See Ballet Blanc and Black Art in this journal.
From the former:
"A blank underlies the trials of device."
— Wallace Stevens
From the latter:
See Ballet Blanc and Black Art in this journal.
From the former:
"A blank underlies the trials of device."
— Wallace Stevens
From the latter:
Margaret Atwood on Lewis Hyde's "Trickster is among other things the gatekeeper who opens the door into the next world; those who mistake him for a psychopath never even know such a door exists." (159) What is "the next world"? It might be the Underworld…. The pleasures of fabulation, the charming and playful lie– this line of thought leads Hyde to the last link in his subtitle, the connection of the trickster to art. Hyde reminds us that the wall between the artist and that American favourite son, the con-artist, can be a thin one indeed; that craft and crafty rub shoulders; and that the words artifice, artifact, articulation and art all come from the same ancient root, a word meaning "to join," "to fit," and "to make." (254) If it’s a seamless whole you want, pray to Apollo, who sets the limits within which such a work can exist. Tricksters, however, stand where the door swings open on its hinges and the horizon expands: they operate where things are joined together, and thus can also come apart. |
"As a Chinese jar . . . ."
— Four Quartets
Rosalind Krauss "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,
"And it's whispered that soon if we all call the tune
The nine engravings of The Club Dumas
An example of the universal— or, according to Krauss,
"This is the garden of Apollo, |
The "Katz" of the August 7 post Art Angles
is a product of Princeton's
Department of Art and Archaeology.
ART —
ARCHAEOLOGY —
"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
See also "True Grid " in this journal.
Rosalind Krauss "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,
"And it's whispered that soon if we all call the tune
The nine engravings of The Club Dumas
An example of the universal— or, according to Krauss,
"This is the garden of Apollo, the field of Reason…." |
See as well . . .
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
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) …
… and the 3×3 grid as a symbol of Apollo
(an Olympian rather than a Titan) —
"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 —
For further background to this morning's post Plan 9 Continues,
see posts tagged Sacred Space —
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.
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 —
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.
“… the object sets up a kind of
frame or space or field
within which there can be epiphany.”
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 —
— 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."
"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:
INT. OFFICE BUILDING – NIGHT
NIGHT WATCHMAN
Bateman wheels around and shoots him.
NEW ANGLE:
INT. PIERCE & PIERCE LOBBY – NIGHT
— AMERICAN PSYCHO |
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
At the exactest point at which it is itself,
The eye made clear of uncertainty, with the sight
Everything, the spirit's alchemicana
The solid, but the movable, the moment,
— Wallace Stevens, "An Ordinary Evening |
"A view of New Haven, say…." —
"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:
Structure vs. Character continued…
Structure |
|
Related vocabulary:
Nick Tosches on the German word “Quell “
* The title is from Heidegger.
"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 above 3×3 grid is embedded in a
somewhat more sophisticated example
of conceptual art from April 1, 2013:
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:
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—
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—
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).
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 |
"… 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 Ninefold Square
See Coxeter and the Aleph and Ayn Sof—
Mathematics and Narrative, Illustrated |
|
Mathematics |
Narrative |
Grid from a post linked to in yesterday's 24 Hour DeLillo—
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
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—
|
Happy day 23 of Mental Health Month.
"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—
A picture of the Hesse configuration–
(See Visualizing GL(2,p), a note from 1985).
Related notes from this journal —
From last November —
From the December 2010 American Mathematical Society Notices—
Related material from this journal— Consolation Prize (August 19, 2010) |
From 2006 —
Sunday December 10, 2006
“Function defined form, expressed in a pure geometry
– J. G. Ballard on Modernism
“The greatest obstacle to discovery is not ignorance –
— Daniel J. Boorstin, |
Also from 2006 —
Sunday November 26, 2006
Rosalind Krauss "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,
"And it's whispered that soon if we all call the tune
The nine engravings of The Club Dumas
An example of the universal*– or, according to Krauss,
"This is the garden of Apollo, the field of Reason…."
