Log24

Thursday, August 9, 2018

True Grids

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

From a search in this journal for "True Grid,"
a fanciful description of  the 3×3 grid

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

A fanciful instance of the 4×2 grid in
a scene from the film "The Master" —

IMAGE- Joaquin Phoenix, corridor scene in 'The Master'

A fanciful novel referring to the number 8,
and a not -so-fanciful reference:

http://www.log24.com/log/pix18/180809-The_EIght-and-coordinates-for-PSL(2,7)-actions-500w.jpg

Illustrated above are Katherine Neville's novel The Eight  and the
"knight" coordinatization of the 4×2 grid from a page on the exceptional
isomorphism between PSL(3,2) (alias GL(3,2)) and PSL(2,7) — groups
of, respectively, degree 7 and degree 8.

Literature related to the above remarks on grids:

Ross Douthat's New York Times  column yesterday purported, following
a 1946 poem by Auden, to contrast students of the humanities with
technocrats by saying that the former follow Hermes, the latter Apollo.

I doubt that Apollo would agree.

Friday, November 24, 2017

The Matrix Meets the Grid

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

The Matrix —

  The Grid

  Picturing the Witt Construction

     "Read something that means something." — New Yorker  ad

Wednesday, March 26, 2014

Grid

Filed under: General — m759 @ 12:00 PM

For some context, see Holy Field
and Krauss Grid.

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…"

Friday, May 11, 2018

A Pure Geometry

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

From posts tagged Modernism

Sunday, December 10, 2006

m759 @ 9:00 PM

A Miniature Rosetta Stone:

The 3x3 grid

“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

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) —

 .

Tuesday, April 4, 2017

Plan 9 Continues

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

"Plan 9 deals with the resurrection of the dead."

— Bill Murray in "Ed Wood"
 

For The Church of Plan 9

(The plan , as well as the elevation ,
of the above structure is a 3×3 grid.)

Wednesday, January 4, 2017

A Drama of Many Forms

Filed under: General,Geometry — Tags: — m759 @ 1:24 PM

According to art historian Rosalind Krauss in 1979,
the grid's earliest employers

"can be seen to be participating in a drama
that extended well beyond the domain of art.
That drama, which took many forms, was staged
in many places. One of them was a courtroom,
where early in this century, science did battle with God,
and, reversing all earlier precedents, won."

The previous post discussed the 3×3 grid in the context of
Krauss's drama. In memory of T. S. Eliot, who died on this date
in 1965, an image of the next-largest square grid, the 4×4 array:

 

See instances of the above image.

Tuesday, January 3, 2017

Cultist Space

Filed under: General,Geometry — Tags: , — m759 @ 6:29 PM

The image of art historian Rosalind Krauss in the previous post
suggests a review of a page from her 1979 essay "Grids" —

The previous post illustrated a 3×3 grid. That  cultist space does
provide a place for a few "vestiges of the nineteenth century" —
namely, the elements of the Galois field GF(9) — to hide.
See Coxeter's Aleph in this journal.

Thursday, July 9, 2015

Man and His Symbols

Filed under: General,Geometry — m759 @ 2:24 PM

(Continued)

A post of July 7, Haiku for DeLillo, had a link to posts tagged "Holy Field GF(3)."

As the smallest Galois field based on an odd prime, this structure 
clearly is of fundamental importance.  

The Galois field GF(3)

It is, however, perhaps too  small  to be visually impressive.

A larger, closely related, field, GF(9), may be pictured as a 3×3 array

hence as the traditional Chinese  Holy Field.

Marketing the Holy Field

IMAGE- The Ninefold Square, in China 'The Holy Field'

The above illustration of China's  Holy Field occurred in the context of
Log24 posts on Child Buyers.   For more on child buyers, see an excellent
condemnation today by Diane Ravitch of the U. S. Secretary of Education.

Saturday, June 13, 2015

The Holy Field

Filed under: General,Geometry — m759 @ 7:45 PM

( A Chinese designation for the 3×3 square )

Plan 9 Continues…

Filed under: General — Tags: — m759 @ 11:32 AM

A version of the song in the previous post that I prefer:

A related meditation —

IMAGE- Joseph Campbell, 'The Inner Reaches of Outer Space,' meditation on the number nine, the Goddess, and the Angelus

For a more abstract version of the
"matrix of the cosmic process,"
see "3×3 Grid" in this journal.

Thursday, December 18, 2014

Platonic Analogy

Filed under: General,Geometry — Tags: , , — m759 @ 2:23 PM

(Five by Five continued)

As the 3×3 grid underlies the order-3 finite projective plane,
whose 13 points may be modeled by
the 13 symmetry axes of the cube,
so the 5×5 grid underlies the order-5 finite projective plane,
whose 31 points may be modeled by
the 31 symmetry axes of the dodecahedron.

See posts tagged Galois-Plane Models.

Tuesday, October 21, 2014

Art as a Tool

Filed under: General,Geometry — m759 @ 12:35 PM

Two news items on art as a tool:

Two Log24 posts related to the 3×3 grid, the underlying structure for China’s
ancient Lo Shu “magic” square:

Finally, leftist art theorist Rosalind Krauss in this journal
on Anti-Christmas, 2010:

Which is the tool here, the grid or Krauss?

