Log24

Wednesday, August 1, 2018

Publish or …

Filed under: General — m759 @ 9:00 PM
 

From The New York Times  online on July 29 —

" Ms. Appelbaum’s favorite authors, she said in an interview with The Internet Writing Journal in 1998, were too many to count, but they included George Eliot, Anthony Trollope, Anne Tyler and Julian Barnes.

'I love to see writers expand our range of understanding, experience, knowledge, even happiness,' she said in that interview. 'Publishing has always struck me as a way to change the world.' "

A version of this article appears in print on , on Page B6 of the New York edition with the headline: Judith Appelbaum, Guru On Publishing, Dies at 78.

See a review of the new Anne Tyler novel Clock Dance
in today's  online New York Times .

For a more abstract dance, see Ballet Blanc .

"A blank underlies the trials of device." — Wallace Stevens

Tuesday, July 31, 2018

Lexicon

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

"A blank underlies the trials of device." — Wallace Stevens

IMAGE- The ninefold square .

Wednesday, June 27, 2018

Stage

Filed under: General,Geometry — Tags: — m759 @ 9:27 AM

See Ballet Blanc 
and Still Point.

Tuesday, February 13, 2018

A Titan of the Field

Filed under: General,Geometry — Tags: — m759 @ 9:45 AM
 

On the late Cambridge astronomer Donald Lynden-Bell —

"As an academic at a time when students listened and lecturers lectured, he had the disconcerting habit of instead picking on a random undergraduate and testing them on the topic. One former student, now a professor, remembered how he would 'ask on-the-spot questions while announcing that his daughter would solve these problems at the breakfast table'.

He got away with it because he was genuinely interested in the work of his colleagues and students, and came to be viewed with great affection by them. He also got away with it because he was well established as a titan of the field."

The London Times  on Feb. 8, 2018, at 5 PM (British time)

Related material —

Two Log24 posts from yesteday, Art Wars and The Void.

See as well the field GF(9)

http://www.log24.com/log/pix12/120220-CoxeterFig10.jpg

and the 3×3 grid as a symbol of Apollo
    (an Olympian rather than a Titan) —

 .

Monday, October 9, 2017

Still Point for a Dance

Filed under: G-Notes,General,Geometry — m759 @ 12:00 PM

"At the still point of the turning world. Neither flesh nor fleshless;
Neither from nor towards; at the still point, there the dance is,
But neither arrest nor movement. And do not call it fixity,
Where past and future are gathered. Neither movement from nor towards,
Neither ascent nor decline. Except for the point, the still point,
There would be no dance, and there is only the dance."

— T. S. Eliot, Four Quartets

See also a recurrent image
from this journal —

IMAGE- The ninefold square .

Tuesday, March 21, 2017

Res Ipsa

Filed under: General,Geometry — Tags: — m759 @ 7:00 PM

From The Poetic Quotidian, a journal of quotations—

See also, in this journal, New Haven + Grid.

The Ninefold Square

Sunday, March 19, 2017

Delos Incorporated* Sunday School

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

The 3x3 square

Click image for a search.

* Parent company of Westworld.
  See also Delos  in this journal.

Friday, December 9, 2016

Sacred Space (continued)

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

See Plan 9 in this journal.

 The 3x3 square 

Monday, June 20, 2016

Sacred Space

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

For further background to this morning's post Plan 9 Continues,
see posts tagged Sacred Space

The 3x3 square .

Sunday, November 29, 2015

There the Dance Is

Filed under: General,Geometry — m759 @ 10:00 PM

In memory of ballet designer
Yolanda Sonnabend, who
reportedly died at 80 on Nov. 9,
see posts on Apollo, Ballet Blanc,
maps of New Haven, etc., etc., etc.

Wednesday, September 9, 2015

Plan 9 Continues

Filed under: General,Geometry — m759 @ 2:56 PM

See posts tagged Clooney Omega in this journal.

Saturday, June 20, 2015

Conceptual Art for Basel

Filed under: General,Geometry — Tags: — m759 @ 7:59 PM

The previous post's link to The Lindbergh Manifesto
and Thursday's post on Basel-born artist Wolf Barth 
suggest the following —

See as well a June 14 New York Times
piece on Art Basel.

The logo of the University of Basel 

suggests a review of The Holy Field —

 .

Tuesday, May 12, 2015

Writing Well*

Filed under: General,Geometry — Tags: — m759 @ 11:00 PM

See Stevens + New Haven.

* The above figure may be viewed as
   the Chinese "Holy Field" or as the
   Chinese character for "Well"
   inscribed in a square.

Friday, February 14, 2014

Epiphany

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

“… the object sets up a kind of 
 frame or space or field 
 within which there can be epiphany.”

Charles Taylor

A frame or space or field —

IMAGE- The ninefold square

Related material —

Star Wars (January 11, 2014),

The Lyche Gate Asterisk , from 10:31 AM ET on May 22, 2010,
the date of Martin Gardner's death —

Image-- The Case of the Lyche Gate Asterisk

— and the March 2014 issue of the
Notices of the American Mathematical Society  —

See as well Epiphany 2014 (Jan. 6) in this journal and the
March Notices  on the Shaw prize —

"Established under the auspices of Run Run Shaw
in November 2002, the prize is managed and
administered by the Shaw Prize Foundation
based in Hong Kong." 

Monday, August 26, 2013

Dark Side Tales

Filed under: General — m759 @ 3:00 PM

"Got to keep the loonies on the path."

Lyrics to Dark Side of the Moon

For those who, like Tom Stoppard, prefer the dark side—

NEW ANGLE:
He runs, panting, until he ends up
in front of a tall, brilliantly lit office building.
As he approaches, the lights in the building
are going off floor by floor.

INT. OFFICE BUILDING – NIGHT
He rushes into
the lobby, running for the elevator.

NIGHT WATCHMAN
Burning the midnight oil, Mr. Smith?
You forgot to sign in.

Bateman wheels around and shoots him.
He runs toward the revolving doors.
As he swings around in the doors, he notices
a JANITOR who has witnessed the shooting.
He revolves back into the lobby and shoots the janitor.

NEW ANGLE:
He runs out of the building
and across the street to an identical office building,
the one that houses Pierce & Pierce.

INT. PIERCE & PIERCE LOBBY – NIGHT
Bateman nods at the Pierce & Pierce NIGHT WATCHMAN
and signs in. He breathes a sigh of relief as
​the elevator doors close behind him.

— AMERICAN PSYCHO
by Mary Harron and Guinevere Turner
(Based on the novel by Bret Easton Ellis, 
Fourth Draft, November 1998)

Not quite so dark—

"And then one day you find ten years have got behind you."

— Lyrics to Dark Side of the Moon

This journal ten years ago, on August 25, 2003

         … We seek

The poem of pure reality, untouched
By trope or deviation, straight to the word,
Straight to the transfixing object, to the object

At the exactest point at which it is itself,
Transfixing by being purely what it is,
A view of New Haven, say, through the certain eye,

The eye made clear of uncertainty, with the sight
Of simple seeing, without reflection. We seek
Nothing beyond reality. Within it,

Everything, the spirit's alchemicana
Included, the spirit that goes roundabout
And through included, not merely the visible,

The solid, but the movable, the moment,
The coming on of feasts and the habits of saints,
The pattern of the heavens and high, night air.

