Log24

Sunday, March 19, 2017

Delos Incorporated* Sunday School

Filed under: Uncategorized — m759 @ 9:00 AM

The 3x3 square

Click image for a search.

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

Sunday, November 29, 2015

There the Dance Is

Filed under: Uncategorized — 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.

Sunday, May 11, 2014

For the Perplexed

Filed under: Uncategorized — m759 @ 9:48 PM

From a New York Times  obituary by Bruce Weber tonight—

Charles Marowitz, Director and Playwright, Dies at 82

“There are two kinds of bafflement in the theater: the kind that fascinates as it perplexes, and the kind that just perplexes,” he wrote in The Times in 1969 in an essay about Mr. Shepard’s play “La Turista,” which had recently opened in London. “If a play doesn’t make quick sense, but enters into some kind of dialogue with our subconscious, we tend to admit it to that lounge where we entertain interesting-albeit-unfamiliar strangers.

“If it only baffles, there are several courses open to us: we can assume it is ‘above our heads’ or directed ‘to some other kind of person,’ or regretfully conclude that it confuses us because it is itself confused. However, the fear of being proved wrong is so great today that almost every new work which isn’t patently drivel gets the benefit of the doubt.”

 Another play by Sam Shepard mentioned in the obituary suggests a review of…

Tuesday, October 15, 2013

The Field of Reason*

Filed under: Uncategorized — m759 @ 12:00 PM

IMAGE- Adolf Holl on the Roman emperor Julian and Apollo

Or perhaps the Richman.

Roger Richman, agent who represented image rights for the
estates of celebrities, reportedly died on October 9, 2013.

This journal on that date —

* For the title, see Apollo + Outram in this journal.

Wednesday, October 9, 2013

To Apollo*

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

From Log24

From Josefine Lyche's website —

* For the title, see Apollo + Outram in this journal.

 

Monday, August 26, 2013

Dark Side Tales

Filed under: Uncategorized — 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."

Wednesday, October 12, 2011

High White Noon

Filed under: Uncategorized — 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

Sunday, April 17, 2011

Sunday School

Filed under: Uncategorized — Tags: — m759 @ 7:00 AM

Apollo and the Tricksters

From The Story of N (Oct. 15, 2010)—

Roberta Smith on what she calls "endgame art"—

"Fear of form above all means fear of compression— of an artistic focus that condenses experiences, ideas and feelings into something whole, committed and visually comprehensible."

Margaret Atwood on tricksters and art—

"If it’s a seamless whole you want, pray to Apollo."

Here is some related material In memory of CIA officer Clare Edward Petty, who died at 90 on March 18—

A review of a sort of storyteller's MacGuffin — the 3×3 grid. This is, in Smith's terms, an "artistic focus" that appears  to be visually comprehensible but is not as simple as it seems.

The Hesse configuration can serve as more than a sort of Dan Brown MacGuffin. As a post of January 14th notes, it can (rather fancifullly) illustrate the soul—

http://www.log24.com/log/pix11/110417-AlderTilleyColoredSm.jpg

" … I feel I understand
Existence, or at least a minute part
Of my existence, only through my art,
In terms of combinational delight…."

— Vladimir Nabokov, Pale Fire

Saturday, January 8, 2011

True Grid (continued)

Filed under: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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

Thursday, August 19, 2010

Consolation Prize

Filed under: Uncategorized — 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.

Sunday, May 9, 2010

For Miss Prothero (and Dylan Thomas)

Filed under: Uncategorized — Tags: — m759 @ 4:23 AM

 

The Ninth Gate

Friday's post "Religion at Harvard" continues…

Image-- List of nine religions in the chapters of Prothero's 'God is Not One'

This list may be of some use to
Rebecca Newberger Goldstein, who, like Prothero,
spoke recently at Harvard Book Store.

See also Rosalind Krauss on Grids,
An Education, and Plan 9 from Outer Space.

Readers more advanced than Harvard audiences
may wish to compare yesterday's linked-to story
"Loo Ree" with the works of Alison Lurie
in particular, Imaginary Friends and Familiar Spirits.

Wednesday, April 7, 2010

Veritas

Filed under: Uncategorized — 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: Uncategorized — 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

Thursday, February 19, 2009

Thursday February 19, 2009

Filed under: Uncategorized — m759 @ 7:07 AM

A Sunrise
for Sunrise

"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."

Rosalind Krauss, "Grids"

Yesterday's entry featured a rather simple-minded example from Krauss of how the ninefold square (said to be a symbol of Apollo)

The 3x3 grid

may be used to create a graphic design– a Greek cross, which appears also in crossword puzzles:

Crossword-puzzle design that includes Greek-cross elements

Illustration by
Paul Rand
(born Peretz Rosenbaum)

A more sophisticated example
of the ninefold square
in graphic design:

"That old Jew
gave me this here."