Sunday, May 11, 2014

For the Perplexed

Filed under: General — 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…

Saturday, November 16, 2013

Raiders of the Lost Theorem

Filed under: General,Geometry — Tags: , — m759 @ 11:30 AM

IMAGE- The 'atomic square' in Lee Sallows's article 'The Lost Theorem'

Yes. See

The 48 actions of GL(2,3) on a 3×3 coordinate-array A,
when matrices of that group right-multiply the elements of A,
with A =

(1,1) (1,0) (1,2)
(0,1) (0,0) (0,2)
(2,1) (2,0) (2,2)

Actions of GL(2,p) on a pxp coordinate-array have the
same sorts of symmetries, where p is any odd prime.

Note that A, regarded in the Sallows manner as a magic square,
has the constant sum (0,0) in rows, columns, both diagonals, and  
all four broken diagonals (with arithmetic modulo 3).

For a more sophisticated approach to the structure of the
ninefold square, see Coxeter + Aleph.

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

Sunday, March 31, 2013

Plan 9 Sermon

Filed under: General — m759 @ 11:00 AM

IMAGE- 3x3 grid of digits, with ninth square empty as in a Raven's Progressive Matrices test

Saturday, August 18, 2012

Summer Reading

Filed under: General — m759 @ 2:56 PM

On the author of a novel published August 14th,
 "Where'd You Go, Bernadette"—

"Semple moved to the Pacific Northwest several years ago
seeking refuge from Los Angeles, but that doesn't mean
that the Emerald City gets a free pass from Semple's
sharp, satirical eye." 

— Stewart Oksenhorn yesterday in The Aspen Times

Related art

IMAGE- 3x3 grid of movie stills with 'North by Northwest' at center

See also a detail from Thursday's 1.3 MB image
"Search for the Lost Tesseract"—

Update of 9 PM EDT (6 PM LA time) the same day, Saturday, Aug. 18—

IMAGE- Cover of 'This One Is Mine,' a novel by Maria Semple

Actually, that one is hers.

Tuesday, August 7, 2012

The Space of Horizons

Filed under: General,Geometry — m759 @ 4:00 AM

“In the space of horizons that neither love nor hate”
— Wallace Stevens, “Things of August”

Seven years ago yesterday—

IMAGE- 3x3 grid related to Borofsky's 'Four Gods'

For some context, see Rosetta Stone as a Metaphor.

Related material from the University of Western Australia

Projective plane of order 3

(The four points on the curve
at the right of the image are
the points on the line at infinity.)

Art critic Robert Hughes,  who nearly died in Western
Australia in a 1999 car crash, actually met his death
yesterday at Calvary Hospital in the Bronx.

See also Hughes on “slow art” in this journal.

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.

Sunday, April 17, 2011

Sunday School

Filed under: General,Geometry — 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

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.

Thursday, January 7, 2010

Lesson No. One

Filed under: General — Tags: — m759 @ 10:01 AM

 

“Zhu Xi maintained that all things are brought into being by the union of two universal aspects of reality: qi, sometimes translated as vital (or physical, material) force; and li, sometimes translated as rational principle (or law).” —Wikipedia

 

“Drop off the key, Lee” — Paul Simon

The 3x3 Grid

Reference frame (Click for details.)

According to Chu Hsi [Zhu Xi],

The word 'Li'

“Li” is “the principle or coherence or order or pattern underlying the cosmos.”

– Smith, Bol, Adler, and Wyatt, Sung Dynasty Uses of the I Ching,
Princeton University Press, 1990

Related material:

Dynasty and

Lesson Number One.

Wednesday, January 6, 2010

Epiphany Revisited

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

January 06, 2007
ART WARS: Epiphany

Picture of Nothing
On Kirk Varnedoe’s
2003 Mellon Lectures,
Pictures of Nothing“–

“Varnedoe’s lectures were ultimately about faith, about his faith in the power of abstraction, and abstraction as a kind of anti-religious faith in itself….”

Related material:

The more industrious scholars will derive considerable pleasure from describing how the art-history professors and journalists of the period 1945-75, along with so many students, intellectuals, and art tourists of every sort, actually struggled to see the paintings directly, in the old pre-World War II way, like Plato’s cave dwellers watching the shadows, without knowing what had projected them, which was the Word.”

— Tom Wolfe, The Painted Word

Log24, Aug. 23, 2005:

“Concept (scholastics’ verbum mentis)–  theological analogy of Son’s procession  as Verbum Patris, 111-12″ — Index to Joyce and Aquinas, by William T. Noon, S.J., Yale University Press 1957,  second printing 1963, page 162

“So did God cause the big bang? Overcome by metaphysical lassitude, I finally reach over to my bookshelf for The Devil’s Bible. Turning to Genesis I read: ‘In the beginning there was nothing. And God said, ‘Let there be light!’ And there was still nothing, but now you could see it.'”
— Jim Holt, Big-Bang Theology, from Slate‘s “High Concept” department

'In the beginning' according to Jim Holt

“Bang.”

“…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….”