— Wallace Stevens, "An Ordinary Evening
     in New Haven," Canto IX
    (Collected Poems , pp. 471-472)


"A view of New Haven, say…." —

The image “http://www.log24.com/theory/images/grid3x3.gif” cannot be displayed, because it contains errors.

"This is the garden of Apollo,
the field of Reason…."
John Outram, architect 


A similar version of this Apollonian image —

  Detail:

Related material for the loonies:

"the spirit's alchemicana."

Sunday, May 5, 2013

Night of Lunacy*

Filed under: General — m759 @ 9:00 PM

Structure vs. Character continued

   IMAGE- The 3x3 square   

Structure

IMAGE- Chinese character for 'well' and I Ching Hexagram 48, 'The Well'


Character

Related vocabulary:

Nick Tosches on the German word "Quell "

and Heidegger on Hölderlin.

* The title is from Heidegger.

Wednesday, April 10, 2013

Caution: Slow Art

Filed under: General,Geometry — Tags: , — m759 @ 9:00 PM

"Of course, DeLillo being DeLillo,
it’s the deeper implications of the piece —
what it reveals about the nature of
film, perception and time — that detain him."

— Geoff Dyer, review of Point Omega

Related material:

A phrase of critic Robert Hughes,
"slow art," in this journal.

A search for that phrase yields the following
figure from a post on DeLillo of Oct. 12, 2011:

The 3x3 square

The above 3×3 grid is embedded in a 
somewhat more sophisticated example
of conceptual art from April 1, 2013:

IMAGE- A Galois-geometry key to Desargues' theorem

Update of April 12, 2013

The above key uses labels from the frontispiece
to Baker's 1922 Principles of Geometry, Vol. I ,
that shows a three-triangle version of Desargues's theorem.

A different figure, from a site at National Tsing Hua University,
shows the three triangles of Baker's figure more clearly:

IMAGE- Desargues' theorem with three triangles (the large Desargues configuration) and Galois-geometry version

Monday, November 5, 2012

Sitting Specially

Filed under: General,Geometry — Tags: — m759 @ 5:01 AM

Some webpages at finitegeometry.org discuss
group actions on Sylvester’s duads and synthemes.

Those pages are based on the square model of
PG(3,2) described in the 1980’s by Steven H. Cullinane.

A rival tetrahedral model of PG(3,2) was described
in the 1990’s by Burkard Polster.

Polster’s tetrahedral model appears, notably, in
a Mathematics Magazine  article from April 2009—

IMAGE- Figure from article by Alex Fink and Richard Guy on how the symmetric group of degree 5 'sits specially' in the symmetric group of degree 6

Click for a pdf of the article.

Related material:

The Religion of Cubism” (May 9, 2003) and “Art and Lies
(Nov. 16, 2008).

This  post was suggested by following the link in yesterday’s
Sunday School post  to High White Noon, and the link from
there to A Study in Art Education, which mentions the date of
Rudolf Arnheim‘s death, June 9, 2007. This journal
on that date

Cryptology

IMAGE- The ninefold square

— The Delphic Corporation

The Fink-Guy article was announced in a Mathematical
Association of America newsletter dated April 15, 2009.

Those who prefer narrative to mathematics may consult
a Log24 post from a few days earlier, “Where Entertainment is God”
(April 12, 2009), and, for some backstory, The Judas Seat
(February 16, 2007).

Friday, October 26, 2012

High White Noon

Filed under: General — m759 @ 12:00 PM

(Continued)

Today's 11:01 AM post discussed time concepts
in Eliot's Four Quartets.

For the temporally challenged, here is
a somewhat simpler conceptual framework—

Three Trios

From a post of Columbus Day
(i.e., Oct. 12), 2011, titled
    "High White Noon" (after DeLillo) —

The 3x3 square

A Study in Art Education

Sunday, January 1, 2012

Sunday Shul

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

"… myths are stories, and like all narratives
they unravel through time, whereas grids
are not only spatial to start with,
they are visual structures that explicitly reject
a narrative or sequential reading of any kind."

— Rosalind Krauss in "Grids,"
October  (Summer 1979), 9: 50-64.

Counterexample—

The image “http://www.log24.com/theory/images/grid3x3.gif” cannot be displayed, because it contains errors.

The Ninefold Square

See Coxeter and the Aleph and Ayn Sof

Mathematics and Narrative, Illustrated
http://www.log24.com/log/pix11/110107-The1950Aleph-Sm.jpg

Mathematics
http://www.log24.com/log/pix11/110107-ScriptAlephSm.jpg
Narrative

Wednesday, October 12, 2011

High White Noon

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

Grid from a post linked to in yesterday's 24 Hour DeLillo

The 3x3 square

A Study in Art Education

For an example of this grid as slow art , consider the following—

"One can show that the binary tetrahedral group
is isomorphic to the special linear group SL(2,3)—
the group of all 2×2 matrices over the finite field F3
with unit determinant." —Wikipedia

As John Baez has noted, these two groups have the same structure as the geometric 24-cell.

For the connection of the grid to the groups and the 24-cell, see Visualizing GL(2,p).

Related material—

The 3×3 grid has been called a symbol of Apollo (Greek god of reason and of the sun).

"This is where we sat through his hushed hour,
a torchlit sky, the closeness of hills barely visible
at high white noon." — Don DeLillo, Point Omega

Monday, May 23, 2011

The Stoner Series

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

A reader comments on yesterday afternoon's New York Times
"The Stone" column by Justin E.H. Smith—

"I did indeed appreciate Mr. Smith’s essay.
And I’m curious as to what future contributions of his,
to the Stoner series, that we can look forward to."

From August 24, 2010

Der Einsatz

Motto of Plato's Academy: 'Let no one ignorant of geometry enter'

The Ninefold Square (a 3x3 grid)

Nichts ist wie es scheint.

See also the film
"23— Nichts ist so wie es scheint."

Happy day 23 of Mental Health Month.

Saturday, January 8, 2011

True Grid (continued)

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

"Rosetta Stone" as a Metaphor
  in Mathematical Narratives

For some backgound, see Mathematics and Narrative from 2005.

Yesterday's posts on mathematics and narrative discussed some properties
of the 3×3 grid (also known as the ninefold square ).

For some other properties, see (at the college-undergraduate, or MAA, level)–
Ezra Brown, 2001, "Magic Squares, Finite Planes, and Points of Inflection on Elliptic Curves."

His conclusion:

When you are done, you will be able to arrange the points into [a] 3×3 magic square,
which resembles the one in the book [5] I was reading on elliptic curves….

This result ties together threads from finite geometry, recreational mathematics,
combinatorics, calculus, algebra, and number theory. Quite a feat!

5. Viktor Prasolov and Yuri Solvyev, Elliptic Functions and Elliptic Integrals ,
    American Mathematical Society, 1997.

Brown fails to give an important clue to the historical background of this topic —
the word Hessian . (See, however, this word in the book on elliptic functions that he cites.)

Investigation of this word yields a related essay at the graduate-student, or AMS, level–
Igor Dolgachev and Michela Artebani, 2009, "The Hesse Pencil of Plane Cubic Curves ."