For more on the field of reason, see
A reasonable set of "strange correspondences" 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 ,
"For every kind of vampire, |
And from last October —
Friday, October 8, 2010
Starting Out in the Evening 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 .
"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,
* The Web version's title has a misprint— |
Leading today's New York Times obituaries —
— 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.
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),
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.
A picture one might view as
related to the novel An Object of Beauty
and the film "The Object of Beauty" —
"If it's a seamless whole you want,
pray to Apollo." — Margaret Atwood
From Doonesbury today—
From this journal (September 20, 2009)—
scheinen
Quine, Pursuit of Truth,
Google search:
|
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—
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.
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–
Adapted from
Ad Reinhardt
Midnight in the Garden, Autumn 2009
Review:
|
From this journal on the following day, Sept. 21:
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.
Images from Log24 on
December 10, 2006 —
Nobel Prize Day, and
the day after
Kirk Douglas’s birthday—
Kirk Douglas ad for
the film “Diamonds”
(2000)
Images from
Google News
at noon today —
“The serpent’s eyes shine
as he wraps around the vine…”
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 —
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:
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:
Selah.
Introduction
For details of the story,
click on the images.
Chapter I:
Chapter II:
Chapter III:
and the following quotation:
"There is no landing fee in Avalon,
or anywhere else in Catalina."
A figure from
Nobel Prize day, December 10,
and from Eugene Wigner‘s
birthday, November 17:
Also on December 10:
the death of Constantine–
(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:
“Ay que bonito es volar
A las dos de la mañana….”
— “La Bruja“
“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
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.)
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
Happy birthday
to the late
Eugene Wigner
… and a belated
Merry Christmas
to Paul Newman:
— Eugene Wigner, Nobel Prize Lecture, December 12, 1963
(continued from
June 15, 2007)
Today is the anniversary
of the 1955 death of poet
Wallace Stevens.
Related material:
an essay on the
relationships between
poets and philosophers —
“Bad Blood,” by
Leonard Michaels —
and
"But not, perhaps,
in the Garden of Apollo":
— "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):
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.'"
Here, yet again, is an illustration that has often appeared in Log24– notably, on the date of Arnheim's death:
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."
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.
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:
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."
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 —
“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.
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:
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:
(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:
Let us hope that Mackey,
a devotee of reason,
is now enjoying the company
of Apollo rather than that of
Tom O’Bedlam:
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.
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
"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:
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:
(continued from
January 9, 2003)
George Balanchine
|
"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:
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:
— Richard Kearney, 2005,
in The New Arcadia Review
Kearney (right) with
Martin Scorsese (left)
and Gregory Peck
in 1997.
— 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 and logical possibility with the diamond
"The possibilia that exist,
— Michael Sudduth, |
Click on picture
for related symbolism.
“This is the garden of Apollo,
the field of Reason….”
John Outram, architect
I need a photo-opportunity
In memory of Joseph Barbera–
co-creator ot the Flintstones–
who died yesterday, a photo
from today’s Washington Post:
Playing the role of
recording angel —
Halle Berry as
Rosetta Stone:
Related material:
“Citizen Stone“
and
“Putting the X in Xmas.”
"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
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…. "
"… 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."
"This is the garden of Apollo,
the field of Reason…."
— John Outram, architect
(Click on picture for details.)
Related material:
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:
"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
“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)
In the punctual centre of all circles white
Stands truly….
… 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.
On Beauty
“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 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
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:
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:
This structure may seem too simple to support movements of interest, but consider the following (click to enlarge):
As Sir Arthur Quiller-Couch, paraphrasing Horace, remarks in his Whitsun, 1939, preface to the new edition of the Oxford Book of English Verse, "tamen usque recurret Apollo."
The alert reader will note that in the above diagrams, only eight of the positions move.
Which muse remains at the center?
Consider the remark of T. S. Eliot, "At the still point, there the dance is," and the fact that on the day Eliot turned 60, Olivia Newton-John was born. How, indeed, in the words of another "sixty-year-old smiling public man," can we know the dancer from the dance?
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