For properties of the “nothing” represented by the 3×3 grid, see The Field of Reason. For religious material related to the above and to Epiphany, a holy day observed by some, see Plato, Pegasus, and the Evening Star and Shining Forth.


Some Context:

Quaternions in Finite Geometry

Click to enlarge.

See also Nativity.

Tuesday, October 13, 2009

Tuesday October 13, 2009

Filed under: General — m759 @ 7:00 AM
Wakes


This morning’s New York Times
reports the deaths of Nuremberg interrogator Richard W. Sonnenfeldt and of avant-garde novelist and Beckett scholar Raymond Federman.

Symbols from this journal on the dates of their deaths:

For Sonnenfeldt, who died
 on Friday, Oct. 9,
a symbol from that date:

The 3x3 grid as religious symbol


For connotations of the symbol appropriate to the name Sonnenfeldt, see the link to A Sunrise for Sunrise in the entry of Saturday, Oct. 10.

For Federman, who died
 on Tuesday, Oct. 6,
a symbol from that date:

Black monolith

A quotation that appeared here on Wednesday, Oct. 7, seems relevant to Federman:

But I am a worker, a tombstone mason, anxious to pleace averyburies and jully glad when Christmas comes his once ayear. You are a poorjoist, unctuous to polise nopebobbies….

— James Joyce, Finnegans Wake

Friday, October 9, 2009

Friday October 9, 2009

Filed under: General — m759 @ 9:00 AM

Identity:

The 3x3 grid as religious symbol

"…strict grids of nine pictures
    establish an egalitarian
        framework
…."

Christopher Knight

Some are more
egalitarian
than others.

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

Thursday, February 19, 2009

Thursday February 19, 2009

Filed under: General — 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

Sunday, March 9, 2008

Sunday March 9, 2008

Filed under: General — m759 @ 1:00 AM

Identity

The 3x3 grid as religious symbol

Click for context.

Related material:

Pictures of Nothing

Art Wars: Epiphany

Tilting at Whirligigs

Down the Up Staircase

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 — Tags: — 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.”

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.

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 — Tags: — 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

The image “http://www.log24.com/log/pix07/070119-Confession.jpg” cannot be displayed, because it contains errors.

Tuesday, January 9, 2007

Tuesday January 9, 2007

Filed under: General,Geometry — m759 @ 9:00 PM
Logos and Logic
(private, cut from prev. entry)

The diamond is used in modal logic to symbolize possibility.

  The 3×3 grid may also be used
to illustrate “possibility.”  It leads,
as noted at finitegeometry.org, to
the famed “24-cell,” which may be
pictured either as the diamond
figure from Plato’s Meno

The image “http://www.log24.com/theory/images/poly-24cell-sm.jpg” cannot be displayed, because it contains errors.

Click for details.

  — or as a figure
with 24 vertices:

The image “http://www.log24.com/theory/images/poly-24cell-02sm.jpg” cannot be displayed, because it contains errors.

Click for details.

The “diamond” version of the
24-cell seems unrelated to the
second version that shows all
vertices and edges, yet the
second version is implicit,
or hidden, in the first.
Hence “possibility.”

Neither version of the 24-cell
seems related in any obvious
way to the 3×3 grid, yet both
versions are implicit,
or hidden, in the grid.
Hence “possibility.”

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)

Saturday, January 6, 2007

Saturday January 6, 2007

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


Picture of Nothing

On Kirk Varnedoe’s
2003 Mellon Lectures,
Pictures of Nothing“–

“Varnedoe’s lectures were ultimately
about faith, about his faith in
the power of abstraction,
and abstraction as a kind of
anti-religious faith in itself….”

The Washington Post

Related material:

The more industrious scholars
will derive considerable pleasure
from describing how the art-history
professors and journalists of the period
1945-75, along with so many students,
intellectuals, and art tourists of every
sort, actually struggled to see the
paintings directly, in the old
pre-World War II way,
like Plato’s cave dwellers
watching the shadows, without
knowing what had projected them,
which was the Word.”

— Tom Wolfe, The Painted Word

Log24, Aug. 23, 2005:

“Concept (scholastics’ verbum mentis)–
theological analogy of Son’s procession
as Verbum Patris, 111-12″

— Index to Joyce and Aquinas,
by William T. Noon, S.J.,
Yale University Press 1957,
second printing 1963, page 162

“So did God cause the big bang?
Overcome by metaphysical lassitude,
I finally reach over to my bookshelf
for The Devil’s Bible.
Turning to Genesis I read:
‘In the beginning
there was nothing.
And God said,
‘Let there be light!’
And there was still nothing,
but now you could see it.'”

— Jim Holt, Big-Bang Theology,
Slate‘s “High Concept” department

The image “http://www.log24.com/log/pix07/070106-Bang.jpg” cannot be displayed, because it contains errors.


“Bang.”

“…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….”

Rosalind Krauss, “Grids

For properties of the
“nothing” represented
by the 3×3 grid, see
The Field of Reason.

For religious material related
to the above and to Epiphany,
a holy day observed by some,
see Plato, Pegasus, and the
Evening Star
and Shining Forth.