From the Dolgachev-Artebani introduction–

In this paper we discuss some old and new results about the widely known Hesse
configuration
  of 9 points and 12 lines in the projective plane P2(k ): each point lies
on 4 lines and each line contains 3 points, giving an abstract configuration (123, 94).

PlanetMath.org on the Hesse configuration

http://www.log24.com/log/pix11/110108-PlanetMath.jpg

A picture of the Hesse configuration–

The image “http://www.log24.com/log/pix05B/grid3x3med.bmp” cannot be displayed, because it contains errors.

(See Visualizing GL(2,p), a note from 1985).

Related notes from this journal —

From last November —

Saturday, November 13, 2010

Story

m759 @ 10:12 PM

From the December 2010 American Mathematical Society Notices

http://www.log24.com/log/pix10B/101113-Ono.gif

Related material from this  journal—

Mathematics and Narrative and

Consolation Prize (August 19, 2010)

From 2006 —

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

Also from 2006 —

Sunday November 26, 2006

 

m759 @ 7:26 AM

Rosalind Krauss
in "Grids," 1979:

"If we open any tract– Plastic Art and Pure Plastic Art  or The Non-Objective World , for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter.  They are talking about Being or Mind or Spirit.  From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete.

Or, to take a more up-to-date example…."

"He was looking at the nine engravings and at the circle,
checking strange correspondences between them."
The Club Dumas ,1993

"And it's whispered that soon if we all call the tune
Then the piper will lead us to reason."
Robert Plant ,1971

The nine engravings of The Club Dumas
(filmed as "The Ninth Gate") are perhaps more
an example of the concrete than of the universal.

An example of the universal*– or, according to Krauss,
a "staircase" to the universal– is the ninefold square:

The image “http://www.log24.com/theory/images/grid3x3.gif” cannot be displayed, because it contains errors.

"This is the garden of Apollo, the field of Reason…."
John Outram, architect    

For more on the field of reason, see
Log24, Oct. 9, 2006.

A reasonable set of "strange correspondences"
in the garden of Apollo has been provided by
Ezra Brown in a mathematical essay (pdf).

Unreason is, of course, more popular.

* The ninefold square is perhaps a "concrete universal" in the sense of Hegel:

"Two determinations found in all philosophy are the concretion of the Idea and the presence of the spirit in the same; my content must at the same time be something concrete, present. This concrete was termed Reason, and for it the more noble of those men contended with the greatest enthusiasm and warmth. Thought was raised like a standard among the nations, liberty of conviction and of conscience in me. They said to mankind, 'In this sign thou shalt conquer,' for they had before their eyes what had been done in the name of the cross alone, what had been made a matter of faith and law and religion– they saw how the sign of the cross had been degraded."

– Hegel, Lectures on the History of Philosophy ,
   "Idea of a Concrete Universal Unity"

"For every kind of vampire,
there is a kind of cross."
– Thomas Pynchon   

And from last October —

Friday, October 8, 2010

 

m759 @ 12:00 PM
 

Starting Out in the Evening
… and Finishing Up at Noon

This post was suggested by last evening's post on mathematics and narrative and by Michiko Kakutani on Vargas Llosa in this morning's New York Times .

http://www.log24.com/log/pix10B/101008-StartingOut.jpg

 

Above: Frank Langella in
"Starting Out in the Evening"

Right: Johnny Depp in
"The Ninth Gate"

http://www.log24.com/log/pix10B/101008-NinthGate.jpg

"One must proceed cautiously, for this road— of truth and falsehood in the realm of fiction— is riddled with traps and any enticing oasis is usually a mirage."

– "Is Fiction the Art of Lying?"* by Mario Vargas Llosa,
    New York Times  essay of October 7, 1984

* The Web version's title has a misprint—
   "living" instead of "lying."

"You've got to pick up every stitch…"

Monday, December 13, 2010

Plan 9 Revisited

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

Leading today's New York Times  obituaries —

http://www.log24.com/log/pix10B/101213-NYTobits.jpg

— is that of Nassos Daphnis, a painter of geometric abstractions
who in 1995 had an exhibition at a Leo Castelli gallery
titled "Energies in Outer Space." (See pictures here.)

Daphnis died, according to the Times, on November 23.
See Art Object, a post in this journal on that date—

There is more than one way
to look at a cube.

http://www.log24.com/log/pix10B/101123-plain_cube_200x227.gif

Some context— this morning's previous post (Apollo's 13,
on the geometry of the 3×3×3 cube), yesterday's noon post
featuring the 3×3 square grid (said to be a symbol of Apollo),

The 3x3 square

and, for connoisseurs of the Ed Wood school of cinematic art,
a search in this journal for the phrase "Plan 9."

You can't make this stuff up.

Sunday, December 12, 2010

Brightness at Noon continued…

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

A picture one might view as
related to the novel An Object of Beauty
and the film "The Object of Beauty" —

The 3x3 square

Click for some background.

"If it's a seamless whole you want,
 pray to Apollo." — Margaret Atwood

Tuesday, August 24, 2010

Magnificent Load

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

From Doonesbury today—

"What a magnificent load"

From this journal (September 20, 2009)—

scheinen
German verb:

  1. to shine; to gleam     
  2. to seem; to appear….

Quine, Pursuit of Truth,
Harvard U. Press, 1990, epigraphs:

http://www.log24.com/log/pix09A/090920-QuineEpigraph.jpg

Google search:

http://www.log24.com/log/pix09A/090920-SozeinChi.jpg

Der Einsatz

Motto of Plato's Academy: 'Let no one ignorant of geometry enter'

The 3x3 grid

Nichts ist wie es scheint.

See also the film
"23— Nichts ist so wie es scheint."

Thursday, August 19, 2010

Consolation Prize

Filed under: General,Geometry — Tags: — m759 @ 9:04 AM

For Kathrin Bringmann, who has been mentioned as a possible candidate for a Fields Medal.

The four Fields medal winners were announced today at the International Congress of Mathematicians in Hyderabad, India. Bringmann was not among them.

Bringmann was, however, the winner of the 2009 SASTRA Ramanujan Prize

See The Hindu  of September 30, 2009 and this journal on that date

Motto of Plato's Academy: 'Let no one ignorant of geometry enter'

The 3x3 grid

A Symbol of Apollo

For more about Bringmann's work, see an article on what has been called Ramanujan's "final problem."

For another problem with a claim to this title, see "Mathematician Untangles Legendary Problem" and search in this journal for Dyson + crank.

Wednesday, April 7, 2010

Veritas

Filed under: General — m759 @ 9:00 AM

Some historians consider today's date, April 7, to be the date of the Crucifixion in the Roman calendar (a solar calendar, as opposed to the Jewish lunar scheme).

Since the ninefold square has been called both a symbol of Apollo and the matrix of a cross, it will serve as an icon for today–

The 3x3 square

The image “http://www.log24.com/log/pix05B/051202-Cross.gif” cannot be displayed, because it contains errors.
Adapted from
Ad Reinhardt

Wednesday, September 30, 2009

Wednesday September 30, 2009

Filed under: General,Geometry — m759 @ 9:48 AM

Midnight in the Garden, Autumn 2009

 

Review:

Der Einsatz

Motto of Plato's Academy: 'Let no one ignorant of geometry enter'

The Ninefold Square (a 3x3 grid)

The New York Times Magazine
on Sunday, Sept. 20, 2009:
 

From this journal on the following day, Sept. 21:

Pearl Jam 'Backspacer' album released Sept. 20, 2009

Happy birthday, Stephen King.