— A Flag for Sunrise  

The 3x3 grid as an organizing frame for Chinese calligraphy. Example-- the character for 'sunrise'
From Paul-Rand.com

Wednesday, December 10, 2008

Wednesday December 10, 2008

Filed under: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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
.

Monday, July 30, 2007

Monday July 30, 2007

Filed under: Uncategorized — 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.

Thursday, May 10, 2007

Thursday May 10, 2007

Filed under: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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

Tuesday, December 19, 2006

Tuesday December 19, 2006

Filed under: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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)

Saturday, September 16, 2006

Saturday September 16, 2006

Filed under: Uncategorized — m759 @ 11:07 AM

Pandora's Box

Part I:
The Pandora Cross

"There is no painter in the West who can be unaware of the symbolic power of the cruciform shape and the Pandora's box of spiritual reference that is opened once one uses it."

— Rosalind Krauss in "Grids"


The image “http://www.log24.com/log/pix06A/060916-Art.jpg” cannot be displayed, because it contains errors.

(See Log24, Sept. 13)

Part II:
The Opening

Remarks by the Pope on Sept. 12,
as reported by the Vatican:

Faith, Reason, and the University:
Memories and Reflections

For the result of
the Pope's remarks, see
a transcript of
 yesterday's Google News
and the following
from BBC today:

The image “http://www.log24.com/log/pix06A/060916-Benedict16.jpg” cannot be displayed, because it contains errors.
Click to enlarge the screenshot.

Part III:
Hope

The New Yorker (issue of June 5, 2006) on the late Oriana Fallaci:

"In September [2005], she had a private audience with Pope Benedict XVI at Castel Gandolfo, his summer residence outside Rome. She had criticized John Paul II for making overtures to Muslims, and for not condemning terrorism heartily enough, but she has hopes for Joseph Ratzinger."

For further details, see yesterday's Log24.


Part IV:
The Sibyl's Song

The image “http://www.log24.com/log/pix06A/060916-MC7.GIF” cannot be displayed, because it contains errors.

— From The Magic Circle,
 a spiritual narrative
 by Katherine Neville

For more on "the long-mute voice
of the past," on "darkness beneath
the volcano," and on uncorking,
see Glory Season and Harrowing.

Related material from
Log24 on Dec. 2, 2005:

Benedict XVI, before he became Pope:

"… a purely harmonious concept of beauty is not enough…. Apollo, who for Plato's Socrates was 'the God' and the guarantor of unruffled beauty as 'the truly divine' is absolutely no longer sufficient."

A symbol of Apollo:

IMAGE- The ninefold square

and a related
Christian symbol,

The image �http://www.log24.com/log/pix05B/051202-Cross.gif� cannot be displayed, because it contains errors.

the Greek Cross
(adapted from
Ad Reinhardt).

Moral of the Pandora Cross:

"Nine is a very powerful Nordic number."
— Katherine Neville in The Magic Circle…

quoted in The Nine, a Log24 entry
for Hermann Weyl's birthday,
November 9, 2004.
 

Friday, August 4, 2006

Friday August 4, 2006

Filed under: Uncategorized — m759 @ 2:56 AM

ART WARS
continued from
previous entry

In memory of
Elisabeth Schwarzkopf:

"Who is the fairest of them all?"

This question might
well be posed by…

The image “http://www.log24.com/log/pix05/050501-Krauss.jpg” cannot be displayed, because it contains errors.

Rosalind Krauss,
Meyer Schapiro Professor
of Modern Art and Theory
at Columbia University
(Ph.D., Harvard U., 1969).

"The grid is a staircase to the Universal….
We could think about Ad Reinhardt, who,
despite his repeated insistence that
'Art is art,'
ended up by painting a series of…
nine-square grids in which the motif
that inescapably emerges is
a Greek cross.

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

There is no painter in the West
who can be unaware of
the symbolic power
of the cruciform shape and the
Pandora's box of spiritual reference
that is opened once one uses it."

— Rosalind Krauss in "Grids"

"Nine is a very powerful Nordic number."
— Katherine Neville, author of The Eight

Related material:

Balanchine's Birthday,

Apollo and Christ.
 

Thursday, January 26, 2006

Thursday January 26, 2006

Filed under: Uncategorized — 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.”

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

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:

The image “http://www.log24.com/theory/images/Quaternions2.jpg” cannot be displayed, because it contains errors.

 

Friday, December 2, 2005

Friday December 2, 2005

Filed under: Uncategorized — m759 @ 8:05 PM
For St. Robertson Davies,
whose feast is today:

Apollo and Christ

Benedict XVI, before he became Pope:

“… a purely harmonious concept of beauty is not enough…. Apollo, who for Plato’s Socrates was ‘the God’ and the guarantor of unruffled beauty as ‘the truly divine’ is absolutely no longer sufficient.”