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, April 27, 2003

Sunday April 27, 2003

Filed under: General,Geometry — 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.

Monday, February 17, 2020

RIP Charles Portis

Filed under: General — Tags: , — m759 @ 4:04 PM

     See also "True Grid " in this  journal.

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    

See as well . . .

Monday, June 3, 2019

Jar Story

Filed under: General — Tags: , — m759 @ 3:41 PM

(Continued)

  ". . . Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness.”

— T. S. Eliot, Four Quartets

From Writing Chinese Characters:

“It is practical to think of a character centered
within an imaginary square grid . . . .
The grid can be subdivided, usually to
9 or 16 squares. . . ."

The image “http://www.log24.com/log/pix04B/041119-ZhongGuo.jpg” cannot be displayed, because it contains errors.

These “Chinese jars” (as opposed to their contents)
are as follows:    

Grids, 3x3 and 4x4 .

See as well Eliot's 1922 remarks on "extinction of personality"
and the phrase "ego-extinction" in Weyl's Philosophy of Mathematics

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.

Friday, December 1, 2017

The Architect and the Matrix

Filed under: General,Geometry — m759 @ 1:00 PM

In memory of Yale art historian Vincent Scully, who reportedly
died at 97 last night at his home in Lynchburg, Va., some remarks
from the firm of architect John Outram and from Scully —

Update from the morning of December 2 —

The above 3×3 figure is of course not unrelated to
the 4×4 figure in The Matrix for Quantum Mystics:

 .

See as well Tsimtsum in this journal.

Harold Bloom on tsimtsum as sublimation

Monday, November 27, 2017

The Golay Code via Witt’s Construction

Filed under: General,Geometry — Tags: — m759 @ 2:08 PM

Reference

Hansen, Robert Peter, "Construction and Simplicity of
the Large Mathieu Groups" (2011). Master's Theses. 4053. 
http://scholarworks.sjsu.edu/etd_theses/4053.

See also The Matrix Meets the Grid (Log24, Nov. 24).
More generally, see SPLAG in this journal.

Monday, October 9, 2017

Still Point for a Dance

Filed under: G-Notes,General,Geometry — Tags: — 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 .

Monday, August 14, 2017

Sound Track

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

Two songs by Chuck Berry on Chess Records in 1958

Sweet Little Sixteen  and  Sweet Little Rock and Roller .

Rock and Roller  begins

She's 9 years old and sweet as she can be
All dressed up like a downtown Christmas tree
Dancin' and hummin' a rock-roll melody

For meditations on Sixteen , see Berry + Sixteen in this journal.

A meditation on Rock and Roller —

Related material — From the above post's date,
March 21, 2017, a memoir by one Siva Vaidhyanathan,
"Robertson Professor of Media Studies and Director of
the Center for Media and Citizenship at the University of Virginia."

Thursday, July 27, 2017

Keeping It Simple

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

Michiko Kakutani in The New York Times

"The detective story genre concerns the finding of clues
and the search for hidden designs, and its very form
underscores Mr. Pynchon’s obsession with conspiracies
and the existence of systems too complicated to understand."

Review of Pynchon's Bleeding Edge , Sept. 10, 2013

Background:  "Moss on the Wall," this  journal on that date.

A less complicated system —

"Plan 9 deals with the resurrection of the dead."

— Bill Murray in "Ed Wood"
 

For The Church of Plan 9

(The plan , as well as the elevation ,
of the above structure is a 3×3 grid.)

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.

Sunday, October 18, 2015

Coordinatization Problem

Filed under: General,Geometry — Tags: — m759 @ 1:06 AM

There are various ways to coordinatize a 3×3 array
(the Chinese "Holy Field'). Here are some —

See  Cullinane,  Coxeter,  and  Knight tour.

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.

Saturday, October 25, 2014

Foundation Square

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

In the above illustration of the 3-4-5 Pythagorean triangle,
the grids on each side may be regarded as figures of
Euclidean  geometry or of Galois  geometry.

In Euclidean geometry, these grids illustrate a property of
the inner triangle.

In elementary Galois geometry, ignoring the connection with
the inner triangle, the grids may be regarded instead as
illustrating vector spaces over finite (i.e., Galois) fields.
Previous posts in this journal have dealt with properties of
the 3×3 and 4×4 grids.  This suggests a look at properties of
the next larger grid, the 5×5 array, viewed as a picture of the
two-dimensional vector space (or affine plane) over the finite
Galois field GF(5) (also known as ℤ5).

The 5×5 array may be coordinatized in a natural way, as illustrated
in (for instance) Matters Mathematical , by I.N. Herstein and
Irving Kaplansky, 2nd ed., Chelsea Publishing, 1978, p. 171:

See Herstein and Kaplansky for the elementary Galois geometry of
the 5×5 array.

For 5×5 geometry that is not so elementary, see…

Hafner's abstract:

We describe the Hoffman-Singleton graph geometrically, showing that
it is closely related to the incidence graph of the affine plane over ℤ5.
This allows us to construct all automorphisms of the graph.