Today's previous entry is based on a song, "Unthought Known,"
from the above album; the cover of the album uses the 3×3 grid
shown in Sept. 20's midnight review. For related material
on the unconscious, see June 13-15, 2005.

I know more than Apollo,
For oft when he lies sleeping
I see the stars at mortal wars
In the wounded welkin weeping.

Tom O'Bedlam's Song

Sunday, September 20, 2009

Sunday September 20, 2009

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

Der Einsatz

Motto of Plato's Academy: 'Let no one ignorant of geometry enter'

The 3x3 grid

Nichts ist wie es scheint.

Tuesday, August 4, 2009

Tuesday August 4, 2009

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

Images from Log24 on
December 10, 2006
Nobel Prize Day, and
the day after
  Kirk Douglas’s birthday

Kirk Douglas promoting his film 'Diamonds'

Kirk Douglas ad for
the film “Diamonds”
(2000)

Motto of Plato's Academy: 'Let no one ignorant of geometry enter'

The 3x3 grid

Images from
Google News
   at noon today —

(Click for details.)

3x3 array of Cameron Douglas images from Google News, August 4, 2009

“The serpent’s eyes shine
    as he wraps around the vine…”

Don Henley on a California hotel

Friday, February 20, 2009

Friday February 20, 2009

Filed under: General,Geometry — Tags: , , — m759 @ 11:01 PM

The Cross
of Constantine

mentioned in
this afternoon's entry
"Emblematizing the Modern"
was the object of a recent
cinematic chase sequence
(successful and inspiring)
starring Mira Sorvino
at the Metropolitan
Museum of Art.

In memory of
Dr. Hunter S. Thompson,
dead by his own hand
on this date
four years ago

Rolling Stone memorial to Dr. Hunter S. Thompson

Click for details.

There is
another sort of object
we may associate with a
different museum and with
a modern Constantine
See "Art Wars for MoMA"
(Dec. 14, 2008).

This object, modern
rather than medieval,
is the ninefold square:

The ninefold square

It may suit those who,
like Rosalind Krauss
(see "Emblematizing"),
admire the grids of modern art
but view any sort of Christian
cross with fear and loathing.

For some background that
Dr. Thompson might appreciate,
see notes on Geometry and Death
in this journal, June 1-15, 2007,
and the five Log24 entries
 ending at 9 AM Dec. 10. 2006,
which include this astute
observation by J. G. Ballard:

"Modernism's attempt to build a better world with the aid of science and technology now seems almost heroic. Bertolt Brecht, no fan of modernism, remarked that the mud, blood and carnage of the first world war trenches left its survivors longing for a future that resembled a white-tiled bathroom."

Selah.

Saturday, January 10, 2009

Saturday January 10, 2009

Filed under: General — m759 @ 10:10 AM
A Russian Doll

Introduction

The 3x3 square

For details of the story,
click on the images.

Chapter I:

'The Power Of The Center: A Study of Composition in the Visual Arts,' by Rudolf Arnheim

Chapter II:

Cover of 'Nine Stories' with 'Dinghy' at center

Chapter III:

Natasha’s Dance

Orson Welles with chessboard


and the following quotation:

There is no landing fee in Avalon,
 or anywhere else in Catalina.”

Sunday, December 14, 2008

Sunday December 14, 2008

Filed under: General — m759 @ 2:00 AM
Symmetry
and
Reflections

A figure from
Nobel Prize day, December 10,
and from Eugene Wigner‘s
birthday, November 17:

The 3x3 square

Also on December 10:
  the death of Constantine–

Mildred Constantine, 95, MoMA Curator, Is Dead

(Click for details.)

Related material:

Tina Modotti: A Fragile Life,
Photos by Tina Modotti,
Art Wars for Trotsky’s Birthday,
as well as
Art Wars, June 1-15, 2007:

Diego Rivera and Frida Kahlo

  “Ay que bonito es volar  
    A las dos de la mañana
….”
— “La Bruja

Wednesday, December 10, 2008

Wednesday December 10, 2008

Filed under: General,Geometry — m759 @ 3:26 PM
Symbol

“If it’s a seamless whole you want,
 pray to Apollo, who sets the limits
  within which such a work can exist.”

Margaret Atwood,
quoted here on
November 17, 2008

The 3x3 square

A symbol of Apollo

Related material:

A web page by
Nick Wedd at Oxford

with a neater version
of pictures I drew on
March 26, 1985

(Recall that Apollo is the god
   of, among other things, reason.)

Monday, November 17, 2008

Monday November 17, 2008

Filed under: General,Geometry — m759 @ 9:00 AM
Limits

From the previous entry:

“If it’s a seamless whole you want,
 pray to Apollo, who sets the limits
  within which such a work can exist.”

— Margaret Atwood,
author of Cat’s Eye

The 3x3 square

Happy birthday
to the late
Eugene Wigner

… and a belated
Merry Christmas
 to Paul Newman:

Elke Sommer, former Erlangen Gymnasium student, in 'The Prize' with Paul Newman, released Christmas Day, 1963

“The laws of nature permit us to foresee events on the basis of the knowledge of other events; the principles of invariance should permit us to establish new correlations between events, on the basis of the knowledge of established correlations between events. This is exactly what they do.”

— Eugene Wigner, Nobel Prize Lecture, December 12, 1963

Saturday, August 2, 2008

Saturday August 2, 2008

Filed under: General,Geometry — m759 @ 2:02 PM
Geometry and Death

(continued from
June 15, 2007)

Today is the anniversary
of the 1955 death of poet
Wallace Stevens.

Related material:

A poem by Stevens,

an essay on  the
relationships between
poets and philosophers —
“Bad Blood,” by
Leonard Michaels

and

The ninefold square, a symbol of Apollo

the Log24 entries
of June 14-15, 2007
.

Sunday, December 23, 2007

Sunday December 23, 2007

Filed under: General — m759 @ 9:00 AM
The image “http://www.log24.com/theory/images/grid3x3.gif” cannot be displayed, because it contains errors.

  “Nine is a Vine.”

Monday, July 30, 2007

Monday July 30, 2007

Filed under: General — m759 @ 9:00 AM
Garden Party
 
"And the serpent's eyes shine    
As he wraps around the vine…"

In The Garden of Allah

"But not, perhaps,
in the Garden of Apollo":

The Garden of Apollo: The 3x3 Grid

— "Garden Party" —
Log24, April 9, 2007

Related material:

"When, on the last day of February 1953 Francis told her excitedly of the double helix discovery, she took no notice: 'He was always saying that kind of thing.' But when nine years later she heard the news of the Nobel Prize while out shopping, she immediately rushed to the fishmonger for ice to fill the bath and cool the champagne: a party was inevitable."

— Matt Ridley on Odile Crick (The Independent, July 20, 2007), who drew what "may be the most famous [scientific] drawing of the 20th century, in that it defines modern biology," according to Terrence J. Sejnowski, a neuroscientist at the Salk Institute for Biological Studies in La Jolla quoted by Adam Bernstein in The Washington Post, July 21, 2007

See also "Game Boy"
(Log24 on the Feast
of the Transfiguration–
August 6, 2006):

The image “http://www.log24.com/log/pix06A/060806-Einsatz.jpg” cannot be displayed, because it contains errors.