Discuss the following
symbol of Apollo

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

as the source of
a Christian symbol–

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

the Greek Cross.

Related material:

The play
now then again

(Time-Bending Love Story
Comes Home to Fermilab
)

The image “http://www.log24.com/log/pix05B/051202-NowThen2.jpg” cannot be displayed, because it contains errors.

with cover art by Roz Francis

illustrating the time 8:05:19,

Hexagram 19 in the
Cullinane series

The image “http://www.log24.com/log/pix05B/051202-Hex19.jpg” cannot be displayed, because it contains errors.

(“A dance results”
— Marie-Louise von Franz),

and

Paul Preuss on Apollo,
quantum physics, and
the isle of Delos.

Saturday, July 30, 2005

Saturday July 30, 2005

Filed under: Uncategorized — Tags: , — m759 @ 11:21 AM

Born today: Laurence Fishburne

Matrix

"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, sheltered by the Gate from the turmoil of the Delta, with its endless cycles of erasure and reinscription. This is the Temple of Solomon, as inscribed, for example, by a nine-fold compartmentation to provide the ground plan of Yale…."

— Architects John Outram Associates
    on work at Rice University

Yale Daily News, Jan. 11, 2001:  

    "When New Haven was founded, the city was laid out into a grid of nine squares surrounded by a great wilderness.
    Last year History of Art Professor Emeritus Vincent Scully said the original town plan reflected a feeling that the new city should be sacred.
    Scully said the colony's founders thought of their new Puritan settlement as a 'nine-square paradise on Earth, heaven on earth, New Haven, New Jerusalem.'"

"Real and unreal are two in one:
    New Haven
 Before and after one arrives…."

 — Wallace Stevens,
    "An Ordinary Evening in New Haven,' XXVIII
 

Related material:
 Log24 entries on
St. Peter's Day, 2004

Thursday, April 22, 2004

Thursday April 22, 2004

Filed under: Uncategorized — Tags: — m759 @ 10:07 PM

Minimalism

"It's become our form of modern classicism."

— Nancy Spector in 
   the New York Times of April 23, 2004

Part I: Aesthetics

In honor of the current Guggenheim exhibition, "Singular Forms" — A quotation from the Guggenheim's own website

"Minimalism refers to painting or sculpture

  1. made with an extreme economy of means
  2. and reduced to the essentials of geometric abstraction….
  3. Minimalist art is generally characterized by precise, hard-edged, unitary geometric forms….
  4. mathematically regular compositions, often based on a grid….
  5. the reduction to pure self-referential form, emptied of all external references….
  6. In Minimal art what is important is the phenomenological basis of the viewer’s experience, how he or she perceives the internal relationships among the parts of the work and of the parts to the whole….
  7. The repetition of forms in Minimalist sculpture serves to emphasize the subtle differences in the perception of those forms in space and time as the spectator’s viewpoint shifts in time and space."

Discuss these seven points
in relation to the following:

 
Form,
by S. H. Cullinane

Logos and Logic

Mark Rothko's reference
to geometry as a "swamp"
and his talk of "the idea" in art

Michael Kimmelman's
remarks on ideas in art 

Notes on ideas and art

Geometry
of the 4×4 square

The Grid of Time

ART WARS:
Judgment Day
(2003, 10/07)

Part II: Theology

Today's previous entry, "Skylark," concluded with an invocation of the Lord.   Of course, the Lord one expects may not be the Lord that appears.


 John Barth on minimalism:

"… the idea that, in art at least, less is more.

It is an idea surely as old, as enduringly attractive and as ubiquitous as its opposite. In the beginning was the Word: only later came the Bible, not to mention the three-decker Victorian novel. The oracle at Delphi did not say, 'Exhaustive analysis and comprehension of one's own psyche may be prerequisite to an understanding of one's behavior and of the world at large'; it said, 'Know thyself.' Such inherently minimalist genres as oracles (from the Delphic shrine of Apollo to the modern fortune cookie), proverbs, maxims, aphorisms, epigrams, pensees, mottoes, slogans and quips are popular in every human century and culture–especially in oral cultures and subcultures, where mnemonic staying power has high priority–and many specimens of them are self-reflexive or self-demonstrative: minimalism about minimalism. 'Brevity is the soul of wit.' "


Another form of the oracle at Delphi, in minimalist prose that might make Hemingway proud:

"He would think about Bert.  Bert was an interesting man.  Bert had said something about the way a gambler wants to lose.  That did not make sense.  Anyway, he did not want to think about it.  It was dark now, but the air was still hot.  He realized that he was sweating, forced himself to slow down the walking.  Some children were playing a game with a ball, in the street, hitting it against the side of a building.  He wanted to see Sarah.