The remarks of Brouwer on graphs connect the 5×5-related geometry discussed
by Hafner with the 4×4 geometry related to the Steiner system S(5,8,24).
(See the Miracle Octad Generator of R. T. Curtis and the related coordinatization
by Cullinane of the 4×4 array as a four-dimensional vector space over GF(2).)

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." 

Saturday, October 5, 2013

For Baron Samedi

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

IMAGE- Symbol in an Arcade Fire video from Haiti

Click on the image for a video.

See also Josefine Lyche's "Grids, you say?"

I prefer Lyche's versions of the diagonal
3×3 grid. Her versions have no lettering.

(This post was suggested by a photo of magical sigils 
that Lyche posted a few hours ago at Facebook.
The above seems to be another such sigil that may
or may not be intended to function like those posted
today by Lyche.)

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."

Tuesday, July 16, 2013

Child Buyers

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

The title refers to a classic 1960 novel by John Hersey.

“How do you  get young people excited about space?”

— Megan Garber in The Atlantic , Aug. 16, 2012
(Italics added.) (See previous four posts.)

Allyn Jackson on “Simplicity, in Mathematics and in Art,”
in the new August 2013 issue of Notices of the American
Mathematical Society

“As conventions evolve, so do notions of simplicity.
Franks mentioned Gauss’s 1831 paper that
established the respectability of complex numbers.”

This suggests a related image by Gauss, with a
remark on simplicity—

IMAGE- Complex Grid, by Gauss

Here Gauss’s diagram is not, as may appear at first glance,
a 3×3 array of squares, but is rather a 4×4 array of discrete
points (part of an infinite plane array).

Related material that does  feature the somewhat simpler 3×3 array
of squares, not  seen as part of an infinite array—

Marketing the Holy Field

IMAGE- The Ninefold Square, in China 'The Holy Field'

Click image for the original post.

For a purely mathematical view of the holy field, see Visualizing GL(2,p).

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.

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, October 21, 2012

Multispeech

Filed under: General — m759 @ 9:00 PM

IMAGE- The 3x3 ('ninefold') square as Chinese 'Holy Field'

     Click image above
     for some background.

IMAGE- Snow White Imprisoned- An image for the Snow Queen

IMAGE- Charlize Theron as Ravenna: 'She drifted.'

Saturday, March 24, 2012

The David Waltz…

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

In Turing's Cathedral

"At the still point…" — T. S. Eliot

In memory of David L. Waltz, artificial-intelligence pioneer,
who died Thursday, March 22, 2012—

  1. The Log24 post of March 22 on the square-triangle theorem
  2. The March 18 post, Square-Triangle Diamond
  3. Remarks from the BBC on linguistic embedding
    that begin as follows—
         "If we draw a large triangle and embed smaller triangles in it,
          how does it look?"—
    and include discussion of a South American "tribe called Piranha" [sic ]
  4. The result of a Cartoon Bank search suggested by no. 3 above—
    (Click image for some related material.)
  5. A suggestion from the Cartoon Bank—
    IMAGE- 'Try our new grid view.'
  6. The following from the First of May, 2010

    The Nine Divisions of Heaven–

    Image-- Routledge Encyclopedia of Taoism, Vol. I, on the Nine Heavens, 'jiutian,' ed. by Fabrizio Pregadio

    Some context–

    IMAGE- The 3x3 ('ninefold') square as Chinese 'Holy Field'

    "This pattern is a square divided into nine equal parts.
    It has been called the 'Holy Field' division and
    was used throughout Chinese history for many
    different purposes, most of which were connected
    with things religious, political, or philosophical."

    – The Magic Square: Cities in Ancient China,
    by Alfred Schinz, Edition Axel Menges, 1996, p. 71

  7. The phrase "embedding the stone" —

Sunday, January 1, 2012

Sunday Shul

Filed under: General,Geometry — Tags: — 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

Monday, June 6, 2011

Tree of Life — Jewish Version

Filed under: General,Geometry — Tags: — m759 @ 6:48 PM

Today's midday NY Lottery number was 753, the number of a significant page in Gravity's Rainbow .

An excerpt from that page ((Penguin Classics paperback, June 1, 1995)—

http://www.log24.com/log/pix11A/110606-Countdown753.gif

"… the Abyss had crept intolerably close, only an accident away…."

Midrash— See Ben Stein in this journal. 

But seriously… See "Geometry and Death" in this journal.

See also 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. .

Some context— A Study in Art Education.

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

Saturday, May 1, 2010

An Education

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

天鈞

 

Made famous by Ursula K. Le Guin
as the book title "Lathe of Heaven,"
this Chinese phrase, tianjun, apparently
means something more like "Scales of Heaven"–
an appropriate image for Law Day 2010.

Image--Scales (the legal symbol)

An anonymous forum user says that

"…if you switch the two characters around,
you get: 鈞天, which is one of
the nine heavens, more specifically,
the middle heaven."

This is supported by a
non-anonymous source:

"I follow A.C. Graham’s translation of
Juntian as 'Level Heaven (the innermost
of the nine divisions of heaven)';
he renders Juntian guangyue as
'the mighty music of the innermost heaven.'"