Tuesday, July 10, 2007

Tuesday July 10, 2007

Filed under: General — m759 @ 9:00 AM
Fewer frames
for Mary Karr

3x3 grid

Mary Karr was “an unfashionably bookish kid whose brain wattage was sapped by a consuming inner life others didn’t seem to bear the burden of. I just seemed to have more frames per second than other kids.”

Friday, June 15, 2007

Friday June 15, 2007

Filed under: General,Geometry — Tags: , — m759 @ 1:00 PM
A Study in
Art Education

Rudolf Arnheim, a student of Gestalt psychology (which, an obituary notes, emphasizes "the perception of forms as organized wholes") was the first Professor of the Psychology of Art at Harvard.  He died at 102 on Saturday, June 9, 2007.

The conclusion of yesterday's New York Times obituary of Arnheim:

"… in The New York Times Book Review in 1986, Celia McGee called Professor Arnheim 'the best kind of romantic,' adding, 'His wisdom, his patient explanations and lyrical enthusiasm are those of a teacher.'"

A related quotation:

"And you are teaching them a thing or two about yourself. They are learning that you are the living embodiment of two timeless characterizations of a teacher: 'I say what I mean, and I mean what I say' and 'We are going to keep doing this until we get it right.'"

Tools for Teaching

Here, yet again, is an illustration that has often appeared in Log24– notably, on the date of Arnheim's death:
 

The 3x3 square

Related quotations:

"We have had a gutful of fast art and fast food. What we need more of is slow art: art that holds time as a vase holds water: art that grows out of modes of perception and whose skill and doggedness make you think and feel; art that isn't merely sensational, that doesn't get its message across in 10 seconds, that isn't falsely iconic, that hooks onto something deep-running in our natures. In a word, art that is the very opposite of mass media. For no spiritually authentic art can beat mass media at their own game."

Robert Hughes, speech of June 2, 2004

"Whether the 3×3 square grid is fast art or slow art, truly or falsely iconic, perhaps depends upon the eye of the beholder."

Log24, June 5, 2004

If the beholder is Rudolf Arnheim, whom we may now suppose to be viewing the above figure in the afterlife, the 3×3 square is apparently slow art.  Consider the following review of his 1982 book The Power of the Center:

"Arnheim deals with the significance of two kinds of visual organization, the concentric arrangement (as exemplified in a bull's-eye target) and the grid (as exemplified in a Cartesian coordinate system)….

It is proposed that the two structures of grid and target are the symbolic vehicles par excellence for two metaphysical/psychological stances.  The concentric configuration is the visual/structural equivalent of an egocentric view of the world.  The self is the center, and all distances exist in relation to the focal spectator.  The concentric arrangement is a hermetic, impregnable pattern suited to conveying the idea of unity and other-worldly completeness.  By contrast, the grid structure has no clear center, and suggests an infinite, featureless extension…. Taking these two ideal types of structural scaffold and their symbolic potential (cosmic, egocentric vs. terrestrial, uncentered) as given, Arnheim reveals how their underlying presence organizes works of art."

— Review of Rudolf Arnheim's The Power of the Center: A Study of Composition in the Visual Arts (Univ. of Calif. Press, 1982). Review by David A. Pariser, Studies in Art Education, Vol. 24, No. 3 (1983), pp. 210-213

Arnheim himself says in this book (pp. viii-ix) that "With all its virtues, the framework of verticals and horizontals has one grave defect.  It has no center, and therefore it has no way of defining any particular location.  Taken by itself, it is an endless expanse in which no one place can be distinguished from the next.  This renders it incomplete for any mathematical, scientific, and artistic purpose.  For his geometrical analysis, Descartes had to impose a center, the point where a pair of coordinates [sic] crossed.  In doing so he borrowed from the other spatial system, the centric and cosmic one."

Students of art theory should, having read the above passages, discuss in what way the 3×3 square embodies both "ideal types of structural scaffold and their symbolic potential."

We may imagine such a discussion in an afterlife art class– in, perhaps, Purgatory rather than Heaven– that now includes Arnheim as well as Ernst Gombrich and Kirk Varnedoe.

Such a class would be one prerequisite for a more advanced course– Finite geometry of the square and cube.

Thursday, June 14, 2007

Thursday June 14, 2007

Filed under: General — Tags: — m759 @ 9:00 PM
A Time
for Remembering

June 9, the birthday of
Aaron Sorkin, a writer
mentioned in recent
Log24 entries, was also
the birthday of writer
Patricia Cornwell.

An illustration
from that date:

The image “http://www.log24.com/theory/images/grid3x3.gif” cannot be displayed, because it contains errors.

Cornwell's first book was
a biography of
Ruth Bell Graham,
A Time for Remembering.

"Seven is heaven,
Eight is a gate,
Nine is a vine."
 

Saturday, June 9, 2007

Saturday June 9, 2007

Filed under: General — m759 @ 9:00 AM
Cryptology

The image “http://www.log24.com/theory/images/grid3x3.gif” cannot be displayed, because it contains errors.

The Delphic Corporation

Thursday, May 10, 2007

Thursday May 10, 2007

Filed under: General — m759 @ 10:00 PM
Riverdance

The image “http://www.log24.com/log/pix07/070510-Dance.jpg” cannot be displayed, because it contains errors.

The above scene from
The Best of Riverdance
furnishes an exercise in
what Victor Turner has called
comparative symbology.”

The circular symbol at top
may be seen as representing
the solar deity Apollo,
Leader of the Muses.

The nine female dancers
may be seen as
the nine muses,
with Jean Butler
at the center
as Terpsichore,
Muse of Dance.

Related Material —

ART WARS:
To Apollo

The image “http://www.log24.com/theory/images/grid3x3.gif” cannot be displayed, because it contains errors.

“This is the garden of Apollo,
the field of Reason….”
John Outram, architect

For another look at
Terpsichore in action,
see Jean Butler at
CRC Irish Dance Camp.

For those who prefer
a different sort of camp
there is of course
Xanadu.

I prefer Butler.

Monday, April 9, 2007

Monday April 9, 2007

Filed under: General — m759 @ 9:00 PM
Garden Party

“And the fruit is rotten;
 the serpent’s eyes shine
 as he wraps around the vine
in the Garden of Allah.
 
— Don Henley

But not, perhaps,
in the Garden of Apollo:

The Garden of Apollo: The 3x3 Grid

Click on the image
for further details.

Saturday, March 10, 2007

Saturday March 10, 2007

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

The Logic of Dreams

From A Beautiful Mind–

“How could you,” began Mackey, “how could you, a mathematician, a man devoted to reason and logical proof…how could you believe that extraterrestrials are sending you messages? How could you believe that you are being recruited by aliens from outer space to save the world? How could you…?”

Nash looked up at last and fixed Mackey with an unblinking stare as cool and dispassionate as that of any bird or snake. “Because,” Nash said slowly in his soft, reasonable southern drawl, as if talking to himself, “the ideas I had about supernatural beings came to me the same way that my mathematical ideas did. So I took them seriously.”