When he came in, she was reading a book, a tumbler of dark whiskey beside her on the end table.  She did not seem to see him and he sat down before he spoke, looking at her and, at first, hardly seeing her.  The room was hot; she had opened the windows, but the air was still.  The street noises from outside seemed almost to be in the room with them, as if the shifting of gears were being done in the closet, the children playing in the bathroom.  The only light in the room was from the lamp over the couch where she was reading.

He looked at her face.  She was very drunk.  Her eyes were swollen, pink at the corners.  'What's the book,' he said, trying to make his voice conversational.  But it sounded loud in the room, and hard.

She blinked up at him, smiled sleepily, and said nothing.

'What's the book?'  His voice had an edge now.

'Oh,' she said.  'It's Kierkegaard.  Soren Kierkegaard.' She pushed her legs out straight on the couch, stretching her feet.  Her skirt fell back a few inches from her knees.  He looked away.

'What's that?' he said.

'Well, I don't exactly know, myself."  Her voice was soft and thick.

He turned his face away from her again, not knowing what he was angry with.  'What does that mean, you don't know, yourself?'

She blinked at him.  'It means, Eddie, that I don't exactly know what the book is about.  Somebody told me to read it once, and that's what I'm doing.  Reading it.'

He looked at her, tried to grin at her — the old, meaningless, automatic grin, the grin that made everbody like him — but he could not.  'That's great,' he said, and it came out with more irritation than he had intended.

She closed the book, tucked it beside her on the couch.  She folded her arms around her, hugging herself, smiling at him.  'I guess this isn't your night, Eddie.  Why don't we have a drink?'

'No.'  He did not like that, did not want her being nice to him, forgiving.  Nor did he want a drink.

Her smile, her drunk, amused smile, did not change.  'Then let's talk about something else,' she said.  'What about that case you have?  What's in it?'  Her voice was not prying, only friendly, 'Pencils?'

'That's it,' he said.  'Pencils.'

She raised her eyebrows slightly.  Her voice seemed thick.  'What's in it, Eddie?'

'Figure it out yourself.'  He tossed the case on the couch."

— Walter Tevis, The Hustler, 1959,
    Chapter 11


See, too, the invocation of Apollo in

A Mass for Lucero, as well as 

GENERAL AUDIENCE OF JOHN PAUL II
Wednesday 15 January 2003
:

"The invocation of the Lord is relentless…."

and

JOURNAL ENTRY OF S. H. CULLINANE
Wednesday 15 January 2003
:

Karl Cullinane —
"I will fear no evil, for I am the
meanest son of a bitch in the valley."

Sunday, April 27, 2003

Sunday April 27, 2003

Filed under: Uncategorized — m759 @ 3:24 PM

ART WARS:

Graphical Password

From a summary of “The Design and Analysis of Graphical Passwords“:

“Results from cognitive science show that people can remember pictures much better than words….

The 5×5 grid creates a good balance between security and memorability.”

 Ian Jermyn, New York University; Alain Mayer, Fabian Monrose, Michael K. Reiter, Bell Labs, Lucent Technologies; Aviel Rubin, AT&T Labs — Research

Illustration — Warren Beatty as
a graphical password:

Town & Country,”
released April 27, 2001

Those who prefer the simplicity of a 3×3 grid are referred to my entry of Jan. 9, 2003, Balanchine’s Birthday.  For material related to the “Town & Country” theme and to Balanchine, see Leadbelly Under the Volcano (Jan. 27, 2003). (“Sometimes I live in the country, sometimes I live in town…” – Huddie Ledbetter).  Those with more sophisticated tastes may prefer the work of Stephen Ledbetter on Gershwin’s piano preludes or, in view of Warren Beatty’s architectural work in “Town & Country,” the work of Stephen R. Ledbetter on window architecture.

As noted in Balanchine’s Birthday, Apollo (of the Balanchine ballet) has been associated by an architect with the 3×3, or “ninefold” grid.  The reader who wishes a deeper meditation on the number nine, related to the “Town & Country” theme and more suited to the fact that April is Poetry Month, is referred to my note of April 27 two years ago, Nine Gates to the Temple of Poetry.

Intermediate between the simplicity of the 3×3 square and the (apparent) complexity of the 5×5 square, the 4×4 square offers an introduction to geometrical concepts that appears deceptively simple, but is in reality fiendishly complex.  See Geometry for Jews.  The moral of this megilla?

32 + 42 = 52.

But that is another story.

Thursday, January 9, 2003

Thursday January 9, 2003

Filed under: Uncategorized — 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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