— "Music in the World of Su Shi (1037-1101):
Terminology
," by Stuart H. Sargent,
Colorado State University,
Journal of Sung-Yuan Studies 32 (2002), 39-81

The Nine Divisions of Heaven–

Image-- Routledge Encyclopedia of Taoism, Vol. I, on the Nine Heavens, 'jiutian,' ed. by Fabrizio Pregadio

Some context–

The 3x3 ('ninefold') square

"This pattern is a square divided into nine equal parts.
It has been called the 'Holy Field' division and
was used throughout Chinese history for many
different purposes, most of which were connected
with things religious, political, or philosophical."

The Magic Square: Cities in Ancient China,
by Alfred Schinz, Edition Axel Menges, 1996, p. 71

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

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
.

Wednesday, June 18, 2008

Wednesday June 18, 2008

Filed under: General,Geometry — m759 @ 3:00 PM
CHANGE
 FEW CAN BELIEVE IN

What I Loved, a novel by Siri Hustvedt (New York, Macmillan, 2003), contains a paragraph on the marriage of a fictional artist named Wechsler–

Page 67 —

“… Bill and Violet were married. The wedding was held in the Bowery loft on June 16th, the same day Joyce’s Jewish Ulysses had wandered around Dublin. A few minutes before the exchange of vows, I noted that Violet’s last name, Blom, was only an o away from Bloom, and that meaningless link led me to reflect on Bill’s name, Wechsler, which carries the German root for change, changing, and making change. Blooming and changing, I thought.”

For Hustvedt’s discussion of Wechsler’s art– sculptured cubes, which she calls “tightly orchestrated semantic bombs” (p. 169)– see Log24, May 25, 2008.

Related material:

Wechsler cubes

(after David Wechsler,
1896-1981, chief
psychologist at Bellevue)

Wechsler blocks for psychological testing

These cubes are used to
make 3×3 patterns for
psychological testing.

Related 3×3 patterns appear
in “nine-patch” quilt blocks
and in the following–

Don Park at docuverse.com, Jan. 19, 2007:

“How to draw an Identicon

Designs from a web page on Identicons

A 9-block is a small quilt using only 3 types of patches, out of 16 available, in 9 positions. Using the identicon code, 3 patches are selected: one for center position, one for 4 sides, and one for 4 corners.

Positions and Rotations

For center position, only a symmetric patch is selected (patch 1, 5, 9, and 16). For corner and side positions, patch is rotated by 90 degree moving clock-wise starting from top-left position and top position respectively.”

    

From a weblog by Scott Sherrill-Mix:

“… Don Park came up with the original idea for representing users with geometric shapes….”

Claire | 20-Dec-07 at 9:35 pm | Permalink

“This reminds me of a flash demo by Jarred Tarbell
http://www.levitated.net/daily/lev9block.html

ScottS-M | 21-Dec-07 at 12:59 am | Permalink

    

Jared Tarbell at levitated.net, May 15, 2002:

“The nine block is a common design pattern among quilters. Its construction methods and primitive building shapes are simple, yet produce millions of interesting variations.

Designs from a web page by Jared Tarbell
Figure A. Four 9 block patterns,
arbitrarily assembled, show the
grid composition of the block.

Each block is composed of 9 squares, arranged in a 3 x 3 grid. Each square is composed of one of 16 primitive shapes. Shapes are arranged such that the block is radially symmetric. Color is modified and assigned arbitrarily to each new block.

The basic building blocks of the nine block are limited to 16 unique geometric shapes. Each shape is allowed to rotate in 90 degree increments. Only 4 shapes are allowed in the center position to maintain radial symmetry.

Designs from a web page by Jared Tarbell

Figure B. The 16 possible shapes allowed
for each grid space. The 4 shapes allowed
in the center have bold numbers.”

   
Such designs become of mathematical interest when their size is increased slightly, from square arrays of nine blocks to square arrays of sixteen.  See Block Designs in Art and Mathematics.

(This entry was suggested by examples of 4×4 Identicons in use at Secret Blogging Seminar.)

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.”

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.

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

Wednesday, March 7, 2007

Wednesday March 7, 2007

Filed under: General,Geometry — m759 @ 8:24 AM
In the Labyrinth
of Time:


8:24:48
AM EST


Related material–


Symbols:

The image “http://www.log24.com/log/pix07/070307-Symbols.gif” cannot be displayed, because it contains errors.

and

The False Artaxerxes:
Borges and the
Dream of Chess

This entry was inspired by
Xanga footprints yesterday
from Virginia:

 
1. Virginia
Weblog
ART WARS:
Time and the Grid
3/6/2007
9:48 AM
2. Virginia
Weblog
Sequel
3/6/2007
11:38 AM
3. Virginia
Weblog
Games and Truth
3/6/2007
1:25 PM
4. Virginia
/item.aspx?user=m759&ta…
The Transcendent Signified
3/6/2007
5:15 PM
5. Virginia
/item.aspx?user=m759&ta…
Zen and Language Games
3/6/2007
5:16 PM
6. Virginia
/item.aspx?user=m759&ta…
Balanchine’s Birthday
3/6/2007
6:12 PM
7. Virginia
/item.aspx?user=m759&ta…
The Agony and the Ya-Ya
3/6/2007
6:12 PM
8. Virginia
/item.aspx?user=m759&ta…
Directions Out
3/6/2007
6:13 PM
9. Virginia
/item.aspx?user=m759&ta…
The Four Last Things
3/6/2007
6:13 PM

Sunday, December 10, 2006

Sunday December 10, 2006

Filed under: General,Geometry — Tags: — m759 @ 9:00 PM
The Librarian
on Nobel Prize Day

"Time and chance
happeneth to them all."
— Ecclesiastes  

PA Lottery Dec. 10, 2006: Mid-day 569, Evening 048

Timeline Index:

Pythagoras, born ca. 569 B.C.