Ideas:

The image “http://www.log24.com/log/pix07/070309-NYlottery.jpg” cannot be displayed, because it contains errors.

The image “http://www.log24.com/log/pix07/070309-PAlottery.jpg” cannot be displayed, because it contains errors.

These numbers may, in the mad way so well portrayed by Sylvia Nasar in the above book, be regarded as telling a story… a story that should, of course, not be taken too seriously.

Friday’s New York numbers (midday 214, evening 711) suggest the dates 2/14 and 7/11.  Clicking on these dates will lead the reader to Log24 entries featuring, among others, T. S. Eliot and Stephen King– two authors not unacquainted with the bizarre logic of dreams.

A link in the 7/11 entry leads to a remark of Noel Gray on Plato’s Meno and “graphic austerity as the tool to bring to the surface, literally and figuratively, the inherent presence of geometry in the mind of the slave.”

Also Friday: an example of graphic austerity– indeed, Gray graphic austerity– in Log24:

Chessboard (Detail)

This illustration refers to chess rather than to geometry, and to the mind of an addict rather than to that of a slave, but chess and geometry, like addiction and slavery, are not unrelated.


Friday’s Pennsylvania numbers, midday 429 and evening 038, suggest that the story includes, appropriately enough in view of the above Beautiful Mind excerpt, Mackey himself.  The midday number suggests the date 4/29, which at Log24 leads to an entry in memory of Mackey.

(Related material: the Harvard Gazette of April 6, 2006, “Mathematician George W. Mackey, 90: Obituary“–  “A memorial service will be held at Harvard’s Memorial Church on April 29 at 2 p.m.“)

Friday’s Pennsylvania evening number 038 tells two other parts of the story involving Mackey…

As Mackey himself might hope, the number may be regarded as a reference to the 38 impressive pages of Varadarajan’s “Mackey Memorial Lecture” (pdf).

More in the spirit of Nash, 38 may also be taken as a reference to Harvard’s old postal address, Cambridge 38, and to the year, 1938, that Mackey entered graduate study at Harvard, having completed his undergraduate studies at what is now Rice University.

Returning to the concept of graphic austerity, we may further simplify the already abstract chessboard figure above to obtain an illustration that has been called both “the field of reason” and “the Garden of Apollo” by an architect, John Outram, discussing his work at Mackey’s undergraduate alma mater:

The image “http://www.log24.com/theory/images/grid3x3.gif” cannot be displayed, because it contains errors.

Let us hope that Mackey,
a devotee of reason,
is now enjoying the company
of Apollo rather than that of
Tom O’Bedlam:

The image “http://www.log24.com/log/pix05A/050613-Crowe.jpg” cannot be displayed, because it contains errors.

For John Nash on his birthday:

I know more than Apollo,
For oft when he lies sleeping
I see the stars at mortal wars
In the wounded welkin weeping.

Tom O’Bedlam’s Song

Thursday, March 1, 2007

Thursday March 1, 2007

Filed under: General — m759 @ 9:00 AM
A stich in time
 saves…


A 3x3 grid

Click on picture
for further details.

Friday, January 19, 2007

Friday January 19, 2007

Filed under: General — m759 @ 7:00 PM

Semantic Transparency

“… semantic transparency … would allow disparate systems to share some understanding of the actual concepts that are represented…”

IBM Developer Works on October 7, 2003

From Wikipedia’s
“Upper Ontology”
and
Epiphany 2007:

“There is no neutral ground
that can serve as
a means of translating between
specialized (lower) ontologies.”

 There is, however,
“the field of reason”–
the 3×3 grid:

The image “http://www.log24.com/theory/images/grid3x3.gif” cannot be displayed, because it contains errors.

Click on grid
for details.

From a Log24 entry of January 7, 2007:

“One of the primary critiques of modernism that Learning from Las Vegas was engaged in, as Frederic [sic] Jameson clearly noted, was the dialectic between inside and outside and the assumption that the outside expressed the interior. Let’s call this the modernist drive for ‘expressive transparency.'”

Aron Vinegar of Ohio State U., “Skepticism and the Ordinary: From Burnt Norton to Las Vegas

From this week’s New Yorker (issue dated Jan. 22, 2007)–

“A Life,” by Zbigniew Herbert
(translated from the Polish by Alissa Vales):

I was a quiet boy a little sleepy and–amazingly–
unlike my peers–who were fond of adventures–
I didn’t expect much–didn’t look out the window
At school more diligent than able–docile stable

For the rest of the poem, click here.

From the Wikipedia article on Zbigniew Herbert:

“In modern poetry, Herbert advocated semantic transparence. In a talk given at a conference organized by the journal Odra he said: ‘So not having pretensions to infallibility, but stating only my predilections, I would like to say that in contemporary poetry the poems that appeal to me the most are those in which I discern something I would call a quality of semantic transparency (a term borrowed from Husserl’s logic). This semantic transparency is the characteristic of a sign consisting in this: that during the time when the sign is used, attention is directed towards the object denoted, and the sign itself does not hold the attention. The word is a window onto reality.'”

(Wikipedia cites as the source–
Herbert’s talk at the meeting “Poet in face of the present day,” organized by the “Odra” journal. Print version: Preface to: Zbigniew Herbert “Poezje,” Panstwowy Instytut Wydawniczy, Warszawa 1998, ISBN 83-06-02667-5.)

Fom Nabokov’s Transparent Things (pdf):

“Its ultimate vision was the incandescence of a book or a box grown completely transparent and hollow.  This is, I believe, it: not the crude anguish of physical death but the incomparable pangs of the mysterious mental maneuver needed to pass from one state of being to another.  Easy, you know, does it, son.”

Related material:

Confession

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Tuesday, January 9, 2007

Tuesday January 9, 2007

Filed under: General — Tags: — m759 @ 12:00 PM
For Balanchine's Birthday

(continued from
January 9, 2003)

George Balanchine

Encyclopædia Britannica Article

born January 22
[January 9, Old Style], 1904,
St. Petersburg, Russia
died April 30, 1983, New York,
New York, U.S.

Photograph:George Balanchine.
George Balanchine.
©1983 Martha Swope

original name 
Georgy Melitonovich Balanchivadze

most influential choreographer of classical ballet in the United States in the 20th century.  His works, characterized by a cool neoclassicism, include The Nutcracker (1954) and Don Quixote (1965), both pieces choreographed for the New York City Ballet, of which he was a founder (1948), the artistic director, and the…


Balanchine,  George… (75 of 1212 words)

"What on earth is
a concrete universal?"
— Robert M. Pirsig

Review:

From Wikipedia's
"Upper Ontology"
and
Epiphany 2007:

"There is no neutral ground
that can serve as
a means of translating between
specialized (lower) ontologies."

There is, however,
"the field of reason"–

the 3×3 grid:

The image “http://www.log24.com/theory/images/grid3x3.gif” cannot be displayed, because it contains errors.

Click on grid
for details.

As Rosalind Krauss
has noted, some artists
regard the grid as

"a staircase to
  the Universal."

Other artists regard
Epiphany itself as an
approach to
the Universal:

"Epiphany signals the traversal
of the finite by the infinite,
of the particular by the universal,
of the mundane by the mystical,
of time by eternity.
"

Richard Kearney, 2005,
in The New Arcadia Review

The image “http://www.log24.com/log/pix07/070109-Kearney2.jpg” cannot be displayed, because it contains errors.

