday, May 25, 2016

Kummer and Dirac

From "Projective Geometry and PT-Symmetric Dirac Hamiltonian,"
Y. Jack Ng  and H. van Dam, 
Physics Letters B , Volume 673, Issue 3,
23 March 2009, Pages 237–239

(http://arxiv.org/abs/0901.2579v2, last revised Feb. 20, 2009)

" Studies of spin-½ theories in the framework of projective geometry
have been undertaken before. See, e.g., Ref. [4]. 1 "

1 These papers are rather mathematical and technical.
The authors of the first two papers discuss the Dirac equation
in terms of the Plucker-Klein correspondence between lines of
a three-dimensional projective space and points of a quadric
in a five-dimensional projective space. The last paper shows
that the Dirac equation bears a certain relation to Kummer’s
surface, viz., the structure of the Dirac ring of matrices is 
related to that of Kummer’s 166 configuration . . . ."


O. Veblen
Proc. Natl. Acad. Sci. USA , 19 (1933), p. 503
Full Text via CrossRef

E.M. Bruins
Proc. Nederl. Akad. Wetensch. , 52 (1949), p. 1135

F.C. Taylor Jr., Master thesis, University of North Carolina
at Chapel Hill (1968), unpublished

A remark of my own on the structure of Kummer’s 166 configuration . . . .

See that structure in this  journal, for instance —

See as well yesterday morning's post.

Wednesday, February 11, 2015

Dead Reckoning

Filed under: General — Tags: — m759 @ 5:28 PM

Continued from yesterday evening

IMAGE- Bogart in 'Casablanca' with chessboard

Today's mathematical birthday — 

Claude Chevalley, 11 Feb. 1909 – 28 June 1984.

From MacTutor —

Chevalley's daughter, Catherine Chevalley, wrote about
her father in "Claude Chevalley described by his daughter"

For him it was important to see questions as a whole, to see the necessity of a proof, its global implications. As to rigour, all the members of Bourbaki cared about it: the Bourbaki movement was started essentially because rigour was lacking among French mathematicians, by comparison with the Germans, that is the Hilbertians. Rigour consisted in getting rid of an accretion of superfluous details. Conversely, lack of rigour gave my father an impression of a proof where one was walking in mud, where one had to pick up some sort of filth in order to get ahead. Once that filth was taken away, one could get at the mathematical object, a sort of crystallized body whose essence is its structure. When that structure had been constructed, he would say it was an object which interested him, something to look at, to admire, perhaps to turn around, but certainly not to transform. For him, rigour in mathematics consisted in making a new object which could thereafter remain unchanged.

The way my father worked, it seems that this was what counted most, this production of an object which then became inert— dead, really. It was no longer to be altered or transformed. Not that there was any negative connotation to this. But I must add that my father was probably the only member of Bourbaki who thought of mathematics as a way to put objects to death for aesthetic reasons.

Recent scholarly news suggests a search for Chapel Hill
in this journal. That search leads to Transformative Hermeneutics.
Those who, like Professor Eucalyptus of Wallace Stevens's
New Haven, seek God "in the object itself" may contemplate
yesterday's afternoon post on Eightfold Design in light of the
Transformative post and of yesterday's New Haven remarks and
Chapel Hill events.

Thursday, April 2, 2009

Thursday April 2, 2009

Filed under: General,Geometry — Tags: — m759 @ 12:25 PM


In memory of
physics historian
Martin J. Klein,
(June 25, 1924-
March 28, 2009)

"… in physics itself, there was what appeared, briefly, to be an ending, which then very quickly gave way to a new beginning: The quest for the ultimate building-blocks of the universe had been taken down to the molecular level in nineteenth-century kinetic theory… and finally to the nuclear level in the second and third decades of the twentieth century. For a moment in the 1920s the quest appeared to have ended…. However… this paradise turned out to be, if not exactly a fool's paradise, then perhaps an Eden lost."

No Truth Except in the Details: Essays in Honor of Martin J. Klein, introduction by A.J. Kox and Daniel Siegel, June 25, 1994

New York Times obituary dated April 1, 2009:

"Martin J. Klein, a historian of modern physics…. died Saturday, [March 28, 2009] in Chapel Hill, N.C. He was 84 and lived in Chapel Hill."

Klein edited, among other things, Paul Ehrenfest: Collected Scientific Papers (publ. by North-Holland, Amsterdam, 1959).