The number 048
may be interpreted
as referring to…

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

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.”

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.

 

Saturday, August 6, 2005

Saturday August 6, 2005

Filed under: General,Geometry — Tags: — m759 @ 9:00 AM
For André Weil on
the seventh anniversary
of his death:

 A Miniature
Rosetta Stone

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

In a 1940 letter to his sister Simone,  André Weil discussed a sort of “Rosetta stone,” or trilingual text of three analogous parts: classical analysis on the complex field, algebraic geometry over finite fields, and the theory of number fields.  

John Baez discussed (Sept. 6, 2003) the analogies of Weil, and he himself furnished another such Rosetta stone on a much smaller scale:

“… a 24-element group called the ‘binary tetrahedral group,’ a 24-element group called ‘SL(2,Z/3),’ and the vertices of a regular polytope in 4 dimensions called the ’24-cell.’ The most important fact is that these are all the same thing!”

For further details, see Wikipedia on the 24-cell, on special linear groups, and on Hurwitz quaternions,

The group SL(2,Z/3), also known as “SL(2,3),” is of course derived from the general linear group GL(2,3).  For the relationship of this group to the quaternions, see the Log24 entry for August 4 (the birthdate of the discoverer of quaternions, Sir William Rowan Hamilton).

The 3×3 square shown above may, as my August 4 entry indicates, be used to picture the quaternions and, more generally, the 48-element group GL(2,3).  It may therefore be regarded as the structure underlying the miniature Rosetta stone described by Baez.

“The typical example of a finite group is GL(n,q), the general linear group of n dimensions over the field with q elements. The student who is introduced to the subject with other examples is being completely misled.”

 — J. L. Alperin, book review,
    Bulletin (New Series) of the American
    Mathematical Society 10 (1984), 121

Friday, November 19, 2004

Friday November 19, 2004

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

From Tate to Plato

In honor of Allen Tate‘s birthday (today)
and of the MoMA re-opening (tomorrow)

“For Allen Tate the concept of tension was the most useful formal tool at the critic’s disposal, as irony and paradox were for Brooks. The principle of tension sustains the whole structure of meaning, and, as Tate declares in Tension in Poetry (1938), he derives it from lopping the prefixes off the logical terms extension and intension (which define the abstract and denotative aspect of the poetic language and, respectively, the concrete and connotative one). The meaning of the poem is ‘the full organized body of all the extension and intension that we can find in it.’  There is an infinite line between extreme extension and extreme intension and the readers select the meaning at the point they wish along that line, according to their personal drives, interests or approaches. Thus the Platonist will tend to stay near the extension end, for he is more interested in deriving an abstraction of the object into a universal….”

— from Form, Structure, and Structurality,
   by Radu Surdulescu

“Eliot, in a conception comparable to Wallace Stevens’ ‘Anecdote of the Jar,’ has suggested how art conquers time:

        Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness.”

F. O. Matthiessen
   in The Achievement of T.S. Eliot,
   Oxford University Press, 1958

From Writing Chinese Characters:

“It is practical to think of a character centered within an imaginary square grid…. The grid can… be… subdivided, usually to 9 or 16 squares….”

The image “http://www.log24.com/log/pix04B/041119-ZhongGuo.jpg” cannot be displayed, because it contains errors.

These “Chinese jars”
(as opposed to their contents)
are as follows:

The image “http://www.log24.com/log/pix04B/041119-Grids.gif” cannot be displayed, because it contains errors.

Various previous Log24.net entries have
dealt with the 3×3 “form” or “pattern”
(to use the terms of T. S. Eliot).

For the 4×4 form, see Poetry’s Bones
and Geometry of the 4×4 Square.

Saturday, June 5, 2004

Saturday June 5, 2004

Filed under: General,Geometry — m759 @ 11:11 AM
A Form,
 continued…

Some cognitive uses
of the 3×3 square
are discussed in

From Lullus to Cognitive Semantics:
The Evolution of a Theory of Semantic Fields

by Wolfgang Wildgen and in

Another Page in the Foundation of Semiotics:
A Book Review of On the Composition of Images, Signs & Ideas, by Giordano Bruno…
by Mihai Nadin

“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.

For a meditation on the related 4×4 square grid as “art that holds time,” see Time Fold.

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?
 