Kearney (right) with
Martin Scorsese (left)
and Gregory Peck
in 1997.

"… one of the things that worried me about traditional metaphysics, at least as I imbibed it in a very Scholastic manner at University College Dublin in the seventies, is that philosophy was realism and realism was truth. What disturbed me about that was that everything was already acquired; truth was always a systematic given and it was there to be learned from Creation onwards; it was spoken by Jesus Christ and then published by St. Thomas Aquinas: the system as perfect synthesis. Hence, my philosophy grew out of a hunger for the 'possible' and it was definitely a reaction to my own philosophical formation. Yet that wasn't my only reaction. I was also reacting to what I considered to be the deep pessimism, and even at times 'nihilism' of the postmodern turn."

— Richard Kearney, interview (pdf) in The Leuven Philosophy Newsletter, Vol. 14, 2005-2006

For more on "the possible," see Kearney's The God Who May Be, Diamonds Are Forever, and the conclusion of Mathematics and Narrative:

 

"We symbolize
logical necessity
with the box (box.gif (75 bytes))
and logical possibility
with the diamond (diamond.gif (82 bytes))."

 

Keith Allen Korcz 

The image “http://www.log24.com/log/pix05B/050802-Stone.gif” cannot be displayed, because it contains errors.

"The possibilia that exist,
and out of which
the Universe arose,
are located in
     a necessary being…."

Michael Sudduth,
Notes on
God, Chance, and Necessity
by Keith Ward,
Regius Professor of Divinity,
Christ Church College, Oxford
(the home of Lewis Carroll)

Tuesday, December 19, 2006

Tuesday December 19, 2006

Filed under: General — Tags: — m759 @ 9:00 AM
Joseph Barbera
at the Apollo


The 3x3 Grid

Click on picture
for related symbolism.

“This is the garden of Apollo,
the field of Reason….”
John Outram, architect

I need a photo-opportunity
I want a shot at redemption
Don’t want to end up a cartoon
In a cartoon graveyard
— Paul Simon

In memory of Joseph Barbera–
co-creator ot the Flintstones–
who died yesterday, a photo
from today’s Washington Post:

Joseph Barbera in Washington Post

Playing the role of
recording angel —

Halle Berry as
Rosetta Stone:

Halle Berry as Rosetta Stone

Related material:

Citizen Stone
and
Putting the X in Xmas.”

Sunday, December 10, 2006

Sunday December 10, 2006

Filed under: General,Geometry — m759 @ 9:00 AM
The Librarian

"Like all men of the Library,
I have traveled in my youth."
— Jorge Luis Borges,
The Library of Babel

"Papá me mandó un artículo
de J. G. Ballard en el que
se refiere a cómo el lugar
de la muerte es central en
nuestra cultura contemporánea
."

— Sonya Walger,
interview dated September 14
(Feast of the Triumph of the Cross),
Anno Domini 2006

The image “http://www.log24.com/log/pix06B/061210-Quest.gif” cannot be displayed, because it contains errors.

Sonya Walger,
said to have been
born on D-Day,
the sixth of June,
in 1974

 

Walger's father is, like Borges,
from Argentina.
She "studied English Literature
at Christ Church College, Oxford,
where she received
    a First Class degree…. "

Wikipedia

"… un artículo de J. G. Ballard…."–

A Handful of Dust
, by J. G. Ballard

(The Guardian, March 20, 2006):

"… The Atlantic wall was only part of a huge system of German fortifications that included the Siegfried line, submarine pens and huge flak towers that threatened the surrounding land like lines of Teutonic knights. Almost all had survived the war and seemed to be waiting for the next one, left behind by a race of warrior scientists obsessed with geometry and death.

Death was what the Atlantic wall and Siegfried line were all about….

… modernism of the heroic period, from 1920 to 1939, is dead, and it died first in the blockhouses of Utah beach and the Siegfried line…

Modernism's attempt to build a better world with the aid of science and technology now seems almost heroic. Bertolt Brecht, no fan of modernism, remarked that the mud, blood and carnage of the first world war trenches left its survivors longing for a future that resembled a white-tiled bathroom.  Architects were in the vanguard of the new movement, led by Le Corbusier and the Bauhaus design school. The old models were thrown out. Function defined form, expressed in a pure geometry that the eye could easily grasp in its entirety."

The image “http://www.log24.com/theory/images/motto2.jpg” cannot be displayed, because it contains errors.
 
The image “http://www.log24.com/theory/images/grid3x3.gif” cannot be displayed, because it contains errors.

"This is the garden of Apollo,
the field of Reason…."
John Outram, architect 

(Click on picture for details.)

The image “http://www.log24.com/log/pix06B/061210-Holl.gif” cannot be displayed, because it contains errors.
The Left Hand of God, by Adolf Holl

Related material:

The Lottery of Babylon
and
the previous entry.
 

Sunday, November 26, 2006

Sunday November 26, 2006

Filed under: General — m759 @ 7:26 AM

Rosalind Krauss
in "Grids," 1979:

"If we open any tract– Plastic Art and Pure Plastic Art or The Non-Objective World, for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter.  They are talking about Being or Mind or Spirit.  From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete.

Or, to take a more up-to-date example…."

"He was looking at
the nine engravings
and at the circle,
checking strange
correspondences
between them."
The Club Dumas,1993

"And it's whispered that soon
if we all call the tune
Then the piper will lead us
to reason."
Robert Plant,1971

The nine engravings of
The Club Dumas
(filmed as "The Ninth Gate")
are perhaps more an example
of the concrete than of the
universal.

An example of the universal*–
or, according to Krauss, a
"staircase" to the universal–
is the ninefold square:

The image “http://www.log24.com/theory/images/grid3x3.gif” cannot be displayed, because it contains errors.

"This is the garden of Apollo,
the field of Reason…."
John Outram, architect    

For more on the field
of reason, see
Log24, Oct. 9, 2006.

A reasonable set of
"strange correspondences"
in the garden of Apollo
has been provided by Ezra Brown
in a mathematical essay (pdf).

Unreason is, of course,
more popular.

* The ninefold square is perhaps a "concrete universal" in the sense of Hegel:

"Two determinations found in all philosophy are the concretion of the Idea and the presence of the spirit in the same; my content must at the same time be something concrete, present. This concrete was termed Reason, and for it the more noble of those men contended with the greatest enthusiasm and warmth. Thought was raised like a standard among the nations, liberty of conviction and of conscience in me. They said to mankind, 'In this sign thou shalt conquer,' for they had before their eyes what had been done in the name of the cross alone, what had been made a matter of faith and law and religion– they saw how the sign of the cross had been degraded."

— Hegel, Lectures on the History of Philosophy, "Idea of a Concrete Universal Unity"

"For every kind of vampire,
there is a kind of cross."
— Thomas Pynchon   
 

Monday, October 9, 2006

Monday October 9, 2006

Filed under: General,Geometry — m759 @ 9:00 AM
 
ART WARS:
To Apollo
 
The image “http://www.log24.com/theory/images/grid3x3.gif” cannot be displayed, because it contains errors.