"It seems, as one becomes older,
 That the past has another pattern,
 and ceases to be a mere sequence…."

— T. S. Eliot, Four Quartets

A Walsh function and a corresponding finite-geometry hyperplane

"Note that at first, you can see
 the 'arrow of time.'
 After a long period, however,
 the direction of time
 is no longer evident."

— "The Ehrenfest Chains,"
     by Kyle Siegrist, ex. 16

Related material:

"Almost every famous chess game
is a well-wrought urn
in Cleanth Brooks’ sense."

— John Holbo,
Now We See
Wherein Lies the Pleasure

"The entire sequence of moves in these… chapters reminds one– or should remind one– of a certain type of chess problem where the point is not merely the finding of a mate in so many moves, but what is termed 'retrograde analysis'…."

— Vladimir Nabokov, foreword to The Defense

Sunday, June 8, 2003

Sunday June 8, 2003

Filed under: General — m759 @ 3:04 AM

Of Time and the River

Today is the feast day of Saint Gerard Manley Hopkins, “immortal diamond.”

“At that instant he saw, in one blaze of light, an image of unutterable conviction, the reason why the artist works and lives and has his being–the reward he seeks–the only reward he really cares about, without which there is nothing. It is to snare the spirits of mankind in nets of magic, to make his life prevail through his creation, to wreak the vision of his life, the rude and painful substance of his own experience, into the congruence of blazing and enchanted images that are themselves the core of life, the essential pattern whence all other things proceed, the kernel of eternity.”

Thomas Wolfe, Of Time and the River

Thomas Wolfe

“entered the university at Chapel Hill at fifteen ‘an awkward, unhappy misfit.’ By the time he graduated, he was editor of the college newspaper….”

Jeff MacNelly, who died on this date in the Year of Our Lord 2000,

“in 1977 started drawing the comic strip ‘Shoe‘…. The strip was named in honor of the legendary Jim Shumaker, for whom MacNelly worked at the Chapel Hill Weekly.” 

From my Monday, June 2, 2003 entry:

Two quotations from “The Diamond Project“:

“We all know that something is eternal,” the Stage Manager says. “And it ain’t houses and it ain’t names, and it ain’t earth, and it ain’t even stars—everybody knows in their bones that something is eternal, and that something has to do with human beings.”
— John Lahr, review of “Our Town 

“Set me as a seal upon your heart, as a seal upon your arm; for love is strong as death, passion fierce as the grave.  Its flashes are flashes of fire, a raging flame.”
Song of Solomon

Here are some other thoughts from the same date, but a different time, fictional time, Faulkner time:

June Second, 1910

Where the shadow of the bridge fell I could see down for a long way, but not as far as the bottom. When you leave a leaf in water a long time after a while the tissue will be gone and the delicate fibers waving slow as the motion of sleep. They dont touch one another, no matter how knotted up they once were, no matter how close they lay once to the bones. And maybe when He says Rise the eyes will come floating up too, out of the deep quiet and the sleep, to look on glory.

— William Faulkner, The Sound and the Fury

The concluding link from my June 2, 2003, entry furnishes a clue to the timelessness of Quentin Compson‘s thoughts above:

Glory… Song of Songs 8. 7-8

From the King James Bible‘s rendition of the Song of Songs:

8:7  Many waters cannot quench love, neither can the floods drown it: if a man would give all the substance of his house for love, it would utterly be contemned.
8:8  We have a little sister, and she hath no breasts: what shall we do for our sister in the day when she shall be spoken for?

For Quentin Compson’s thoughts on his little sister Caddy, consult the online hypertext edition of

Friday, February 7, 2003

Friday February 7, 2003

Filed under: General — m759 @ 5:00 AM

Saint’s Day

Today is the birthday of Thomas More, an alleged Catholic saint, and the date of death of Dale Evans, Protestant saint.

As yesterday’s note implies, we should not look to saints, or indeed to religion generally, for truth.  Those who mistake the stories of the Church or the Bible for truth have done, and continue to do, a great deal of harm in this world.  But those who seek, not truth, but values, in stories may sometimes be among the blessed — as Dale Evans certainly was, and as Thomas More, after centuries of atoning for his sins in Purgatory, may, by this time, be.

Let us pray that young Catholics (like the girl pictured at St. Thomas More Catholic School in Chapel Hill, N. C.) learn the proper uses of stories, as well as of more respectable intellectual disciplines.

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