Saturday, July 20, 2002

Saturday July 20, 2002

Filed under: General,Geometry — Tags: , — m759 @ 10:13 PM
 

ABSTRACT: Finite projective geometry explains the surprising symmetry properties of some simple graphic designs– found, for instance, in quilts. Links are provided for applications to sporadic simple groups (via the "Miracle Octad Generator" of R. T. Curtis), to the connection between orthogonal Latin squares and projective spreads, and to symmetry of Walsh functions.
We regard the four-diamond figure D above as a 4×4 array of two-color diagonally-divided square tiles.

Let G be the group of 322,560 permutations of these 16 tiles generated by arbitrarily mixing random permutations of rows and of columns with random permutations of the four 2×2 quadrants.

THEOREM: Every G-image of D (as at right, below) has some ordinary or color-interchange symmetry.

Example:


For an animated version, click here.

Remarks:

Some of the patterns resulting from the action of G on D have been known for thousands of years. (See Jablan, Symmetry and Ornament, Ch. 2.6.) It is perhaps surprising that the patterns' interrelationships and symmetries can be explained fully only by using mathematics discovered just recently (relative to the patterns' age)– in particular, the theory of automorphism groups of finite geometries.

Using this theory, we can summarize the patterns' properties by saying that G is isomorphic to the affine group A on the linear 4-space over GF(2) and that the 35 structures of the 840 = 35 x 24 G-images of D are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2).

This can be seen by viewing the 35 structures as three-sets of line diagrams, based on the three partitions of the four-set of square two-color tiles into two two-sets, and indicating the locations of these two-sets of tiles within the 4×4 patterns. The lines of the line diagrams may be added in a binary fashion (i.e., 1+1=0). Each three-set of line diagrams sums to zero– i.e., each diagram in a three-set is the binary sum of the other two diagrams in the set. Thus, the 35 three-sets of line diagrams correspond to the 35 three-point lines of the finite projective 3-space PG(3,2).

For example, here are the line diagrams for the figures above:

Shown below are the 15 possible line diagrams resulting from row/column/quadrant permutations. These 15 diagrams may, as noted above, be regarded as the 15 points of the projective 3-space PG(3,2).


The symmetry of the line diagrams accounts for the symmetry of the two-color patterns. (A proof shows that a 2nx2n two-color triangular half-squares pattern with such line diagrams must have a 2×2 center with a symmetry, and that this symmetry must be shared by the entire pattern.)

Among the 35 structures of the 840 4×4 arrays of tiles, orthogonality (in the sense of Latin-square orthogonality) corresponds to skewness of lines in the finite projective space PG(3,2). This was stated by the author in a 1978 note. (The note apparently had little effect. A quarter-century later, P. Govaerts, D. Jungnickel, L. Storme, and J. A. Thas wrote that skew (i.e., nonintersecting) lines in a projective space seem "at first sight not at all related" to orthogonal Latin squares.)

We can define sums and products so that the G-images of D generate an ideal (1024 patterns characterized by all horizontal or vertical "cuts" being uninterrupted) of a ring of 4096 symmetric patterns. There is an infinite family of such "diamond" rings, isomorphic to rings of matrices over GF(4).

The proof uses a decomposition technique for functions into a finite field that might be of more general use.

The underlying geometry of the 4×4 patterns is closely related to the Miracle Octad Generator of R. T. Curtis– used in the construction of the Steiner system S(5,8,24)– and hence is also related to the Leech lattice, which, as Walter Feit has remarked, "is a blown up version of S(5,8,24)."

For a movable JavaScript version of these 4×4 patterns, see The Diamond 16 Puzzle.

The above is an expanded version of Abstract 79T-A37, "Symmetry invariance in a diamond ring," by Steven H. Cullinane, Notices of the American Mathematical Society, February 1979, pages A-193, 194.

For a discussion of other cases of the theorem, click here.

Related pages:

The Diamond 16 Puzzle

Diamond Theory in 1937:
A Brief Historical Note

Notes on Finite Geometry

Geometry of the 4×4 Square

Binary Coordinate Systems

The 35 Lines of PG(3,2)

Map Systems:
Function Decomposition over a Finite Field

The Diamond Theorem–
The 2×2, the 2x2x2, the 4×4, and the 4x4x4 Cases

Diamond Theory

Latin-Square Geometry

Walsh Functions

Inscapes

The Diamond Theory of Truth

Geometry of the I Ching

Solomon's Cube and The Eightfold Way

Crystal and Dragon in Diamond Theory

The Form, the Pattern

The Grid of Time

Block Designs

Finite Relativity

Theme and Variations

Models of Finite Geometries

Quilt Geometry

Pattern Groups

The Fano Plane Revisualized,
or the Eightfold Cube

The Miracle Octad Generator

Kaleidoscope

Visualizing GL(2,p)

Jung's Imago

Author's home page

AMS Mathematics Subject Classification:

20B25 (Group theory and generalizations :: Permutation groups :: Finite automorphism groups of algebraic, geometric, or combinatorial structures)

05B25 (Combinatorics :: Designs and configurations :: Finite geometries)

51E20 (Geometry :: Finite geometry and special incidence structures :: Combinatorial structures in finite projective spaces)




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Page created Jan. 6, 2006, by Steven H. Cullinane      diamondtheorem.com

 

Initial Xanga entry.  Updated Nov. 18, 2006.

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