"This is the garden of Apollo,
the field of Reason…."
John Outram, architect

To Apollo (10/09/02)
Art Wars: Apollo and Dionysus
(10/09/02)
Balanchine's Birthday
(01/09/03)

Art Theory for Yom Kippur
(10/05/03)

A Form
(05/22/04)
Ineluctable
(05/27/04)

A Form, continued
(06/05/04)
Parallelisms
(06/06/04)
Ado
(06/25/04)

Deep Game
(06/26/04)
Gameplayers of Zen
(06/27/04)
And So To Bed
(06/29/04)
Translation Plane for Rosh Hashanah
(09/15/04)
Derrida Dead
(10/09/04)
The Nine
(11/09/04)
From Tate to Plato
(11/19/04)
Art History
(05/11/05)
A Miniature Rosetta Stone
(08/06/05)
High Concept
(8/23/05) 
High Concept, Continued
(8/24/05)
Analogical Train of Thought
(8/25/05)
Today's Sermon: Magical Thinking
(10/09/05)
Balance
(10/31/05)
Matrix
(11/01/05)
Seven is Heaven, Eight is a Gate
(11/12/05)
Nine is a Vine
(11/12/05)
Apollo and Christ
(12/02/05)
Hamilton's Whirligig
(01/05/06)
Cross
(01/06/06)
On Beauty
(01/26/06)
Sunday Morning
(01/29/06)
Centre
(01/29/06)
New Haven
(01/29/06) 
Washington Ballet
(02/05/06)
Catholic Schools Sermon
(02/05/06)
The Logic of Apollo
(02/05/06)
Game Boy
(08/06/06)
Art Wars Continued: The Krauss Cross
(09/13/06)
Art Wars Continued: Pandora's Box
(09/16/06)
The Pope in Plato's Cave
(09/16/06)
Today's Birthdays
(09/26/06)
Symbology 101
(09/26/06)

Tuesday, September 26, 2006

Tuesday September 26, 2006

Filed under: General — m759 @ 9:00 AM
Symbology 101

The image “http://www.log24.com/theory/images/grid3x3.gif” cannot be displayed, because it contains errors.

See Balance and
A Form.

Sunday, January 29, 2006

Sunday January 29, 2006

Filed under: General — m759 @ 9:00 AM
Centre

In the punctual centre of all circles white
     Stands truly….

The image “http://www.log24.com/theory/images/grid3x3.gif” cannot be displayed, because it contains errors.


… and Bloom with his vast accumulation
Stands and regards and repeats the primitive lines.

— Wallace Stevens,
“From the Packet of Anacharsis”

Related material:
Balanchine’s Birthday.

Thursday, January 26, 2006

Thursday January 26, 2006

Filed under: General,Geometry — m759 @ 9:00 AM
In honor of Paul Newman’s age today, 81:

On Beauty

Elaine Scarry, On Beauty (pdf), page 21:

“Something beautiful fills the mind yet invites the search for something beyond itself, something larger or something of the same scale with which it needs to be brought into relation. Beauty, according to its critics, causes us to gape and suspend all thought. This complaint is manifestly true: Odysseus does stand marveling before the palm; Odysseus is similarly incapacitated in front of Nausicaa; and Odysseus will soon, in Book 7, stand ‘gazing,’ in much the same way, at the season-immune orchards of King Alcinous, the pears, apples, and figs that bud on one branch while ripening on another, so that never during the cycling year do they cease to be in flower and in fruit. But simultaneously what is beautiful prompts the mind to move chronologically back in the search for precedents and parallels, to move forward into new acts of creation, to move conceptually over, to bring things into relation, and does all this with a kind of urgency as though one’s life depended on it.”

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The above symbol of Apollo suggests, in accordance with Scarry’s remarks, larger structures.   Two obvious structures are the affine 4-space over GF(3), with 81 points, and the affine plane over GF(32), also with 81 points.  Less obvious are some related projective structures.  Joseph Malkevitch has discussed the standard method of constructing GF(32) and the affine plane over that field, with 81 points, then constructing the related Desarguesian projective plane of order 9, with 92 + 9 + 1 = 91 points and 91 lines.  There are other, non-Desarguesian, projective planes of order 9.  See Visualizing GL(2,p), which discusses a spreadset construction of the non-Desarguesian translation plane of order 9.  This plane may be viewed as illustrating deeper properties of the 3×3 array shown above. To view the plane in a wider context, see The Non-Desarguesian Translation Plane of Order 9 and a paper on Affine and Projective Planes (pdf). (Click to enlarge the excerpt beow).

The image “http://www.log24.com/theory/images/060126-planes2.jpg” cannot be displayed, because it contains errors.

See also Miniquaternion Geometry: The Four Projective Planes of Order 9 (pdf), by Katie Gorder (Dec. 5, 2003), and a book she cites:

Miniquaternion geometry: An introduction to the study of projective planes, by T. G. Room and P. B. Kirkpatrick. Cambridge Tracts in Mathematics and Mathematical Physics, No. 60. Cambridge University Press, London, 1971. viii+176 pp.

For “miniquaternions” of a different sort, see my entry on Visible Mathematics for Hamilton’s birthday last year:

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Thursday, January 9, 2003

Thursday January 9, 2003

Filed under: General — Tags: — m759 @ 4:48 PM

Balanchine's Birthday

Today seems an appropriate day to celebrate Apollo and the nine Muses.

From a website on Balanchine's and Stravinsky's ballet, "Apollon Musagete":

In his Poetics of Music (1942) Stravinsky says: "Summing up: What is important for the lucid ordering of the work– for its crystallization– is that all the Dionysian elements which set the imagination of the artist in motion and make the life-sap rise must be properly subjugated before they intoxicate us, and must finally be made to submit to the law: Apollo demands it."  Stravinsky conceived Apollo as a ballet blanc– a "white ballet" with classical choreography and monochromatic attire. Envisioning the work in his mind's eye, he found that "the absence of many-colored hues and of all superfluities produced a wonderful freshness." Upon first hearing Apollo, Diaghilev found it "music somehow not of this world, but from somewhere else above." The ballet closes with an Apotheosis in which Apollo leads the Muses towards Parnassus. Here, the gravely beautiful music with which the work began is truly recapitulated "on high"– ceaselessly recycled, frozen in time.

— Joseph Horowitz

 

Another website invoking Apollo:

The icon that I use… is the nine-fold square…. The nine-fold square has centre, periphery, axes and diagonals.  But all are present only in their bare essentials.  It is also a sequence of eight triads.  Four pass through the centre and four do not.  This is the garden of Apollo, the field of Reason…. 

In accordance with these remarks, here is the underlying structure for a ballet blanc:

A version of 'grid3x3.gif.'

This structure may seem too simple to support movements of interest, but consider the following (click to enlarge):

As Sir Arthur Quiller-Couch, paraphrasing Horace, remarks in his Whitsun, 1939, preface to the new edition of the Oxford Book of English Verse, "tamen usque recurret Apollo."

The alert reader will note that in the above diagrams, only eight of the positions move.

Which muse remains at the center?

Consider the remark of T. S. Eliot, "At the still point, there the dance is," and the fact that on the day Eliot turned 60, Olivia Newton-John was born.  How, indeed, in the words of another "sixty-year-old smiling public man," can we know the dancer from the dance?
 

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